Master's thesis of Applied Science: The transmission of forced waves at a junction

The transmission of forced waves at a junction Alexandra Irwin Master of Applied Science RMIT

The transmission of forced waves at a junction A thesis submitted in fulfillment of the requirements for the degree of Master of Applied Science Alexandra Laura Irwin BSc(Hons)(Physics) School of Applied Sciences College of Science, Engineering and Health RMIT Universit y February 2013

Declaration

I d ecl ar e t ha t t h e wo r k p r es e nted i n t hi s t he si s is t hat o f m y o wn , e x cep t wh er e d u e

ac k no wl ed g me n t ha s b ee n ma d e, a nd ha s no t b ee n s ub mi t ted p r e v io u sl y, in wh o le o r

p ar t, to q ua li f y f o r a n y o th er a cad e mic a wa r d .

T he co n te nt o f t h e t h es i s i s t he r e s ul t o f wo r k whi c h ha s b ee n c ar r i ed o u t s i nc e 2 8t h

Feb r ua r y, 2 0 1 1, t hi s b e i n g t he o f f ic ial d at e o f c o mme n ce me n t o f t h is p r o gr a m me.

Si g n at ur e : …… … …… … …… .. Na me : Ale x a nd r a I r wi n Dat e: Feb r u ar y, 2 0 1 3.

Acknowledgements

I wi s h to a c k no wl ed ge t he co nt i n ued s up p o r t a n d e nco ur a g e me n t I ha v e r ece i ved fr o m

m y p r o j e ct s up er v i so r , Ad j u nc t. P r o fe s so r J o h n Da v y. T ha n k s fo r t he m an y h o ur s sp e nt

o n m y p r o j ect, I wa s r ea ll y i mp r e s sed wi t h yo ur in n o vat i ve id ea s a nd b o u nd le ss

k no wled g e.

I wo uld al so li k e to t ha n k m y s eco nd s up er v i so r As so c P r o f P h il ip W i l k s ch.

I mu s t a lso t ha n k V lad i mi r P a va so v ic fo r h i s wo nd er f u l t he si s P a v aso v i c ( 2 0 0 6) , a so ur c e o f i n sp ir a tio n.

Declaration .............................................................................................. i

Acknowledgements .................................................................................. ii

Table of Contents ................................................................................... iii

Abstract ................................................................................................ xvi

Chapter 1 Introduction .............................................................................. 1

Chapter 2 Derivation of the fluid media sound equations ............................. 4

2.1 Introduction ..................................................................................... 4

2.2 Normal Incidence with the same media on both sides of the junction ... 4

2.3 Media same for x<0 and x>0, normal incidence forced wave ............... 8

2.4 The total intensity when x =0 (at the junction of the two media) ......... 11

2.5 The transmitted intensit y (I t(x)) when x>0 ....................................... 12

2.6 The intensit y carried b y the reflected wave (x<0) if propagating alone. .......................................................................................................... 13

2.7 Media different for x<0 and x>0. A normall y incident forced wave is firstl y considered. The transmitted intensit y and the incident and reflected intensit y is found. ................................................................................ 14

2.8 The acoustic particle velocit y in the oblique incidence case. ............. 17

2.9 Proof of no power flow in a nearfield .............................................. 19

2.10 Derivation of transmitted and reflected pressures for oblique incidence. ........................................................................................... 21

2.11 Derivation of total and forced intensit y for oblique incidence. Different Media. .................................................................................. 26

2.12 Diffuse field incidence. Media the same. Anal ytical calculation of intensit y for r less than or equal to 1..................................................... 30

2.13 Diffuse field incidence. Media the same. r greater than or equal to 1. .......................................................................................................... 37

2.14 Diffuse incidence when the media are different. In terms of Z ,k. ...... 38

2.15 Summary ..................................................................................... 41

Chapter 3 The fluid media sound results ................................................... 43

3.1 Introduction ................................................................................... 43

3.2 Normal incidence, the same media ................................................... 43

3.3 Normal incidence, different media. .................................................. 49

3.4 Oblique incidence, same media. ...................................................... 49

3.5 Oblique incidence, different impedances, same wave numbers ........... 51

3.6 Oblique incidence, different wave numbers ...................................... 53

3.7 Summary ....................................................................................... 63

Chapter 4 The transmission of bending waves between two panels at a pinned joint....................................................................................................... 64

iii

4.1 Introduction ................................................................................... 64

4.2 The transmitted and reflected wave equations for normal incidence with a freel y propagating incident wave. ....................................................... 64

4.3 Derivation of wave numbers for the obliquel y incident forced wave case. ................................................................................................... 70

4.4 The angular velocit y and the torsional moment for the obliquel y incident wave case. .............................................................................. 76

4.5 Derivation of the transmitted and reflected waves for obliquel y incident freel y propagating waves. ..................................................................... 79

4.6 Obliquel y incident forced waves ..................................................... 81

4.7 Transmitted bending wave power..................................................... 84

4.8 The power per unit length transmitted by a diffuse bending wave field. .......................................................................................................... 88

4.9 The transmitted power when plate 1 is excited by a diffuse acoustic field. .................................................................................................. 90

4.10 Summary ..................................................................................... 93

Chapter 5 The results for pinned plates. ................................................... 94

Chapter 6 Conclusion ............................................................................ 132

Appendix 1 ........................................................................................... 135

Matlab code ...................................................................................... 135

References ........................................................................................... 139

List of figures

Figure Page Fi g ur e 2 .1 T h e t wo d i me n sio n al ca se. ......................................................................... 9

Fi g ur e 2 .2 : A f i g ur e s ho wi n g t he co - o r d i na te f i g ur e s u sed i n t hi s se ct io n . .................... 21

Fi g ur e 2 .3 Gr ap h ica l d i a gr a m o f eq u at io n ( 2 .2 5 8 ) . ..................................................... 32

Fi g ur e 3 .1 : T he tr a n s mi t ted i n te n s it y d u e to a no r mal l y i n cid e n t fo r ced wav e. .............. 45

Fi g ur e 3 .2 : T he tr a n s mi t te d , r e fle ct ed , i n cid e nt a nd s u m o f i nc id e n t a nd r ef le cted

in te n s it ie s d ue to a no r ma ll y i nc id e n t fo r c ed wav e. ............................................. 47

Fi g ur e 3 .3 : T he no r mal i sed tr a n s mi t ted , r e fl ect e d a nd i nc id e n t so u nd p r e s s ur e s a s a

f u nc tio n o f t h e r a tio r o f t he fo r ced i nc id e nt wa ve n u mb er to t h e fr e el y p r o p ag at i n g

wa v e n u mb er . .................................................................................................. 48

Fi g ur e 3 .4 T h e no r ma li s ed tr a n s mi t ted i nt e ns it y d ue to fo r ced p la ne wa v es i nc id e nt at

an g le s o f i nc id e nc e fr o m 0 to 9 0 d e gr ee s i n 1 5 d egr ee i ncr e me nt s. ........................ 50

Fi g ur e 3 .5 T h e tr a n s mi tt ed i nt e n si t y d ue to a p la ne so u nd wa v e i n cid e n t at a n g le s r a n g i n g

fr o m 0 to 9 0 d e gr e es i n 1 5 d e gr ee i ncr e me n ts . T he r e s ul ts ar e g r ap hed a s a f u n ct io n

o f t he r at io r o f t h e fo r c ed i nc id e n t wa v e n u mb e r to t h e wa v e n u mb er o f a fr ee l y

p r o p ag at i n g wa v e i n t h e f ir st me d i u m. T he r at io o f t he wa ve n u mb er o f a fr eel y

p r o p ag at i n g wa v e i n t h e f ir st me d i u m to t ha t i n th e seco nd med i u m ( α ) i s ½. T h e

r atio o f t h e i mp ed a nc e o f t he c har a cte r i st ic i mp e d an ce o f t he f ir st med i u m to t h at o f

th e seco nd med i u m ( β) i s ½. .............................................................................. 55

Fi g ur e 3 .6 T h e tr a n s mi tt ed i nt e n si t y d ue t o a p la ne so u nd wa v e i n cid e n t at a n g le s r a n g i n g

fr o m 0 to 9 0 d e gr e es i n 1 5 d e gr ee i ncr e me n ts . T he r e s ul ts ar e g r ap hed a s a f u n ct io n

o f t he r at io r o f t h e fo r c ed i nc id e n t wa v e n u mb e r to t h e wa v e n u mb er o f a fr ee l y

p r o p ag at i n g wa v e i n t h e f ir st me d i u m. T he r at io o f t he wa ve n u mb er o f a fr eel y

p r o p ag at i n g wa v e i n t h e f ir st me d i u m to t ha t i n th e seco nd med i u m ( α ) i s ½. T h e

r atio o f t h e i mp ed a nc e o f t he c har a cte r i st ic i mp e d an ce o f t he f ir st med i u m to t h at o f

th e seco nd med i u m ( β) i s 1 . .............................................................................. 56

Fi g ur e 3 .7 T h e tr a n s mi tt ed i nt e n si t y d ue to a p la ne so u nd wa v e i n cid e n t at a n g le s r a n g i n g

fr o m 0 to 9 0 d e gr e es i n 1 5 d e gr ee i ncr e me n ts . T he r e s ul ts ar e g r ap hed a s a f u n ct io n

o f t he r at io r o f t h e fo r c ed i nc id e n t wa v e n u mb e r to t h e wa v e n u mb er o f a fr ee l y

p r o p ag at i n g wa v e i n t h e f ir st me d i u m. T he r at io o f t he wa ve n u mb er o f a fr eel y

p r o p ag at i n g wa v e i n t h e f ir st me d i u m to t ha t i n th e seco nd med i u m ( α ) i s ½. T h e

r atio o f t h e c ha r ac ter is ti c i mp ed a nce o f t h e fi r s t med i u m to t h at o f t h e s e co nd

med i u m ( β) i s 2 . .............................................................................................. 57

Fi g ur e 3 .8 T h e tr a n s mi tt ed i nt e n si t y d ue to a p la ne so u nd wa v e i n cid e n t at a n g le s r a n g i n g

fr o m 0 to 9 0 d e gr e es i n 1 5 d e gr ee i ncr e me n ts . T he r e s ul ts ar e g r ap hed a s a f u n ct io n

o f t he r at io r o f t h e fo r c ed i nc id e n t wa v e n u mb e r to t h e wa v e n u mb er o f a fr ee l y

p r o p ag at i n g wa v e i n t h e f ir st me d i u m. T he r at io o f t he wa ve n u mb er o f a fr eel y

p r o p ag at i n g wa v e i n t h e f ir st me d i u m to t ha t i n th e seco nd med i u m ( α ) i s 2 . T he r a tio

o f t he i mp ed a nce o f t he ch ar ac ter i st ic i mp ed a n c e o f t h e fir s t med i u m to th at o f t h e

seco nd med i u m ( β) is 1 / 2 . ................................................................................. 58

Fi g ur e 3 .9 T h e tr a n s mi tt ed i nt e n si t y d ue to a p la ne so u nd wa v e i n cid e n t at a n g le s r a n g i n g

fr o m 0 to 9 0 d e gr e es i n 1 5 d e gr ee i ncr e me n ts . T he r e s ul ts ar e g r ap hed a s a f u n ct io n

o f t he r at io r o f t h e fo r c ed i nc id e n t wa v e n u mb e r to t h e wa v e n u mb er o f a fr ee l y

p r o p ag at i n g wa v e i n t h e f ir st me d i u m. T he r at io o f t he wa ve n u mb er o f a fr eel y

p r o p ag at i n g wa v e i n t h e f ir st me d i u m to t h a t i n th e seco nd med i u m ( α ) i s 2 . T he r a tio

o f t he i mp ed a nce o f t he ch ar ac ter i st ic i mp ed a n c e o f t h e fir s t med i u m to th at o f t h e

seco nd med i u m ( β) is 1 . .................................................................................... 59

Fi g ur e 3 .1 0 T h e tr a n s mi tted i nt e n si t y d ue to a p l an e so u nd wa v e i ncid e nt at a n g le s

r an g i n g fr o m 0 to 9 0 d e gr e e s i n 1 5 d e gr e e i n cr e me n t s. T h e r e s ul t s ar e g r ap h ed a s a

f u nc tio n o f t h e r a tio r o f t he fo r ced i nc id e nt wa ve n u mb er to t h e wa v e n u mb e r o f a

fr e el y p r o p a ga ti n g wa v e i n t h e fir s t med i u m. T h e r at io o f t he wa ve n u m b er o f a

fr e el y p r o p a ga ti n g wa v e i n t h e fir s t med i u m to t ha t i n t he s eco nd me d i u m ( α ) i s 2 .

T he r a tio o f t he i mp ed a nc e o f t he c h ar ac ter i st ic i mp ed a n ce o f t he fir s t med i u m to

th at o f t h e s eco nd me d i u m ( β) i s 2 . .................................................................... 60

Fi g ur e 3 .1 1 T h e tr a n s mi tted i nt e n si t y d ue to a f o r ced d i f f u s e i n cid e n t s o u nd f ie ld . T he

r es u lt s a r e gr ap h ed a s a f u nc tio n o f t h e r a tio r o f t he fo r ced i nc id e nt wa ve n u mb er to

th e wa ve n u mb er o f a fr eel y p r o p a ga ti n g wa ve i n t he f ir st me d i u m. Alp h a i s eq u al to

½. B e ta i s ½, 1 o r 2 . ......................................................................................... 61

Fi g ur e 3 .1 2 T h e tr a n s mi tted i nt e n si t y d ue to a f o r ced d i f f u s e i n cid e n t s o u nd f ie ld . T he

r es u lt s a r e gr ap h ed a s a f u nc tio n o f t h e r a tio r o f t he fo r ced i nc id e nt wa ve n u mb er to

th e wa ve n u mb er o f a fr eel y p r o p a ga ti n g wa ve i n t he f ir st me d i u m. Alp h a i s eq u al to

2 . B eta i s ½, 1 , o r 2 . ........................................................................................ 62

Fi g ur e 4 .1 P l ate 1 a nd P lat e 2 . ................................................................................. 65

Fi g ur e 4 .2 T h e a n g le o f r ef le ct io n eq ua l s t h e a n gl e o f i nc id e nc e fo r a fr eel y p r o p a ga ti n g

wa v e. T he p o s iti v e y- a x is p o i nt s v er t ica ll y o ut o f t he p a ge. ................................. 74

Fi g ur e 4 .3 An i n cid e n t a co u s tic so u nd wa v e. ............................................................. 90

Fi g ur e 4 .4 T h e o c ta nt t h at eq u at io n 4 .1 8 7 is a ve r ag ed o ver . ........................................ 92

Fi g ur e 5 .1 T h e r e la ti ve t r an s mi t ted i nt e ns it y at t he j u nc tio n o f t wo i n f i ni te p a n el s d ue to a

fo r ced wa ve i n t h e f ir s t p an el i nc id e nt at a n a n g l e o f i n cid e n ce to t h e no r ma l o f 0 ˚ .

T he r a tio κ o f t he wa ve n u mb e r i n t he se co nd p a ne l to t ha t i n t h e fir s t p a ne l i s ½.

C ur ve s ar e g i ve n fo r t he r at io ψ eq u al s ½, 1 a nd 2 . ............................................10 3

Fi g ur e 5 .2 . T he r ela ti v e tr a n s mi t t ed i n te n s it y at th e j u n ct io n o f t wo i n fi n ite p a ne ls d ue to

a fo r ced wa v e i n t he f ir s t p a ne l i n cid e nt at a n a n gl e o f i nc id e nc e to t he n o r mal o f 0 ˚ .

T he r a tio κ o f t he wa ve n u mb e r i n t he se co nd p a ne l to t ha t i n t h e fir s t p a ne l i s 1 .

C ur ve s ar e g i ve n fo r t he r at io ψ eq u al s ½, 1 a nd 2 . ............................................10 4

Fi g ur e 5 .3 T h e r e la ti ve t r an s mi t ted i nt e ns it y at t he j u nc tio n o f t wo i n f i ni te p a n el s d ue to a

fo r ced wa ve i n t h e f ir s t p an el i nc id e nt at a n a n g l e o f i n cid e n ce to t h e no r ma l o f 0 ˚ .

T he r a tio κ o f t he wa ve n u mb e r i n t he se co nd p a ne l to t ha t i n t h e fir s t p a ne l i s 2 .

C ur ve s ar e g i ve n fo r t he r at io ψ eq u al s ½, 1 a nd 2 . ............................................10 5

Fi gu r e 5. 4 Th e r e lati v e t ran s mit ted in t en s ity at th e ju n c tio n o f t wo in fi n it e p an e ls d u e to

a fo rc ed w a ve in th e fir s t p an el in cid en t at an a n gl e o f in c id en ce to th e n or ma l o f

15˚ . Th e ra tio κ o f t h e w av e n u mb e r in th e s ec o n d p an el to th at in th e f ir st p an e l i s

1/ 2. Cu r v e s ar e gi v en fo r th e r ati o ψ eq u a l s ½ , 1 an d 2 . .....................................10 6

Fi g ur e 5 .5 T h e r e la ti ve t r an s mi t ted i nt e ns it y at t he j u nc tio n o f t wo i n f i ni te p a n el s d ue to a

fo r ced wa ve i n t h e f ir s t p an el i nc id e nt at a n a n g l e o f i n cid e n ce to t h e no r ma l o f

1 5 ˚ .T he r a tio κ o f t he wav e n u mb er i n t he se co n d p a nel to t hat i n t he f ir st p a ne l i s 1 .

vii

C ur ve s ar e g i ve n fo r t he r at io ψ eq u al s ½, 1 a nd 2 . ............................................10 7

Fi g ur e 5 .6 T h e r e la ti ve t r an s mi t ted i nt e ns it y at t he j u nc tio n o f t wo i n f i ni te p a n el s d ue to a

fo r ced wa ve i n t h e f ir s t p an el i nc id e nt at a n a n g l e o f i n cid e n ce to t h e no r ma l o f 1 5 ˚ .

T he r a tio κ o f t he wa ve n u mb e r i n t he se co nd p a ne l to t ha t i n t h e fir s t p a ne l i s 2 .

C ur ve s ar e g i ve n fo r t he r at io ψ eq u al s ½, 1 a nd 2 . ............................................10 8

Fi g ur e 5 .7 T h e r e la ti ve t r an s mi t ted i nt e ns it y at t he j u nc tio n o f t wo i n f i ni te p a n el s d ue to a

fo r ced wa ve i n t h e f ir s t p an el i nc id e nt at a n a n g l e o f i n cid e n ce to t h e no r ma l o f 3 0 ˚ .

T he r a tio κ o f t he wa ve n u mb e r i n t he se co nd p a ne l to t ha t i n t h e fir s t p a ne l i s 1 /2 .

C ur ve s ar e g i ve n fo r t he r at io ψ eq u al s ½, 1 a nd 2 . ............................................10 9

Fi g ur e 5 .8 T h e r e la ti ve t r an s mi t ted i nt e ns it y at t he j u nc tio n o f t wo i n f i ni te p a n el s d ue to a

fo r ced wa ve i n t h e f ir s t p an el i nc id e nt at a n a n g l e o f i n cid e n ce to t h e no r ma l o f 3 0 ˚ .

T he r a tio κ o f t he wa ve n u mb e r i n t he se co nd p a ne l to t ha t i n t h e fir s t p a ne l i s 1 .

C ur ve s ar e g i ve n fo r t he r at io ψ eq u al s ½, 1 a nd 2 . ............................................11 0

Fi g ur e 5 .9 T h e r e la ti ve t r an s mi t ted i nt e ns it y at t he j u nc tio n o f t wo i n f i ni te p a n el s d ue to a

fo r ced wa ve i n t h e f ir s t p an el i nc id e nt at a n a n g l e o f i n cid e n ce to t h e no r ma l o f 3 0 ˚ .

T he r a tio κ o f t he wa ve n u mb e r i n t he se c o nd p a ne l to t ha t i n t h e fir s t p a ne l i s 2 .

C ur ve s ar e g i ve n fo r t he r at io ψ eq u al s ½, 1 a nd 2 . ............................................11 1

Fi g ur e 5 .1 0 T h e r e la ti ve tr a n s mi tt ed i n te n s it y at th e j u n ct io n o f t wo i n fi n ite p a ne ls d ue to

a fo r ced wa v e i n t he f ir s t p a ne l i n cid e nt at a n a n gl e o f i nc id e nc e to t he n o r mal o f

4 5 ˚ . T he r at io κ o f t he wa v e n u mb er i n t h e seco nd p a nel to t hat i n t he f i r st p a ne l i s

1 /2 . C ur ve s a r e g i ve n fo r t he r at io ψ eq u al s ½, 1 an d 2 . ......................................11 2

Fi g ur e 5 .1 1 . T he r ela ti v e tr a n s mi t ted i nt e ns it y a t t he j u nc tio n o f t wo i n f in it e p a n el s d ue

to a fo r ced wa ve i n t he f ir s t p a n el i nc id e nt at a n an g le o f i nc id e n ce to t h e n o r ma l o f

4 5 ˚ . T he r at io κ o f t he wav e n u mb er i n t he se co n d p a nel to t hat i n t he f ir st p a ne l i s 1 .

C ur ve s ar e g i ve n fo r t he r at io ψ eq u al s ½, 1 a nd 2 . ............................................11 3

Fi g ur e 5 .1 2 T h e r e la ti ve tr a n s mi tt ed i n te n s it y at th e j u n ct io n o f t wo i n fi n ite p a ne ls d ue to

a fo r ced wa v e i n t he f ir s t p a ne l i n cid e nt at a n a n gl e o f i nc id e nc e to t he n o r mal o f

4 5 ˚ . T he r at io κ o f t he wav e n u mb er i n t he se co n d p a nel to t hat i n t he f ir st p a ne l i s 2 .

C ur ve s ar e g i ve n fo r t he r at io ψ eq u al s ½, 1 a nd 2 . ............................................11 4

Fi g ur e 5 .1 3 T h e r e la ti ve tr a n s mi tt ed i n te n s it y at th e j u n ct io n o f t wo i n fi n ite p a ne ls d ue to

viii

a fo r ced wa v e i n t he f ir s t p a ne l i n cid e nt at a n a n gl e o f i nc id e nc e to t he n o r mal o f

6 0 ˚ . T he r at io κ o f t he wav e n u mb er i n t he se co n d p a nel to t hat i n t he f ir st p a ne l i s

1 /2 . C ur ve s a r e g i ve n fo r t he r at io ψ eq u al s ½, 1 an d 2 . ......................................11 5

Fi g ur e 5 .1 4 T h e r e la ti ve tr a n s mi tt ed i n te n s it y at th e j u n ct io n o f t wo i n fi n ite p a ne ls d ue to

a fo r ced wa v e i n t he f ir s t p a ne l i n cid e nt at a n a n gl e o f i nc id e nc e to t he n o r mal o f

6 0 ˚ . T he r at io κ o f t he wav e n u mb er i n t he se co n d p a nel to t hat i n t he f ir st p a ne l i s 1 .

C ur ve s ar e g i ve n fo r t he r at io ψ eq u al s ½, 1 a nd 2 . ............................................11 6

Fi g ur e 5 .1 5 T h e r e la ti ve tr a n s mi tt ed i n te n s it y at th e j u n ct io n o f t wo i n fi n ite p a ne ls d ue to

a fo r ced wa v e i n t he f ir s t p a ne l i n cid e nt at a n a n gl e o f i nc id e nc e to t he n o r mal o f

6 0 ˚ . T he r at io κ o f t he wa v e n u mb er i n t h e s eco nd p a nel to t hat i n t he f i r st p a ne l i s

2 . C ur v es ar e gi ve n fo r t he r at io ψ eq ual s ½, 1 a nd 2 . .........................................11 7

Fi g ur e 5 .1 6 . T he r ela ti v e tr a n s mi t ted i nt e ns it y a t t he j u nc tio n o f t wo i n f in it e p a n el s d ue

to a fo r ced wa ve i n t he f ir s t p a n el i nc id e nt at a n an g le o f i nc id e n ce to t h e n o r ma l o f

7 5 ˚ . T he r at io κ o f t he wav e n u mb er i n t he se co n d p a nel to t hat i n t he f ir st p a ne l i s

1 /2 . C ur ve s a r e g i ve n fo r t he r at io ψ eq u al s ½, 1 an d 2 . ......................................11 8

Fi g ur e 5 .1 7 . T he r ela ti v e tr a n s mi t ted i nt e ns it y a t t he j u nc tio n o f t wo i n f in it e p a n el s d ue

to a fo r ced wa ve i n t he f ir s t p a n el i nc id e nt at a n an g le o f i nc id e n ce to t h e n o r ma l o f

7 5 ˚ . T he r at io κ o f t he wav e n u mb er i n t he se co n d p a nel to t hat i n t he f ir st p a ne l i s 1 .

C ur ve s ar e g i ve n fo r t he r at io ψ eq u al s ½, 1 a nd 2 . ............................................11 9

Fi g ur e 5 .1 8 T h e r e la ti ve tr a n s mi tt ed i n te n s it y at th e j u n ct io n o f t wo i n fi n ite p a ne ls d ue to

a fo r ced wa v e i n t he f ir s t p a ne l i n cid e nt at a n a n gl e o f i nc id e nc e to t he n o r mal o f

7 5 ˚ . T he r at io κ o f t he wav e n u mb er i n t he se co n d p a nel to t hat i n t he f ir st p a ne l i s 2 .

C ur ve s ar e g i ve n fo r t he r at io ψ eq u al s ½, 1 a nd 2 . ............................................12 0

Fi g ur e 5 .1 9 . T he r ela ti v e tr a n s mi t ted i nt e ns it y a t t he j u nc tio n o f t wo i n f in it e p a n el s d ue

to a fo r ced wa ve i n t he f ir s t p a n el i nc id e nt at a n an g le o f i nc id e n ce to t h e n o r ma l o f

9 0 ˚ . T he r at io κ o f t he wav e n u mb er i n t he se co n d p a nel to t hat i n t he f ir st p a ne l i s

1 /2 . C ur ve s a r e g i ve n fo r t he r at io ψ eq u al s ½, 1 an d 2 . ......................................12 1

Fi g ur e 5 .2 0 T h e r e la ti ve tr a n s mi tt ed i n te n s it y at th e j u n ct i o n o f t wo i n fi n ite p a ne ls d ue to

a fo r ced wa v e i n t he f ir s t p a ne l i n cid e nt at a n a n gl e o f i nc id e nc e to t he n o r mal o f

9 0 ˚ . T he r at io κ o f t he wav e n u mb er i n t he se co n d p a nel to t hat i n t he f ir st p a ne l i s 1 .

C ur ve s ar e g i ve n fo r t he r at io ψ eq u al s ½, 1 a nd 2. ............................................12 2

Fi g ur e 5 .2 1 T h e r e la ti ve tr a n s mi tt ed i n te n s it y at th e j u n ct io n o f t wo i n fi n ite p a ne ls d ue to

a fo r ced wa v e i n t he f ir s t p a ne l i n cid e nt at a n a n gl e o f i nc id e nc e to t he n o r mal o f

9 0 ˚ . T he r at io κ o f t he wa v e n u mb er i n t h e seco nd p a nel to t hat i n t he f i r st p a ne l i s

2 . C ur v es ar e gi ve n fo r t he r at io ψ eq ual s ½, 1 a nd 2 . .........................................12 3

Fi g ur e 5 .2 2 T h e i n cid e n t fi eld is a d i f f u se v ib r at io n al f ie ld . T h e i n te gr at io n i s d o ne o ver

all t he p o s sib le a n gl es o f i nc id e nc e. B ec a u se o f s y m me tr y, t he i nt e gr a tio n is o nl y

d o ne fr o m 0 to 9 0 d e gr e es. T he r at io κ o f t he wa ve n u mb er i n t h e seco nd p a nel to

th at i n t he f ir st p a nel i s 1 /2 . C ur ve s a r e g i ve n fo r t he r at io ψ eq u al s ½, 1 an d 2 . ...12 4

Fi g ur e 5 .2 3 T h e i n cid e n t fi eld is a d i f f u s e v ib r at io n al f ie ld . T h e i n te gr at io n i s d o ne o ver

all t he p o s sib le a n gl es o f i nc id e nc e. B ec a u se o f s y m me tr y, t he i nt e gr a tio n is o nl y

d o ne fr o m 0 to 9 0 d e gr e es. T he r at io κ o f t he wa ve n u mb er i n t h e seco nd p a nel to

th at i n t he f ir st p a nel i s 1 . C ur v es ar e g i ve n fo r t he r at io ψ eq ual s ½, 1 a nd 2 . ......12 5

Fi g ur e 5 .2 4 T h e i n cid e n t fi eld is a d i f f u se v ib r at io n al f ie ld . T he i nte g r a tio n is d o n e o ve r

all t he p o s sib le a n gl es o f i nc id e nc e. B ec a u se o f s y m me tr y, t he i nt e gr a tio n is o nl y

d o ne fr o m 0 to 9 0 d e gr e es. T he r at io κ o f t he wa ve n u mb er i n t h e seco nd p a nel to

th at i n t he f ir st p a nel i s 2 . C ur v es ar e gi ve n fo r t he r at io ψ eq ual s ½, 1 a nd 2 . ......12 6

Fi g ur e 5 .2 5 . T he v ib r a ti o na l f ield i n t he 1 st p a n el i s e xc ited b y a d i f f u s e i nc id e n t

aco us ti c fi eld . T he r at io κ o f t h e wa v e n u mb er i n t he se co nd p a ne l to t hat i n t he fir s t

p an el i s 1 /2 . C u r ve s ar e gi v e n fo r t he r at io ψ eq u al s ½, 1 a nd 2 . ..........................12 7

Fi g ur e 5 .2 6 . T he v ib r a ti o na l f ield i n t he 1 st p a n el i s e xc ited b y a d i f f u s e i nc id e n t

aco us ti c fi eld . T he r at io κ o f t h e wa v e n u mb er i n t he f ir st p a nel to t ha t i n t he se co nd

p an el i s 1 . C ur ve s ar e g i ve n fo r t h e r a tio ψ eq ua l s ½, 1 a nd 2 . .............................12 8

Fi g ur e 5 .2 7 . T he v ib r a ti o na l f ield i n t he 1 st p a n el i s e xc ited b y a d i f f u s e i nc id e n t

aco us ti c fi eld . T he r at io κ o f t h e wa v e n u mb er i n t he se co nd p a ne l to t h at i n t he fir s t

p an el i s 2 . C ur ve s ar e g i ve n fo r t h e r a tio ψ eq ua l s ½, 1 a nd 2 . .............................12 9

Fi g ur e 5 .2 8 T h e r e la ti ve i nte n s it y tr a n s mi t ted at a p i n ned j o i n t b e t we e n t wo p a ne l s wh e n

th e t wo p a n el s h a ve t he sa me ma ter ial p r o p er tie s . T he f ir st p a n el i s e x ci t ed o n o ne o f

it s sid e s b y a d i f f u se so u nd f ie ld . T he I r wi n c ur ve s ho ws t h e ca lc u lat io n s mad e i n

th i s t h es i s, wh i le t he Vi llo t & G u i go u - Car ter c u r ve s ho ws t h e ca lc u la tio n s mad e b y

Vil lo t a nd G ui go u - C ar te r ( 2 0 0 0 ) . T he x - a x i s var i ab le, r i s t h e r a tio o f t he wa ve

n u mb e r o f t h e d i f f u s e so u nd f ie ld to t he f r ee b e n d in g wa v e n u mb er o f t h e t wo

id e nt ica l p a n el s. T he c r i tic al fr eq ue nc y o cc ur s whe n r eq ua l s o ne. .......................13 0

List of Symbols a a co ns ta n t

a 1 a co n st a nt

a 2 a co n st a nt

B b e nd i n g st i f f ne s s o f t h e p l at e

B 1 b e nd i n g s ti f f n e s s o f t h e p l at e i n med i u m 1

B 2 b e nd i n g s ti f f n e s s o f t h e p l at e i n med i u m 2

B 0 t he ad iab at ic b ul k mo d ul u s o f t he air

c t he p h as e sp ee d o f t he so u nd wa v e i n a f l uid med i u m

c m t he sp eed o f so u nd i n t h e m t h med i u m

c i fr ee l y p r o p a g at in g b e nd i n g wa v e ve lo c i t y o f t h e i t h p lat e

E( z) var ia tio n o f b e nd i n g wa v e i n z -a x i s d ir ect io n

e e xp o ne n tia l f u nc tio n

exp e xp o n e nti al f u nc tio n

f fr eq u e nc y

f c cr it ica l fr eq ue nc y

F x r ea ct io n fo r c e p er u ni t l e n gt h at a b o u n d ar y

I i nt e ns it y o f t h e aco u st ic wa v e

I i i nc id e n t i n te n s it y

I r r e f lec ted i nte n si t y

I t tr a n s mi t ted i nt en s it y

I ( i + r ) to ta l i n te n si t y at x =0

I t f o r c e d fo r c ed tr a n s mi tted i nt e n si t y d ue to a f o r ced i n cid e nt wa ve

I t f r e e tr a n s mi t ted i n te n si t y o f a no r ma ll y i nc id e nt f r ee l y p r o p a g at i n g wa ve

j i s t he sq uar e r o o t o f -1

k wa v e n u mb er

k a a ir b o r ne wa ve n u mb er

k f fo r c ed wa v e n u mb e r

k 1 wa v e n u mb er in med i u m 1

k 2 wa v e n u mb er in med i u m 2

xii

k t x t r a n s mi tt ed wav e n u mb er i n t he x d ir e ctio n

k t y tr a n s mi tt ed wa v e n u mb er i n the y d ir e ct io n

k m wa v e n u mb er i n t h e m th me d iu m

k= (k x,k y,0 ) wa v e n u mb er v ecto r

k t= ( k t x,k t y,0 ) t r a n s mi tt ed wa v e n u mb er v ect o r

k r= ( k r x,k r y,0 ) r e f le c ted wa ve n u mb er v ect o r

k f= ( k f x,k f y,0 ) fo r ced i n cid e n ce wa ve n u m b er vec tor

k N n ea r f ield wa v e n u mb er

k N 1 ne ar f ie ld wa v e n u mb er i n med i u m 1

k N 2 ne ar f ie ld wa v e n u mb er i n med i u m 2

k x i i nc id e nt wa v e n u mb er i n x - a xi s d ir e ct io n

k x r r e fl ected wa v e n u mb er i n x - ax i s d ir ect io n

k x 2 wa ve n u mb er i n x -a x i s d ir e ctio n i n me d i u m 2

m ma s s p er u ni t ar e a o f p lat e

m i ma s s p er u ni t ar e a o f t he ith p lat e

M x z mo me n t p e r u ni t l e n gt h ab o ut t he z - a xi s e xer t ed o n th e b e a m cr o s s

sec tio n al ar ea no r ma l to t he x - a xi s

M z a n g ul ar mo me n t p er u ni t le n gt h ab o ut t he z - a xi s

p aco u st ic p r es s ur e

p f fo r c ed aco u st ic p r e ss ur e

p r r e f l ected aco u s tic p r e ss u r e

p t tr a n s mi tt ed a co us ti c p r e s s ur e

Q x s he a r fo r ce p er u ni t l e n gt h a cti n g o n a p la ne no r ma l to t he x -a x i s

r r at io o f f o r ced i nc id e n t wa v e no . to fr eel y p r o p a ga ti n g wa ve no . i n

th e i n cid e nt me d i a

r a mp lit ud e o f r e fl ect ed wa v e

r r at io o f air b o r n e wa v e no . to wa ve n u mb er i n p l at e 1

2 κθ =

r j a mp l it ud e o f near f ie ld r e f le c ted wa ve

sin

θ 2

2 χ a

s 2

t ti me

xiii

t a mp l it ud e o f tr a ns mi tted wa v e

t j a mp li t ud e o f tr an s mi t ted n ear f ield wa v e

U vo l u me velo ci t y p er u ni t vo l u me i ns er t ed i nto t he med i u m a t p o si tio n x

u f fo r ced co mp l e x p ar t icl e ve lo c it y

u co mp le x co nj u ga t e o f t h e p ar tic le v elo ci t y

xu co mp le x p a r t icl e ve lo c it y i n t h e x d ir e ct i o n

fxu fo r ced co mp l e x p ar ti cl e ve lo ci t y ( i n t h e d ir ec tio n o f t he x -a x i s)

rxu r e f lec ted co mp le x p ar t icl e ve lo ci ti e s ( i n t he x -a x is d ir ect io n)

txu tr a n s mi t ted co mp l ex p ar t ic le v elo c it ie s ( i n t he x - a xi s d ir e ct io n)

u co mp le x p ar t ic le ve lo c it y

v y tr a n s v er s e ve l o cit y o f t h e p l ate i n t he d ir ec tio n o f t he y -a x i s

v 1 tr a ns v er s e b e n d in g wa v e i n p la te 1

v 2 tr a ns v er s e b e n d in g wa v e i n p la te 2

v 1 + tr a n s ver se v elo cit y a mp l it ud e o f a b e nd in g wa v e

w c an g ula r cr it ica l fr eq ue nc y

w a n g ular velo ci t y

w z an g ul ar ve lo ci t y o f t he p la te ab o ut t he z -a x is

w z 2 an g u lar v elo ci t y a b o ut t he z -a x i s i n p la te 2

z i mp ed a nce

Z 1 i mp ed a nc e i n med ia 1

Z 2 i mp ed a nc e i n med ia 2

Z i i mp ed a nc e e xp er ie n ced b y a fr e el y p r o p ag at i n g b e nd i n g wa v e

θ f i nc id e nt a n gl e

θ r r e fl ect ed a n g le

θ t tr a n s mi t ted a n g le

η 1 d a mp i n g lo s s f acto r o f p la te 1

ρ c ha n ge i n d e n s i t y fr o m t he a mb ie n t d e n si t y

ρ 0 i s t h e a mb ie n t d en s it y o f t he med i u m i n wh i c h t he wa v e i s tr a v ell i n g

Ω∂ ele me nt al s tr ip o f so l id a n g le

xiv

ρ m a mb ie n t d e n s it y o f t he m t h med i u m

< > a ver a g e

ZPPu , f

T he b ar o ver t he se s y mb o ls i nd ica te s t a ki n g t he co mp l e x co nj u ga te

Re r ea l p ar t o f t h e i ma g i nar y n u mb er

κ k 2 /k 1

φ a n gl e to no r m al o f wa ve n u mb er ve cto r o f e xc it i n g aco u s tic p la ne wa v e

χ a r at io o f fo r ced i ncid e nt wa v e n u mb er

2 κ

−

ω a n g ular fr eq ue nc y

)

(

2 kB 22 2 kB 11

k 1 k

Z 1 Z

≥ χκ a

uθ

χκ < a

κ χ a

  

  

π   2    arcsin  

Abstract

T he E ur o p e a n Co o p er at i o n i n Sci e nce a nd T ec h n o lo g y ( CO ST ) Ac tio n F P 0 7 0 2 “Ne t -

Aco u st ic s fo r T i mb e r b a sed Li g ht we i g ht B u ild i n g s a nd E le me nt s ” i s a tt e mp ti n g to

ex te nd t he EN1 2 3 5 4 ser i es o f s ta nd a r d s. T he E ur o p ea n Sta nd ar d s ( E N) C EN ( 2 0 0 0 a) ,

CE N ( 2 0 0 0 b) , C EN ( 2 0 0 0 c) , CEN ( 2 0 0 0 d) , CE N ( 2 0 0 0 e ) , CE N ( 2 0 0 0 f) se r ie s, E N

1 2 3 5 4 g i ve s me t ho d s fo r p r ed ic ti n g t he p r o p a ga t io n o f so u nd a nd v ib r a ti o n i n

b ui ld i n g s. T he f ir st f o ur p ar t s o f t hi s ser ie s ha v e al so b e e n p ub l i s hed a s t he

I n ter na tio n al Or ga n iza ti o n fo r S ta nd ar d iz at io n ( I SO) I S O1 5 7 1 2 ser ie s o f st a nd ar d s

( I SO ( 2 0 0 5 a) , I S O ( 2 0 0 5 b) , I S O ( 2 0 0 5 c) , I SO ( 2 0 0 5 d) ) . T he se se r ie s o f st a nd ar d s a r e

co mp le me n t ed b y t h e I S O 1 0 8 4 8 ser ie s o f st a nd ar d s ( I S O ( 2 0 0 6 a) , I SO ( 2 0 0 6 b) , I S O

( 2 0 0 6) ) , wh i c h sp e ci f ic me t ho d s o f me a s ur i n g t h e fla n k i n g so u nd tr a n s m is s io n i n

b ui ld i n g s.

T hi s r e sea r c h ha s r ai sed t he q ue s tio n o f wh e t her t he v ib r a tio n tr a n s mi s s i o n at a wa ll

j un ct io n d ep e nd s o n ho w t h e wa ll i s e x ci ted ( b y e it h er a so u nd wa v e o r a mec h a ni cal

s ha ke r ) . T h e a i m o f t hi s r es ear c h i s to i n ve s ti ga t e t hi s q u e stio n a nd he n c e co ntr ib ut e to

th e r e v i sio n o f t he s er i e s o f st a nd ar d s. Vi llo t a n d G ui go u - C ar te r ( 2 0 0 0 ) ha v e

co n s id er ed t hi s p r o b le m b ut t he ir eq uat io n s ( 1 0 ) and ( 1 3 ) ap p ear to b e i n er r o r . T h is

r es ear c h wi ll e nd e a vo ur to d er i ve t he co r r ec t eq u atio n s a nd r ep e at Vi llo t and G u i go u -

Car ter ’ s cal c ul at io ns.

T hi s t he s i s s ho ws t h at t he tr a n s mi s s io n o f fo r ce d b e nd i n g wa ve s i s d i f f e r en t fr o m t he

tr a n s mi s sio n o f fr e el y p r o p ag at i n g b e nd i n g wa v es. Ho we v er , V il lo t a nd G ui go u -

Car ter ’ s ( 2 0 0 0) ca lc u lat i o n s, fo r a p i nn ed j u n ct io n b et wee n t wo p a ne ls t h at ar e t h e

sa me , o ver e st i ma te s t h e d i f fer e nce. T he ca se o f a fo r ced wa v e i n a f l uid in cid e nt o n a

p la ne i nt er fac e s ur fa ce wh er e t he p r o p er t ie s o f t he f l u id ma y c ha n ge i s i n ve s ti ga ted

f ir s t. I t i s s ho wn t h at t h e i nt e n si t y p r o p a ga ted t o a nd fr o m t he i nte r fac e s ur fa ce ca n no t

b e ca lc ul at ed sep ar ate l y fo r t he fo r ced i nc id e n t wa v e a nd t h e fr e el y p r o p ag at i n g

r ef le cted wa v e, b e ca u se th e cr o s s ter ms i n t h e i n te n si t y ca lc u lat io n c a n n o t b e c a nce led .

T hi s i mp li es t ha t a tr a n s mi s s io n fa cto r o r co e f f i cie n t ca n no t b e ca lc u lat ed . T hi s i s t h e

xvi

r ea so n fo r t he er r o r i n V illo t a nd G u i go u - Car ter ’ s ( 2 0 0 0 ) eq uat io n ( 1 0 ) .

T he ca lc u la tio n s ar e t h e n e x te nd ed to t he p i n ned j u nct io n b e t we e n t he t wo p a nel s ca se ,

co n s id er ed b y V il lo t a nd G ui go u- C ar te r . I t i s s ho wn t h at t hei r d i f f u se fi el d we i g hti n g

xvii

is al so i n er r o r .

Chapter 1 Introduction

Fla n k i n g so u nd tr a n s mi s sio n i s t h e tr a n s mi s sio n o f so u nd fr o m o ne r o o m to a no t h er

o th er t ha n v ia t he co m m o n wa ll o f t he t wo r o o m s ( E . Ge r r et s e n ( 1 9 8 6 ) , E . Ge r r et se n

( 1 9 7 9) , N i g ht i n gal e ( 1 9 9 5) ) . T he c ur r e n t t h eo r y is o nl y v al id fo r he a v y we i g h t si n g le

lea f wa ll s. Ma n y r es ear c her s h a ve a d d r e ss ed t h i s ar ea o f r e sea r c h, se e t h e p ap er s b y

Vil lo t ( 2 0 0 2) , Ni g h ti n ga le a nd B o s ma n s ( 2 0 0 3) , Da v y, Ma h n, G ui go u - C ar ter , a nd

Vil lo t ( 2 0 1 2) ; ( Ed d y Ge r r et se n, 2 0 0 7 ) , CST C ( 2 0 0 8 ) , Ma h n ( 2 0 0 8 ) , a nd Da v y ( 2 0 0 9) .

Re se ar c her s ar e ab le to me a s ur e t he f la n k i n g tr a n s mi s sio n o f ma n y b ui ld in g s ys t e ms

( B r u n s ko g a nd C h u n g ( 2 0 1 1) , Cr i sp i n, I n gel aer e , Va n Da m me , a nd W u yt s ( 2 0 0 6) a nd

G ui go u - Car ter , V il lo t, a nd Ro la nd ( 2 0 0 6) ) . Li g h t we i g ht b u ild i n g e le me n ts t yp i ca ll y

ha v e cr it ica l fr eq ue nc ie s i n o r ab o v e t h e fr eq u e n c y r a n ge o f i n ter e s t, so t he c ur r e n t

th eo r y d o e s no t ap p l y to t he m. T he E u r o p ea n Co o p er at io n i n S cie n ce a nd T ech no lo g y

( C OST ) Act io n FP 0 7 0 2 “Ne t - Aco us ti cs fo r T i m b er b a sed L i g ht we i g ht B ui ld i n g s a nd

Ele me n t s” h as b ee n at te mp ti n g to e x te nd t he E N 1 2 3 5 4 ser ie s o f s ta nd ar d s ( I SO

( 2 0 0 5 a) , I SO ( 2 0 0 5 b ) , I SO ( 2 0 0 5 c) a nd I S O ( 2 0 0 5 d) ) fo r ca lc u lat i n g t he f la n k i n g so u nd

tr a n s mi s sio n o f si n g le le af hea v y we i g ht wa ll s to li g h t we i g ht wa ll s a nd i n p ar t ic ul ar to

mo r e co mp li cat ed l i g ht we i g h t wal l s. T he r e sea r ch wo r k o f t h is ac tio n h as r ai sed t he

q ue s tio n o f wh e t her t he tr a n s mi s sio n o f vib r at io n fr o m o ne wa ll to a no t h er wa ll

d ep e nd s o n wh e t he r t h e ex ci ted wal l is e xc ited me c ha n ica ll y b y a s ha k e r o r

aco us ti ca ll y b y a so u nd f ield . T he a n s wer i n g o f th i s q ue s tio n wi ll e nab l e t he e x te n s io n

o f EN1 2 3 5 4 to l i g ht we i g ht wa ll s o n a mo r e r a tio na l b a s is . T hi s e xte n s io n is

p ar ti c ul ar l y i mp o r ta nt f o r Au str a li a b e ca u se Au str al ia ha s mu c h mo r e l i g ht we i g ht

co n s tr uct io n t ha n mo s t p ar t s o f E ur o p e.

Vil lo t a nd G ui go u - C ar te r ( 2 0 0 0 ) ha ve co n sid er e d t hi s p r o b le m b u t t h eir eq u at io ns ( 1 0 )

and ( 1 3 ) ap p ear to b e i n er r o r . T he y h a ve p u b li s h ed a no t h er p ap e r o n t h i s s ub j ec t i n

wh ic h t he ir me a s ur e me n t me t ho d s ha ve b ee n r ec o n sid er ed G ui go u - C ar te r et a l. ( 2 0 0 6) .

T hi s r e sea r c h wi ll e nd ea vo ur to d er i ve t he co r r e ct eq u at io ns a nd r ep ea t Vil lo t a nd

G ui go u - Car ter ’ s ca lc ul a tio n s.

Air b o r n e e xc it at io n o f a wa ll wi t h a s i n gle fr eq u en c y o f so u nd a t a s i n gl e a n gl e o f

in cid e nce p r o d uce s a fo r ced b e nd i n g wa v e i n t h e wa ll wi t h a wa ve le n g t h wh ic h i s eq u al

to t he tr ace wa v ele n g t h o f t he i nc id e nt so u nd o n t he wal l. W h e n t h i s fo r ced b e nd i n g

wa v e i s r e f lec ted at t he ed g es o f t h e wa l l, i t p r o d uc es r e so na n t b e nd i n g wa v e s a nd

exp o n e nt ial l y d e ca yi n g ne ar fi eld s. E x ci ta tio n wit h a mec h a ni cal s h a ker p r o d uc e s

r eso n a nt b e n d i n g wa ve s and e xp o ne n tia ll y d ec a y in g near f ie ld s. T h is r e se ar c h wi l l

calc u la te t heo r e ti ca ll y t he d i f fe r e nc e b e t we e n t h e tr a n s mi s s io n o f fo r ced b e nd i n g

wa v e s a nd r e so na n t b e n d in g wa v e s fr o m a n e xc i ted wal l to a wal l co n n e cted to t h e

ex ci ted wal l. Fo r mo s t wa l l s, t h e tr a n s ver se v el o cit y o f t h e r e so na n t b e nd i n g wa v e s i s

lar ger t ha n t he tr a n s ve r s e v elo ci t y o f t he fo r ced b end i n g wa ve s. Vi llo t a nd G ui go u -

Car ter ( 2 0 0 0 ) s u g ge st t h at t h e t r a n s mi s sio n o f fo r ced b e nd i n g wa v e s t hr o u g h t he

j un ct io n c a n b e i g no r ed . I n cid e n t a co us ti c wa v e ex ci tat io n a nd t he mec h an ica ll y

ex ci ted e x ci ta tio n wi ll b e co n sid er ed . Aco u st ic e xc it at io n i nd u ce s fo r c ed an d r e so na n t

b end i n g wa ve s. E xci ta ti o n b y a me c ha ni ca l s ha k er i nd uc es o nl y r e so na nt b e nd i n g

wa v e s. So b y co mp ar i n g t he t wo ca se s o f tr a n s m is s io n o f b e nd i n g wa v e en er g y fr o m

o ne wa ll to a no t her wa ll v ia a co m mo n j u nc tio n , it ca n b e d et er mi n ed i f th e fo r c ed

b end i n g wa ve tr a n s mi s s i o n is s i g ni f ic a nt o r n o t. T he a i m i s to f i nd t h e ve lo ci t y

d i f fer e n ce b et wee n t he t wo wa l ls, wh e n t he wa ll is e xc it ed aco u s tic al l y and

me c ha n ica ll y. W i ll t he v elo c it y d i f fer e nce b e t h e sa me? I t i s s u sp e cted t h at i t wi ll b e

d i f fer e n t b e ca u se i n t h e me c ha n ica l c as e t h er e is o n l y a r e so na nt b e nd i n g wa ve , wh i l e

in t he ai r b o r ne c a se t her e i s b o t h a fo r ced b e nd i n g wa ve a nd r e so n a nt b e nd i n g wa v e.

T he q ue st io n i s b y h o w mu c h wi l l t h e ve lo ci t y t r an s mi s s io n d i f fer ?

T he d er i vat io n o f t he r el ev a nt eq uat io n s i n C r e m er , H ec kl , a nd P e ter s so n ( 2 0 0 5) wi l l

b e s t ud i ed a s wi ll t hei r ex te n sio n to eq uat io n ( 9 ) o f V il lo t a nd G u i go u - Car ter ( 2 0 0 0 ) .

T he d i f f er e nc e b e t we e n th e t r a n s mi s sio n o f fo r c ed a nd r e so n a nt wa ve s wi l l f ir st b e

in v e st i gat ed fo r t he si m p ler c as e o f so u nd wa ve s i n a f l uid me d i u m. T he n t he co r r ec t

ver s io ns o f eq u at io ns ( 1 0 ) a nd ( 1 3 ) o f Vi llo t a n d G ui go u - C ar te r ( 2 0 0 0 ) wi l l b e d er i ved

u si n g t he k no wl ed ge g ai ned i n t he so u nd wa ve c ase . T he se co r r e ct ed eq u atio n s wi ll

th e n b e u s ed to p r ed i ct t he d i f fe r e nc e b e t we e n f o r ced b e nd i n g wa v e tr a n s mi s sio n a nd

r eso n a nt b e n d i n g wa ve t r an s mi s s io n fr o m a n e x c ited wa ll to a n at ta c hed wa l l.

Chapter 2 Derivation of the fluid media sound equations

2.1 Introduction

T he a i m o f t hi s t he si s is to lo o k at t he tr a n s mi s s io n o f fo r c ed b e nd i n g wav es at a

j un ct io n o f t wo p la te s. I n t hi s c h ap t er t h e si mp le r ca se o f f o r ced wa v es i n a fl u id

med i u m ar e co ns id er ed b eca u se t her e ar e le s s v ar iab le s a nd t he eq ua tio n s ar e sl i g ht l y

si mp ler .

I n t hi s c hap t er , t he c as e o f t he tr a n s mi s s io n o f a fo r ced so u nd wa v e fr o m o n e i n f i ni te

ha l f sp ac e f l uid me d i u m to a no t her is co n sid er ed . F ir s t t he c as e o f no r ma l i nc id e n ce ,

wh e n t he t wo me d ia ar e th e sa me i s co n s id er ed .

I t wo u ld no t b e p o s sib le to ge ne r at e a fo r ced wa ve i n a t hr ee d i me n sio n a l fl u id med i u m

b eca u se a d i st r ib u ted t hr ee d i me n s io na l vo l u me ve lo c it y so ur c e wo u ld b e n eed ed .

Ho we v er t h e co n cep t o f a fo r ced wa v e i s t heo r e t ica ll y p o s sib le a nd mo st d er i va tio n s o f

th e t h r ee d i me n s io nal fl uid wa v e eq u at io n d o i n cl ud e a d i s tr ib u ted t hr ee d i me n sio n al

vo l u me ve lo c it y so ur ce. A p o i n t so ur ce i s u s ua ll y i ntr o d u ced b y ma k i n g th e sp a tia l

d is tr ib u tio n o f t h is vo l u me velo ci t y so ur ce a sp a tia l D ir a c d e lta f u nc tio n .

Fo r ced wa v e s i n a fl u id med i u m, d o e xi s t p r a ct i cal l y i n t he t wo d i me n s i o na l c as e o f t he

air c a vi t y i n a d o ub le wall s ys t e m wh e n t he wid th o f t h e c a vi t y i s s ma ll co mp ar ed to

th e wa ve le n gt h o f so u nd . T he y a l so e x is t p r a ct ic all y i n t he o ne d i me n s io na l c as e o f a

mi cr o p ho ne t ur b u le nc e s cr ee n wh e n t he i nt er nal cr o s s se ct io nal d i me n s io n s o f t he

mi cr o p ho ne t ur b u le nc e s cr ee n t ub e ar e s ma ll co mp a r ed to t h e wa v el e n gt h o f so u nd .

2.2 Normal Incidence with the same media on both sides of the junction

T he co mp r es s ib i li t y eq u atio n i s

ρρ = , c

B 0 ρ 0

( 2 .1 )

wh er e ρ i s t he c ha n ge i n d e ns it y f r o m t h e a mb ie nt d e n si t y ( d e n s it y f l uc t ua tio n d u e to

th e so u nd wa v e ) , p i s t h e aco u st ic p r e ss ur e fr o m t he a mb ie n t p r es s ur e . B 0 i s t he

ad iab a ti c b ul k mo d ul u s o f t he air . B 0 i s a me a s u r e o f t h e vo l u me tr ic i nc o mp r es s ib i li t y

0ρ is t he a mb i e nt d e ns it y o f t he med i u m i n wh i c h

o r vo l u me s ti f f n es s o f t h e med i u m.

th e wa ve i s tr a vel i n g. c is t he p h a se sp eed o f t h e so u nd wa ve i n t he ga s e o u s o r liq u id

med i u m. B eca u se t he s e med ia ar e no n -d i sp er si v e , t he gr o up ve lo c it y ( t h e v elo ci t y a t

wh ic h e ner g y i s tr a n sp o r ted ) a nd t he s i g na l ve lo c it y a r e a l so eq ua l to t h e p ha se

ve lo c it y c.

2 c =

B 0 ρ 0

( 2 .2 )

Eq u at io n ( 2 .2 ) r ela te s t h e p ha se sp e ed o f t h e so u nd wa ve to t he ad iab at ic b u l k mo d ul u s

and t he d e n si t y o f t h e m ed i u m.

T he co n ti n u it y eq uat io n is

ρ 0

ρ U 0

ρ ∂ ∂ t

∂ u ∂ x

( 2 .3 )

wh er e u i s t he a co us ti c p ar ti cl e ve lo ci t y o f t he s o u nd wa ve i n t he p o si ti v e x -a x is

∂ u ∂ x

is t he gr ad ie nt i n t he x -a x is d ir e ct io n o f t he aco u st ic p ar t ic l e v elo ci t y. d ir ec tio n.

T he aco u s tic p ar tic le v e lo ci t y i s t he v elo cit y o f th e med i u m ca u sed b y t he so u nd wa v e.

U i s t he vo l u me v elo c it y p er u ni t vo l u me i ns er t e d i nto t he me d i u m at p o si tio n x .

A mb ie n t r e fer s to t he cl i ma ti c co nd it io n s o f t h e med i u m b e fo r e it i s d i s t ur b ed b y t h e

aco us ti c wa v e tr a ve ll i n g t hr o u g h i t.

T he fo l lo wi n g t wo e q uat io n s ar e Ne wto n ’ s i ner ti a eq ua tio n s , wh er e t i s t he ti me .

ρ 0

∂ u ∂ t

∂ p ∂ x

( 2 .4 )

Di f f er e nt ia ti n g ( 2 .3 ) wi t h r esp ect to t gi v e s

)

ρ ( 0

ρ 0

2 ρ 2

∂ ∂ t

∂ u ∂ x

∂ U ∂ t

( 2 .5 )

∂

Di f f er e nt ia ti n g ( 2 .4 ) wi t h r esp ect to x gi v es

)

ρ ( 0

p 2

∂ ∂ x

∂ u ∂ t

∂ x

( 2 .6 )

Di f f er e nt ia ti n g eq ua tio n ( 2 .1 ) t wi ce wit h r e sp e ct to t g i ve s

p 2

2 ρ . 2

1 2 c

∂ ∂ t

( 2 .7 )

2 ∂ ρ 2 t∂

Us e eq ua tio n ( 2 .7 ) to r e p lace t he val u e o f i n t h e fo llo wi n g eq ua tio n g i ve s

−

−=

ρ 0

p 2

∂ p 2 ∂ x

1 2 c

∂ ∂ t

∂ U ∂ t

( 2 .8 )

T hi s i s t he o ne d i me n s io na l wa ve eq uat io n. T o s o lv e t h e o ne d i me n s io na l wa v e

eq u at io n fo r t he fo r ced p r es s ur e , r eq u ir e s t h at it s var iab le s ( p a n d U) ar e d e fi n ed ,

d i f fer e n ti at ed a nd t he n p laced i n to eq ua tio n ( 2 . 8) . T hi s is d o ne b e lo w.

Let t he vo l u me v elo cit y eq u al

−

= UU

exp[

ω ( t

)],

xk f

( 2 .9 )

and t he aco u st ic p r e ss ur e eq ua l

−

exp[

ω ( t

)],

xk f

( 2 .1 0 )

wh er e ω i s t he a n g ular f r eq u e nc y.

Fir s tl y p ( eq ua tio n ( 2 .1 0 ) ) wi ll b e d i f f er e n ti at ed t wi ce wi t h r e sp e ct to t a nd x.

Di f f er e nt ia ti n g p t wic e wi t h r e sp ec t to x g i ve s

2 2 ωω −=

p .

p 2

∂ ∂ t

( 2 .1 1)

Ne xt U is d i f f er e n tia ted wi t h r e sp ec t to t .

ω=

.Uj

∂ U ∂ t

( 2 .1 2)

p 2

∂ U ∂ t

∂ ∂ x

∂ ∂ t

, ( eq uat io n ( 2 .1 1) ) , a nd ( eq u at io n ( 2 .1 2) ) i nto t he S ub s ti t ut i n g va l ue s fo r

o ne d i me n s io na l wa v e e q ua tio n ( ( 2 .8 ) ) g i ve s

−

−=

(

)

ωρ U j 0

2 k f

2 ω 2

( 2 .1 3)

−

T he u se o f eq uat io n ( 2 .9 ) a nd ( 2 .1 0) to r ep lac e p an d U i n eq u at io n ( 2 .1 3 ) g i ve s

ωρ j 0 2 − f

( 2 .1 4)

wh er e

ω ,

( 2 .1 5)

is t he wa v e n u mb er a nd wh er e ω =2 π f.

T he b o u nd ar y o f t he t wo ha l f i n fi n it e fl u id me d i a i n t he p la ne x=0 .

T he fo r c ed so l ut io n o f t he n o n - ho mo ge neo u s eq ua tio n ( 2 .8 ) i s g i ve n b y eq u at io n ( 2 .1 4)

−

ωρ j 0 2 − f

wh er e t he f o r ci n g o nl y o cc ur s i n me d i u m 1 wh e r e x<0 . T h u s U f=0 i f x >0 an d

P f

     0 

( 2 .1 6)

2.3 Media same for x<0 and x>0, normal incidence forced wave

Medium 2

Medium 1

F i gu r e 2 . 1 Th e t wo d i m e n s i o n a l c a s e .

I n t hi s se ct io n t h e s o l ut i o n to t h e ho mo ge no u s v er s io n o f eq u at io n ( 2 .8 ) wi l l b e

calc u la ted . T he r e fle ct e d a nd tr a ns mit ted p r e ss u r es wi ll al so b e ca lc u la t ed . F i na ll y t he

in cid e nt tr a n s mi t ted i nte n si t y wi l l b e ca lc u lat ed .

T o o b tai n t h e co n ti n u it y o f p r e ss ur e a nd aco u st i c p ar t ic le v elo c it y at x= 0 , so l ut io ns o f

th e ho mo ge n eo u s ver sio n o f eq ua tio n ( 2 .8 ) need to b e ad d ed . T he ho mo g en eo us ver sio n

o f eq uat io n ( 2 .8 ) i s

−

p 2

∂ p 2 ∂ x

1 2 c

∂ ∂ t

( 2 .1 7)

T o so l ve t he ab o v e eq u a tio n t he p r es s ur e wi l l b e d e fi ned a nd d i f fer e n ti at ed wi t h

r esp ect to x a nd t. T h e n th e se so l u tio n s wi ll b e r ep lac ed i n to eq u at io n ( 2 .1 7 ) .

T o b egi n t he p r e ss ur e i s d e fi ned to b e

−

exp[

ω ( t

)].

( 2 .1 8 )

Eq u at io n ( 2 .1 8) i s d i f f er en ti ated t wic e wi t h r e sp ect to x a nd t a nd t he r e s ul t s ar e p ut

in to eq u at io n ( 2 .1 7) to g iv e

−

2 am

2 ω 2

( 2 .1 9)

I f a i s no t eq ua l to zer o th e n eq u at io n ( 2 .1 9) ma y b e r ear r a n ged to g i ve

±=

( 2 .2 0)

T he r e s ul t o f eq uat io n ( 2 .2 0) i s p ut i n to eq ua tio n ( 2 .1 8 ) to g i ve t he so l u tio n. T h us t he

−

j ([

ω t

)]

j ([

ω t

ge n er a l so l ut io n o f t he h o mo ge n eo u s eq ua tio n i s

.)]

ea 1

ea 2

( 2 .2 1 )

He nce t he so l ut io n fo r t he ca se o f t he o nl y i nc id en t wa ve b ei n g a fo r c ed in cid e nt wa ve

in t he fir s t med i u m is

−

]

exp[

)]

[ exp

)

ω ( t

xk f

p r

−

exp[

)]

ω ( t

p t

    

( 2 .2 2)

−

≤

)]

exp[

)]

ω ( t

exp[

ω t (

xk f

pk f ω

kp r ω

T hu s,

ρ u 0

−

≥

exp[

)]

ω ( t

p t

k ω

      

( 2 .2 3)

At x=0 a nd t =0 wi t h t he sa me med i u m o n b o t h s id e s, t h e co n ti n u it y o f p r es s ur e gi ve s

p r

p .t

( 2 .2 4 )

Us i n g eq u at io n ( 2 .2 3) , t he co nt i n ui t y o f t he p ar t icl e ve lo ci t y a t x =0 a nd t=0 g i ve s

−

k f ωρ 0

k ωρ 0

( 2 .2 5)

T o fi nd t he tr a n s mi tt ed p r es s ur e t he ab o ve eq uat io n i s mu l t ip l ied b y ρ 0ω

−

pk f

kp r

kp .t

( 2 .2 6 )

t he tr a ns mi t ted p r e ss ur e i s

( 2 .2 7)

−

Re ar r a n g i n g t h is gi ve s

p r

f 2

( 2 .2 8)

T he to ta l i n te n si t y p r o p ag ated i n t he p o s it i ve x d ir ec tio n i f x <0 i s

) u(p. .

ri+

= )

(

( 2 .2 9 )

I ( i + r ) is t he to tal i nt e n si t y, R e i s t he r eal p ar t o f th e co mp le x n u mb er , p i s t h e p r e s s ur e

and u is t h e co mp le x p ar t i cle v elo c it y. T he to t al p r es s ur e o n t h e x <0 s id e i s

−

exp[

ω ( t

)]

[ exp

ω ( t

].)

xk f

p r

( 2 .3 0 )

Fr o m eq u at io n ( 2 .2 5) t h e co mp l e x p ar ti cle velo c it y is gi ve n b y

−

exp[

)]

exp[

)].

ω ( t

xk f

p r

k f ωρ 0

k ωρ 0

( 2 .3 1)

T he b ar o v er t h e u i n eq ua tio n ( 2 .3 1) i n d ic ate s t he o p er a tio n o f ta k i n g t he co mp l e x

co nj u ga te ( c ha n gi n g t he si g n o f t h e i ma g i na r y p a r t o f t h e co mp le x n u mb e r ) . T hi s is

al so so met i me s s ho wn wit h a s up er scr ip t *.

Re(

up ).

(

−

exp[

ω t (

)]

[ exp

ω t (

)

) ]

xk f

P ut ti n g eq uat io n ( 2 .3 0) and ( 2 .3 1) i nto eq uat io n ( 2 .2 9) gi v es

−

exp[

ω t (

)]

exp[

ω t (

)]

xk f

k f ωρ 0

k ωρ 0

  

  .   

=+ ri ) (        

( 2 .3 2)

)

( kk

−

pk f

− 2

f 4

−

)

kk (

th i s b e co me s

−

kj (

xk )

[ − exp

(

=+ ) ri

1 ωρ 0

−

)

f k 2 ( k

( kj

) xk

[ exp

f 2

    ]    ]   

         

     .     

     Re     

( 2 .3 3)

T hu s t he to ta l i nte n s it y ha s b ee n c al c ula ted .

2.4 The total intensity when x=0 (at the junction of the two

media)

I n t hi s se ct io n t h e to ta l in te n s it y is ca lc u lat ed a t t he j u nct io n o f t he p la t es ( x=0 ) .

−

kk (

)

kkk (2

)

kk (2

)

Eq u at io n ( 2 .3 3) i s co n si d er ed a g ai n. E q uat io n ( 2 .3 3 ) ca n b e wr it te n as

−

(

=+ ) ri

2 kk 2

f kk 22

f 4

f ωρ 0

   

 .   

( 2 .3 4)

T hu s

(

=+ ) ri

2 f 4

f ωρ 0

   

 .   

( 2 .3 5)

ω ,

Us i n g t h e fo llo wi n g r ela tio n s hip s

c = , kc ω ,

( 2 .3 6)

(

)

gi v e s

=+ ri )

(

k 2

f ρ 0

( 2 .3 7)

and t h u s t h e to ta l i n te n s it y a t t he j u nct io n o f t he t wo med i a i s

(

=+ ) ri

f ρ 0

 + k  

  

( 2 .3 8)

2.5 The transmitted intensity (I t(x)) when x>0

I n t hi s se ct io n t h e tr a ns mi t ted i nte n s it y i s ca lc u lat ed .

T he tr a ns mi tted i n te n si t y is gi ve n b y

Re( up .

It =

( 2 .3 9 )

T he p r e s s ur e i s gi v e n b y

−

exp[

ω ( t

)].

p t

( 2 .4 0 )

T he co mp le x co nj u ga te o f t he co mp l e x p ar tic le ve lo c it y i s gi v e n b y

−

exp[

ω ( t

)].

p t

k ωρ 0

( 2 .4 1)

No w eq uat io n ( 2 .4 0) a n d ( 2 .4 1) ar e p ut i nto eq u atio n ( 2 .3 9) . T h i s gi v es

−

exp[

ω t (

)]

exp[

ω t (

)].

p t

k ωρ 0

( 2 .4 2)

Eq u at io n ( 2 .4 2) i s s i mp l i fied d o wn to

t =

p t

k . ωρ 0

( 2 .4 3)

Fr o m eq u at io n ( 2 .2 7)

( 2 .4 4)

p f = t ρ 0

 + k  

  

( 2 .4 5)

2.6 The intensity carried by the reflected wave (x<0) if

propagating alone.

I n t hi s se ct io n t h e r e f lec ted i n te n s it y is ca lc u lat e d .

T he i n te n si t y o f t he r e fl ected wa v e i n t he x <0 r e gio n i s

.).( up

I r =

( 2 .4 6 )

T he va l ue u sed fo r p i s t he s a me as eq ua tio n ( 2 .3 0 ) .

−

exp[

ω ( t

)]

[ exp

ω ( t

].)

xk f

p r

( 2 .4 7 )

T he va l ue u sed fo r t he c o mp le x p a r t icl e ve lo ci t y u i s t h e sa me a s eq ua tio n ( 2 .3 1)

−

exp[

ω t (

)]

exp[

ω t (

)].

xk f

p r

k f ωρ 0

k ωρ 0

( 2 .4 8)

−

exp[

( kj

pp f r

T he r e f lec ted i nt e ns it y ( I r) i s t he sa me a s t he i nc id e nt i nt e n si t y ( I i)

pk pk f r ωρωρωρ 0

exp[

( kj

]) xk

pp f r

0 k f ωρ 0

  ]) xk  .    

      

( 2 .4 9)

0=fp

I n t hi s ca se t he i ncid e nt fo r ced wa ve i s no t b ei n g c o ns id er e d , t h u s

−=

pk r ωρ 0

−

(

)

( 2 .5 0)

p r

k 2

f 4

( 2 .5 1)

U s i n g t h e r e lat io n s hi p i n eq ua tio n ( 2 .3 6) it i s fo u nd t ha t

−=

f ρ 0

  

 − k  

( 2 .5 2)

2.7 Media different for x<0 and x>0. A normally incident forced

wave is firstly considered. The transmitted intensity and the

incident and reflected intensity is found.

I n t hi s se ct io n t h e med ia ar e d i f fer e n t o n b o t h si d es o f t he j u nc tio n. T he co mp le x

p ar ti cl e ve lo ci t y i s ca lc ul at ed . T h e r e f lec ted a n d tr a n s mi tt ed p r e s s ur e s ar e c alc u la ted .

T he n t h e tr a ns mi tted i nt en s it y is ca lc u lat ed . Fi n all y t he i nc id e n t i n te n si t y is

calc u la ted .

No te t ha t I r i s ne g at i ve b eca u se t he r e fl ect ed i n t en s it y is p r o p a ga ti n g i n th e ne g at i ve x -

ax i s d ir ect io n r at her t ha n i n t h e p o si ti ve x -a x i s d ir ec tio n. I f ρ 1 a nd c 1 a p p l y fo r x <0 a nd

ρ 2 a nd c 2 ap p l y fo r x>0 , th e n we d e f i ne t he i mp e d an ce s o f t he t wo med i a to b e

and ,

c 11

c 22

( 2 .5 3 )

and t he wa v e n u mb er s t o b e

k and

k 1

ω c

ω c 1

−

)

exp[

)]

ω ( t

( 2 .5 4)

[ exp

xk f

p r

xk 1

−

exp[

)]

ω ( t

p t

xk 2

    

T he p r e s s ur e i n t h e t wo med ia i s gi v e n b y ] ( 2 .5 5)

−

exp[

)]

exp[

)]

ω ( t

xk f

xk 1

pk r 1 ωρ 1

pk f f ωρ 1

Af te r mu c h s ub st it u tio n,

−

exp[

)]

ω ( t

xk 2

pk t 2 ωρ 2

      

( 2 .5 6)

T o fi nd t he tr a n s mi tt ed p r es s ur e , eq ua tio n ( 2 .2 4 ) i s co n sid er ed as it ap p lie s at x=0 a nd

t=0 . Eq uat io n ( 2 .2 4) i s

p r

p t

( 2 .5 7 )

Al so u si n g t he v er sio n o f eq uat io n ( 2 .2 5) wi t h t h e ap p r o p r i at e wa v e n u m b er s g i ve s

−

p t

k 1 = p r ωρωρ

k f ωρ 1

( 2 .5 8)

T he co mp le x p ar t icl e ve lo ci t y c a n no w b e e xp r e s sed a s

−

p r

p t

k ρ

f ck 111

k 1 ρ ck 111

k 2 ck 222

( 2 .5 9)

(

)

k 1

T he tr a n s mi t ted p r e s s ur e i s

p t

Z 2 +

k 1

Z 1

( 2 .6 0)

T o fi nd t he r e fle ct ed p r e s s ur e,

−

p t Z

k 1

Z 1

P r Z 1

( 2 .6 1)

−

)

(

Re ar r a n g i n g t h is eq ua tio n g i ve s t h e r e f le cted p r e s s ur e a s

1 )

Zk 1 + Z

Zk f 2 ( Zk 1

( 2 .6 2)

T r ans mi tted i n te n si t y a t x=0 i s

Re( up .

It =

( 2 .6 3 )

gi v e s

−

exp[

ω t (

)]

exp[

ω t (

)]

xk 2

p t Z

  Re  

  .  

( 2 .6 4)

T her e f o r e eq u at io n ( 2 . 6 4) r ed uce s d o wn to

p t Z

  Re  

  .  

( 2 .6 5)

(

)

k 1

Fr o m eq u at io n ( 2 .6 0)

p t

Z 2 + Z

)

(

k 1

Z 1

  

  

( 2 .6 6)

T he tr a ns mi tted i n te n si t y is fo u nd to b e

(

)

k 1

Z 2 + Z

)

(

Z 1

k 1

  

  

( 2 .6 7)

T o tal i n te n si t y a t x =0 i s

Re(

up .

=+ ) ri

(

−

exp[

]) x

( kj 1

f Zk 11

pp r Z 1

( 2 .6 8 )

(

=+ ) ri

−

exp[

x ])

kj ( 1

k 1

pp f r Z 1

p r Z 1

   .    

   Re    

( 2 .6 9)

At x=0

−

=+ ri )

(

f Zk 11

pp r Z 1

k 1

pp f r Z 1

p r Z 1

  Re  

  .  

( 2 .7 0)

−

)

(

Fr o m eq u at io n ( 2 .6 2)

p r

Zk f 2 Zk ( 1

Zk 11 + Z ) 1

k 1

( 2 .7 1)

=+ ) ri

(

Z +

)

(

2 Z 1

k 1

  

  

( 2 .7 2)

He nce t h is i s t h e si mp le st f o r m o f t he to ta l i nte n si t y.

2.8 The acoustic particle velocity in the oblique incidence

case.

I n t hi s se ct io n t h e co mp le x p ar ti cl e vel o ci t y i n t he x d i r ec tio n i s ca lc u la ted .

T he t hr ee d i me n sio n al m o me n t u m eq uat io n is

−=∇ p

∂ u m ∂ 0 t

( 2 .7 3)

0mρ is t he a mb i e nt d e n si t y o f t he

wh er e u i s t he ve cto r al a co u s tic p ar t ic le v elo c it y an d

mt h me d i u m. T he x co m p o ne n t o f eq uat io n ( 2 .7 4 ) i s

−=

∂ p ∂ x

∂ u x 0 m ∂ t

( 2 .7 4)

ω t j

T he co mp le x p ar t icl e ve lo ci t y i n t he x d ir ec tio n is g i ve n b y

eu 0 x

( 2 .7 5 )

Di f f er e nt ia ti n g wi t h r e s p ect to t gi ve s

ω= uj

∂ u x ∂ t

( 2 .7 6)

P ut ti n g eq uat io n ( 2 .7 6) i nto ( 2 .7 4) g i ve s

ωρ−= j

m u 0 x

∂ p ∂ x

( 2 .7 7)

gi ve s

ρ−= jk

uc xmmm 0

∂ p ∂ x

( 2 .7 8)

−

)

+ ( ykxkj

ω t j

T he p r e s s ur e is gi ve n b y

ep 0

( 2 .7 9 )

T hu s t he co mp le x p ar tic le ve lo c it y i n t h e x d ir e c tio n is

p .

x =

1 Z

k k

( 2 .8 0)

wh er e Z m ca n b e Z 1 o r Z 2, k x ca n b e k 1 x, k r x, o r k f x a nd wi ll ei t her b e r ea l o r i ma g i nar y

and k m ca n b e k 1 o r k 2, r d e no te s t he r e f le cted wa v e a nd f t he i nc id e n t fo r ced wa ve.

2.9 Proof of no power flow in a nearfield

I n t hi s se ct io n t h e co mp le x p ar ti cl e vel o ci t y i s c alc u lat ed . T h e n i t i s p r o ved t ha t t h er e

is no p o we r flo w i n a ne ar f ie ld .

W he n to t al i nt er na l r e f l ect io n o cc u r s , a no n -p r o p ag at i n g nea r fie ld is p r o d uc ed i n t he

seco nd med i u m. T h er e i s no p o wer f lo w wi t h a ne ar fi eld . T hi s is p r o ved b elo w fo r a

−=

ρ 0

tr a n s mi tt ed nea r fie ld wa ve. Co ns id er

∂ p ∂ x

∂ u ∂ t

( 2 .8 1)

−

ω t j

T he so u nd p r e s s ur e i n a ne ar fi eld i s gi v e n b y

xk tx e

ep t

( 2 .8 2 )

−

∂ p ∂ x

∂ u ∂ t

1 ρ 0

−

xk tx

( 2 .8 3)

−ω t j e

ep t

− −

∂ u ∂ t

k tx ρ 0

( 2 .8 4)

Eq u at io n ( 2 .1 7 9 ) i s i n te gr a ted wi t h r e sp ect to t to gi ve t he co mp le x p a r t icl e ve lo ci t y

−

ω t j

Eq u at io n ( 2 .1 8 1 ) b eco m es

xk tx e

ep t

− jk tx ωρ 0

( 2 .8 5)

−

ω t j

T aki n g t he co mp l e x co nj u ga te o f t he p ar t ic le v el o cit y g i ve s

xk tx e

ep t

+ jk tx ωρ 0

( 2 .8 6)

T he i n te n si t y i s g i ve n b y

I =

Re( up .

( 2 .8 7 )

−

ω t j

xk tx

Eq u at io n ( 2 .1 7 6 ) a nd ( 2 . 1 8 4 ) g i ve s

up .

(

).(

ep t

+ jk tx ωρ 0

( 2 .8 8)

−

xk tx

Af te r so me si mp li f ic at io n t hi s b e co me s

up .

2 ep t

+ jk tx ωρ 0

( 2 .8 9)

T he r ea l p a r t o f eq ua tio n ( 2 .1 8 7 ) i s z er o b e ca u s e it i s p ur e l y i ma g i nar y. He nc e

Re(

= up 0).

( 2 .9 0 )

T hu s t her e i s no p o wer f lo w fo r a ne ar fi eld .

2.10 Derivation of transmitted and reflected pressures for

oblique incidence.

p t

θ r

θ t

θ f

p i

p r F i gu r e 2 . 2 : A fi gu r e s h o w i n g t h e c o- o r d i n a t e fi g u r e s u s e d i n t h i s s e c t i o n .

I n t hi s se ct io n t h e s i ne s and co si ne s o f t he tr a n s mi t ted a n g le , t he r e f lec t ed a n gl e a nd

th e t r a n s mi tt ed a n g le ar e cal c ula ted . T he tr a n s mi tted wa v e n u mb er i n t he x a nd y

d ir ec tio n i s ca lc u la te d . T he r e f lec ted wa v e n u m b er i n t he x a nd y d ir ect i o n is

calc u la te d . T he fo r ced wa v e n u mb er i n t h e x a n d y d ir ect io n is ca lc ul at ed . T he

tr a n s mi tt ed p r e s s ur e i s c alc u lat ed . Fi n al l y t h e r e f lec ted p r es s ur e i s c al c u lat ed .

(

)0,

x k ,

T he p l a ne wa ve n u mb er k= lie s i n t he x y p la ne and ma ke s a n a n g le o f θ wi t h

th e x - a xi s. T he p o si tio n var iab le is x =(x , y, z) . T he ti me is t a nd ω is t he an g u lar

ve lo c it y. Θ is t he a n gl e b et we e n k a nd x. T he a x es h a ve to b e r o ta ted so th at t her e i s

no co mp o ne nt o f wa v e n u mb e r i n t he z - a xi s d i r e ctio n. T he p la n e aco u s ti c wa v e var ie s

−

ω ( t

cos

θ )

ω (

− xkt ).

−

ω ( t

(

+ ykxk

= e ,))

( 2 .9 1)

T he co mp o n e nt o f t he f o r ced wa v e n u mb er i n t he y a xi s d ir e ct io n i s

θ sin f

( 2 .9 2 )

wh er e k f is t he ma g ni t ud e o f t h e fo r c ed wa v e n u mb e r a nd θ f is t he i nc id e nt a n gl e. I n t he

y a xi s d ir ec tio n t he fo r c ed i nc id e n t ( f) , r e f lec ted ( r) , a nd t r a n s mi tt ed wa ve n u mb er s

mu s t al l b e t he s a me b ec au se o f t h e co n ti n u it y o f t he a co u st ic p r e s s ur e a n d p ar t icl e

ve lo c it y

sin

θ f

θ r

θ t

( 2 .9 3 )

B eca u se t he r e fle ct ed wav e i s fr ee l y p r o p a ga ti n g , t he r e fl ect ed wa v e n u mb e r i s eq u al to

th e wa ve n u mb er i n med iu m o ne , wh er e t h e wa v e p ha se sp e ed i s c 1

ω .

k 1

c 1

( 2 .9 4)

B eca u se t he tr a n s mi tt ed wa v e i s fr e el y p r o p a gat i n g , t he tr a n s mi t ted wa ve n u mb er ( k t)

is eq ua l to t he wa v e n u mb e r ( k 2) i n t he s ec o nd med i u m wh er e t h e wa v e p ha se sp eed is

c 2

( 2 .9 5)

th e co n ti n u it y o f aco u s ti c p r e ss u r e a nd p ar t ic le v elo c it y i s gi v e n b y

sin

θ f

θ r

θ t

f ck 1 1

c 1

c 2

( 2 .9 6)

T he wa v e p ha s e sp eed s i n me d i u ms o n e a nd t wo ar e gi v e n b y

ω ,

and

c 1

c 2

k 1

( 2 .9 7)

sin

θ t

θ f

k 1 ω

c 2 c 1

k 1

f kk 1

So l v i n g fo r si n θ t g i ve s

sin

θ f

( 2 .9 8)

So l v i n g fo r si n θ r g i ve s

sin

θ = r

θ f

k 1

( 2 .9 9)

Fr o m eq u at io n ( 2 .8 0) , t h e co mp o ne n t s o f t h e fo r ced i n cid e n ce, r e fl ec ted and

cos

pk f

θ f

tr a n s mi tt ed co mp le x p ar tic le v elo c it ie s ( i n t h e d ir ec tio n o f t he x -a x i s) ar e g i ve n b y

xpk f Zk 11

f Zk 11

( 2 .1 0 0)

θ r

and

cos Z

pk rx Zk 1

( 2 .1 0 1)

p t

θ t

and

u tx

cos Z

pk tx t Zk 2

( 2 .1 0 2)

T he fo r c ed , r e f le cted a n d tr a n s mi tt ed p r e s s ur e s ar e gi v e n b y

p r

.t p

( 2 .1 0 3 )

T he co mp o n e nt s o f t he f o r ced , r e f le ct ed a nd tr a n s mi tt ed co mp le x p ar t ic l e v elo ci tie s i n

th e d i r ec tio n o f t he x -a x is ar e r el ated b y

−

cos

θ f

θ r

u t

θ t

( 2 .1 0 4 )

b eca u se t he aco u st ic p a r tic le v elo c it y i s co n ti n u o u s a t t h e j u n ct io n o f t h e t wo med ia

cos

pk f

θ f

θ r

p t

p r

θ t

( x=0 ) . Eq u at io n ( 2 .1 0 4) ca n b e wr i tt e n u si n g eq u atio n s ( 2 .1 0 0) , ( 2 .1 0 1) , ( 2 .1 0 2) a s

−

cos Z

f Zk 11

cos Z 1

( 2 .1 0 5)

Fr o m t he ma t he mat ica l i d en ti t y,

cos

sin

( 2 .1 0 6 )

T her e fo llo ws

−

cos

sin

θ f

( 2 .1 0 7 )

Fr o m eq u at io n ( 2 .9 9) ,

−

cos

sin

θ r

θ f

2 f 2 k 1

( 2 .1 0 8)

−

cos

sin

θ t

θ f

2 f 2 2

Fr o m eq u at io n s ( 2 .1 0 7) and ( 2 .9 8) co sθ t i s fo u n d to b e

−

sin

θ f

2 k 1 2 k 2

2 f 2 k 1

( 2 .1 0 9)

T he wa v e n u mb er t h at i s tr a n s mi tt ed t h r o u g h t h e j u nct io n a t a p ar tic u lar an g le to t h e x

−

sin

( k

)

2 2

2 f

θ f

)

(

and y a x i s i s

2 f

−

sin

θ f

k k

2 1 2 2

2 1

  ,  

   

( 2 .1 1 0)

and t he wa v e n u mb er t h at i s r e fl ec ted fr o m t he j u nc tio n at a p ar ti c ula r a n gl e to t he x

and y a x i s i s

−

sin

( k

)

2 k 1

2 f

θ f

)

(

−

sin

k 1

θ f

2 f 2 k 1

   

  ,  

( 2 .1 1 1)

and t he wa v e n u mb er t h at i s fo r ced o n to t he j u n ctio n o f t h e t wo med ia a t a p a r t ic ul ar

−

sin

( k

)

2 f

θ f

)

(

an g le i s

−

sin

θ f

k 1

  . 

  

( 2 .1 1 2)

T he z -a xi s co mp o ne n t h as b ee n o mi t ted i n t h e a b o ve eq ua tio n s b e ca u se it is al wa ys

zer o . B e ca u se t he aco u st ic p r e s s ur e i s co n ti n uo u s a t t h e j u n ct io n o f t he t wo med ia

( x=0 ) ,

p r

p .t

( 2 .1 1 3 )

cos

θ f

θ t

θ r

Eq u at io n ( 2 .1 0 5) i s

−

cos Z

Zk 1

rθ

( 2 .1 1 4)

cos Z

cos

θ f

θ t

θ r

a nd ad d i n g eq u a tio n ( 2 .1 1 4 ) gi v es Mu lt ip l yi n g eq u at io n ( 2 . 1 1 3) b y

p t

cos Z

Zk 11

  

  

  

  

( 2 .1 1 5)

cos

θ r

T hu s t he tr a n s mi tt ed p r e s s ur e a mp l it ud e i s

p t

θ f cos

2 cos

f Zk 11

Z 2 + θ t

Zk 1 Zk 1

θ r

  

  

tθ

( 2 .1 1 6)

cos Z

a nd s ub tr ac ti n g it fr o m eq ua tio n ( 2 .1 1 4 ) g i ve s Mu lt ip l yi n g eq u at io n ( 2 . 1 1 3) b y

cos

θ f

θ t

θ r

−

cos Z

Zk 11

  

  

  

  

( 2 .1 1 7)

−

cos

T hu s t he r e fle ct ed p r e s s ur e i s

θ f cos

cos

[ k f [ Zk 1

Z 2 θ r

Zk 11 Zk 11

] θ t ] θ t

( 2 .1 1 8)

2.11 Derivation of total and forced intensity for oblique

incidence. Different Media.

I n t hi s se ct io n t h e to ta l in te n s it y i n t h e fi r s t me d iu m i s cal c ul ated . T he tr a n s mi tt ed

in te n s it y i n t h e seco nd med i u m i s t h e n ca lc u lat ed . T he to t al a nd tr a n s m itt ed i n te n s it y

ar e fo u nd to b e t he sa me .

T he i n cid e n t i nte n s it y i n t he d ir ec tio n o f t h e x - a xi s i s gi v e n b y

Re(

(

+ xri )

pu . x

( 2 .1 1 9 )

is t he co mp l e x co nj u ga te o f wh er e u x i s t he p ar t icl e ve lo c it y i n t h e x d ir e ct i o n, a nd p

th e p r e s s ur e .

T hu s

−

)(

[ (Re

] , )

(

+ xri )

( 2 .1 2 0)

T he fo r c ed p ar tic le v elo cit y i s

u = f

f Zk 11

( 2 .1 2 1)

and t he f o r ced p ar t ic le v elo c it y i n t h e x d ir e ct io n is

u = fx

fx Zk 11

( 2 .1 2 2)

T he r e f lec ted p ar t ic le ve lo ci t y i n t he x d ir ec tio n is

u = rx

p r

k rx Zk 11

( 2 .1 2 3)

−

pp f

pp r

fx Zk 1

k rx Zk 1

fx Zk 1

  

  Re 

T hu s t he i nte n s it y i s

−

p p

k k

p p

kZ 11

  1  

   

   

   

   

    1Re    

  .  

−1

( 2 .1 2 4)

k k

p p

wil l no w b e co ns id er ed Eq u at io n ( 2 .1 2 4) ca n b e f ur t her s i mp l i fi ed . T he t er m

−

Zk fx

Zk tx 1

Fr o m eq u at io n ( 2 .1 1 8)

p r p

Zk rx

Zk tx 1

k 1 k 2 k 1 k

( 2 .1 2 5)

−

Zk rx

Zk tx 1

k k

k 1 k

He nce

k k

p r p

Zk rx

Zk tx 1

fx k 1 k

( 2 .1 2 6)

−

k k

p p

T hu s

)

Zk tx

k k

Zk rx

Zk tx

k 1 k k 1 k

( 2 .1 2 7)

p p

   

   

−

Zk fx

Zk 1 tx

Ev al ua ti n g gi v es

p r p

   

   

Zk rx

Zk tx 1

k 1 k 2 k 1 k

( 2 .1 2 8)

b eca u se k f x, Z 2, k 1, k 2, Z 1 ar e al l r e al n u mb er s a n d t he co mp le x co nj u g at e o f a r e al

n u mb e r i s t he r eal n u mb er .

2 f

T he co mp o n e nt o f t he r e f lec ted wa ve n u mb er i n th e x - a xi s d ir ec tio n

−

sin

θ f

2 1

sin

θ2 f

( 2 .1 2 9)

2 f 2 k 1

ca n b e gr e ate r th a n 1 , ma k i n g t he ca n b e r ea l o r i ma gi n ar y b eca u se

n u me r ic al v al u e u nd er t he sq u ar e r o o t si g n n e ga ti ve . I n t h i s ca s e t h e sq u ar e r o o t

n u mb e r i s a n i ma g i na r y n u mb e r .

2 f

T he co mp o n e nt o f t he tr an s mi t ted wa ve n u mb er in t he x -a x i s d ir ec tio n

−

sin

θ f

2 2

( 2 .1 3 0)

ca n b e r ea l o r i ma gi n ar y , b eca u se t he val u e u nd e r t he sq uar e r o o t s i g n c a n b e ne ga ti v e,

ma k i n g t he sq uar e r o o t an i ma g i nar y n u mb er .

( k

T hu s

p p

   

   

Zk rx

Zk tx

) Zk rx k k

( 2 .1 3 1)

He nce p ut ti n g eq uat io n s ( 2 .1 2 7) a nd ( 2 .1 3 1 ) to g et her g i ve s

−

p p

k k

p p

   

  1  

   

   

  1  

    Z

(

1)((

)

Zk tx

k k

k 1 k

Zk rx

Zk tx

k 1 k

( 2 .1 3 2)

)

Zk tx

k k

( k

(

+ xri )

kZ 11

Zk rx

Zk tx

Zk rx

Zk tx

) Zk rx k 1 k

k 1 k k 1 k

     

     

      

      

     

     

T her e fo r e t he to tal i n te n si t y i n t he fir st med i u m in t he x -a x i s d ir ec tio n is

kZ 2

( k

Zk rx

kZk tx 1

k 1 k

( 2 .1 3 3)

T he tr a ns mi tted i n te n si t y i n t h e seco nd med i u m in t he x -a x i s d ir ec tio n is g i ve n b y

Re(

I = tx

pu tx

( 2 .1 3 4 )

T he p ar tic le v elo ci t y tr a n s mi tt ed i n t he x d ir ec ti o n is gi ve n b y

u = tx

p t

k tx Zk 2

( 2 .1 3 5)

)

Re(

k tx kZ 2

T hu s

Re(

t kZ 2

( 2 .1 3 6)

) Zk rx

p t

Fr o m eq u at io n ( 2 .1 1 6) t he tr a n s mi t ted so u nd p r e s s ur e is

Zk tx 1

Zk rx

( k 1 k

( 2 .1 3 7)

T hu s t he mo d u l u s sq uar ed o f t h e tr a ns mi tted so u nd p r es s ur e i s

2 2

p t

Zk 1 tx

Zk rx

k 1 k

( 2 .1 3 8)

2 2

Re(

)

kZ 2

Zk rx

Zk tx

k k

T hu s t he tr a n s mi tt ed i n t en s it y ca n b e wr it te n a s

kZ 2

Re(

kZk 1

Zk rx

k k

( 2 .1 3 9)

T hi s i s t he sa me a s t he i nc id e n t i n te n si t y g i ve n b y eq uat io n ( 2 .1 3 3) .

2.12 Diffuse field incidence. Media the same. Analytical calculation of intensity for r less than or equal to 1. I n t hi s se ct io n t h e fo r c e d tr a n s mi tt ed i n te n s it y f o r d i f f u se f ie ld i n cid e n c e i s c alc u la ted

and t he n no r ma liz ed . T h e a ver a ge tr a n s mi tt ed i n te n si t y o ver t he ar ea o f a h e mi sp h er e

o f so lid a n gle 2 π is c a lc ul at ed . T h e a ver a ge fo r c ed tr a n s mi t ted i nt e ns it y is ca lc ul at ed

and t he n no r ma liz ed . T h e t hr e e i n te gr al s i n t h is eq u at io n ar e t h e n e va l ua ted u si n g

Gr ad s ht e yn a nd R yz h i k ( 1 9 8 0) .

T he d i f f u se fi eld i nc id e nc e ca s e ( wh e n t h e med i a ar e t he sa me) is no w c o n sid er ed . T he

let ter r i s d e f i ned to b e th e r a tio o f t h e i n cid e n t fo r ced wa ve n u mb er ( k f) , to t h e fr e el y

p r o p ag at i n g wa v e n u mb er ( k) .

k f= k

( 2 .1 4 0)

B eca u se t he me d ia ar e t he s a me

k 1

( 2 .1 4 1 )

1 Z=

( 2 .1 4 2 )

and

= t θθθ .

( 2 .1 4 3 )

Fr o m eq u at io n ( 2 .1 3 9) , th e fo r c ed t r a n s mi tt ed i nt e n si t y d ue to a fo r ced in cid e nt wa ve

cos

θ i

Re(

θ )

tforced

cos 2

cos k

cos

θ i

f k

Re(cos

θ ).

f Z

cos

( 2 .1 4 4)

Fr o m t he eq u al it y o f t he y a xi s co mp o ne n ts o f t h e wa v e n u mb er s

sin

θ ,

= θ k i

( 2 .1 4 5 )

and

sin

θ i

f k

( 2 .1 4 6)

T he co si n e f u nc tio n s c a n b e ea s il y se e n to b e

−

cos

sin

θ i

( 2 .1 4 7)

and

−

cos

sin

θ i

( 2 .1 4 8 )

−

cos

sin

θ i

T hu s t he fo r ced tr a n s mi t ted i n te n s it y is

−

Re(

sin

θ i

tforced

−

f Z

1(4

sin

)

θ i

≥

sin if

θ i

1 r

( 2 .1 4 9)

tforced

−

cos

sin

θ i

sin if

θ i

1 r

−

1(4

sin

)

θ i

 0      

( 2 .1 5 0)

T he tr a ns mi tted i n te n si t y o f a no r ma ll y i nc id e n t ( θ=0 ) f r ee l y p r o p a g at i n g wa ve ( r =1 )

wi l l b e us ed to no r ma liz e t he f o r ced tr a n s mi t ted in te n s it y.

tfree

=θ ( i

≥

sin if

θ i

1 r

( 2 .1 5 1)

−

cos

(

sin

)

θ i

tforced = θ ( i

tfree

sin if

θ i

1 r

−

1(4

sin

)

θ i

 0      

( 2 .1 5 2)

T he e le me n ta l str ip o f s o lid a n g le b et wee n θ i a n d θ i+d θ i i s g i ve n b y

sin

idθθπ=Ω∂

dθi

θi

F i gu r e 2 . 3 G r a p h i c a l d i a gr a m o f e q u a t i o n ( 2 . 2 5 8) .

( 2 .1 5 3 )

T hu s t he s u m o f t he tr a n s mi tt ed i n te n si t y o ve r a l l a n gl e s o f i n cid e nce i s

2/ I

sin)

(

θθθπ 2 i i

∫

( 2 .1 5 4)

T he ar e a o f a he mi sp h er e o f a so lid a n g le i s 2 π. Di v id e b y 2 π to o b ta i n t he a ver a ge

tr a n s mi tt ed i n te n si t y o v er t he so lid a n gl e.

π 2 I

sin)

(

θθθπ 2 i i

∫

tforced

π 2

( 2 .1 5 5)

Fr o m eq u at io n ( 2 .1 5 0) t he tr a n s mi t ted i nt e ns it y d ue to a fr ee l y p r o p a gat i n g i ncid e nt

wa v e i s

cos

tfree

θ i

f Z

( 2 .1 5 6)

T he a ver a ge v al ue o ver all a n gl es o f i n cid e n ce i s

(

)

cos

sin

tfree

θθθ i

∫

( 2 .1 5 7)

T he ca se wh e n t he r a tio , r , o f t he fo r ced i nc id e n t wa v e n u mb e r to t he f r e el y

p r o p ag at i n g wa v e n u mb er i s no w co n sid er ed . Fr o m eq uat io n s ( 2 .1 5 0) a n d ( 2 .1 5 5) , t h e

−

av er a ge o f t h e tr a ns mi tt ed i nt e n si t y d ue to a fo r ced i n cid e n t wa ve i s

(

cos

sin

)

θ i

sin

Average (

)

θθ d . i i

tforced

∫

f Z

−

sin

θ i

( 2 .1 5 8)

P ut

θ= cos i

( 2 .1 5 9 )

th e n

−=

sin θθd , i

( 2 .1 6 0 )

sin

−= 1

cos

−= 1

θ i

−

−= 1

)

θ i

( 2 .1 6 1 )

sin 2

−= 1

22 xr

( 2 .1 6 2)

T he i n te gr a l li mi t s c h a n ge to

cos(0) = 1,

( 2 .1 6 3 )

and

cos(

= .0)

( 2 .1 6 4)

S wa p p i n g t he i nte g r al s l i mi ts b ec a us e o f t he – si n θ co n ver ts t he mi n u s si g n to a p l u s

si g n. T he a ver a g e tr a n s mi t ted i nte n s it y i s no r m ali zed b y d i vid i n g b y t h e a ver a ge

−

)

(

)

∫

r 2

f Z

tforced I )

(

−

tfree

222 xr ) 22 xr

)

tr a n s mi tt ed i n te n si t y fo r a fr ee l y p r o p a gat i n g i nc id e nt wa v e.

−

(

)

dx .

∫

r 2

1 2

−

222 xr ) 22 xr

)

( 2 .1 6 5)

−

(

)

222 ) xr

Exp a nd i n g t he to p l i ne o f t he i nt e gr a nd g i ve s

−

)

1((

)

22 xr

( 2 .1 6 6)

tforced

T hu s

−

22 xr

dxx

)

dx .

∫

x 2

r 2

r 2 2

1 2

−

tfree

22 xr

)

( 2 .1 6 7)

T hes e t hr ee i nt e gr al s wi ll b e e va l ua ted i nd i v id u all y.

P ut

,2cx

( 2 .1 6 8 )

wh er e

−= 1(

( 2 .1 6 9 )

and

=∆

−

1(4

2 r )

( 2 .1 7 0 )

T he fir s t i n te gr al

∫

x 2

r 2

−

22 xr

)

( 2 .1 7 1)

Ca n b e e v al ua te u s i n g i n te gr a l n u mb er 2 .2 6 4 .3 o n p a ge 8 3 o f Gr ad s h te yn an d R yz hi k

−

)

(

)

(

∫

b 3 c 8

x c 2

b 3 2 c 4

dx R

( 1 9 8 0)

a c 2 −

)

−

22 xr

)

(

)

∫

r 2 r

1( 2

2 dxx R x r 2

dx R

( 2 .1 7 2)

−

22 xr

2 xr

2ln(

)

2ln(

)

∫

1 c

dx R

Fr o m t he i nte gr al 2 .2 6 1 in Gr ad s h te yn a nd R yz h i k ( 1 9 8 0) .

−

)

−

22 xr

2 xr

)

2ln(

)

r 3 r

x r 2

1( 2

( 2 .1 7 3)

∫

x 2

−

22 xr

)

P ut ti n g eq uat io n ( 2 .1 7 3 ) i nto eq uat io n ( 2 .1 7 2 ) gi ve s

−

)

−

)

22 xr

2ln(

)

22 xr

2 xr

x 2 r

1( 2

r 3 r

( 2 .1 7 4)

∫

−

x ) 2

( 1

2 xr

Ev al ua ti n g t h is i nt e gr a l b et we e n z er o a nd o ne.

)

ln(

− +

− r r

r r

1( 1(

)

1 2 r

  .)  

  1  

( 2 .1 7 5)

T he seco nd i nt e gr a l i s e as il y e val u at ed .

xdx

∫

x 2

1 2

  

1  = 

( 2 .1 7 6)

T he t h ir d i n te gr al i s e va lu at ed u si n g i nt e gr al n u mb e r 2.2 6 2 .1 fr o m Gr ad s ht e yn a nd

2 xr

2 )r-(1

∫

Rb ) c

+ 4

∆ c 8

dx R

R yz h i k ( 1 9 8 0) .

−

2 xr

)

1(4

)

r 2

∫

dx 2

r 2 r

1( 4

− 8

−

)

2 xr

( 2 .1 7 7)

−

2ln(

)

2 xr

∫

dx 2

1 r

−

2 xr

)

No w

)

−

)

2ln(

)

2 xr

1 2

`1 2

− r r

( 2 .1 7 8)

)

−

)

2ln(

)

2 r )2

∫

− r

1 2

)

T hu s

−

2ln(

))

1 2 − r r

1 2

−

)

ln(

)

ln(

− r

1 2

−

r −

 1  

 .)  

   

  1  

( 2 .1 7 9)

tforced

−

xdx

2 xr

)

x 2

∫

r 2

r 2 2

1 2

−

tfree

2 xr

)

(

ln(

)

ln(

) 2

− +

− r

r 2

1 r 2

1( 1(

r r )

r ++ 2

1 2

−

P ut ti n g t he t hr e e i n te gr a ls to ge t her g i ve s

)

   

  1  

)

−

ln(

− +

− r

1( 1(

r r )

r ++ 2

1 4

1( 1(

r r )

1 4

   

  1  

  1   + 2

( 2 .1 8 0)

2.13 Diffuse field inci dence. Media the same. r greater than or

equal to 1.

I n t hi s se ct io n eq ua tio n ( 2 .1 6 7) i s e v al ua ted a ga in b ut wi t h d i f f er e n t li m it s o n t he

−

in te gr al. Fr o m eq ua tio n ( 2 .1 5 0) , i f r i s gr e ater t ha n o r eq ua l to o ne, t he in te gr al o n l y

sin

=θ i

−

ne ed s to b e e va l ua te o v e r v al ue s o f x fo r wh i c h is l e ss t ha n o r eq ua l to

2 x <

T his 1 /r. T h u s t h e i n te gr a l o n l y need s to b e e va l uat ed fo r val u es wh e n

−

−>

/11

i mp l ie s t h at

− r

( 2 .1 8 1)

Average

−

tforced

(

)

22 )

T hu s t he no r ma l ized i n t egr al ( eq ua tio n ( 2 .1 6 7) ) b eco me s

dx .

r 2

1 1 ∫ 2

Average

−

tfree

2 xr 2 2 xr

1(4

)

− 1

( 2 .1 8 2)

−

)

+ dxx

2 xr

. dx

T hi s i n te gr a l c a n a g ai n b e sp li t i n to t hr ee i nt e gr al s

∫

x 2

1 2

r 2

−

)

2 xr

− 1

( 2 .1 8 3)

Gr ad s ht e yn a nd R yz h i k ( 1 9 8 0) i nt e gr a l n u mb e r 2 .2 6 4 .3 i s u sed to e v al u ate t he

−

)

−

2 xr

)

2ln(

)

fo llo wi n g t hr ee i n te gr a l s . T he f ir s t i nte gr al i s e q ua l to

r 3 r

x r

1( 2

 r  22 

  

2 − 1 r 2

( 2 .1 8 4)

)

1 4

− 1

− r r

r r

   

   

  .  

  1  

( 2 .1 8 5)

(

−

xdx

∫

− 1

− 2 r

x 2

1 2

r 2

  

  

  

  

− 1

Ag ai n , t he s eco nd i nt e gr al i s ea si l y e v al ua ted .

− 2

r −= 2

rr ( 2

− r r 2

1 r 2

( 2 .1 8 6)

Us i n g G r ad s hte yn a nd R yz h i k ( 1 9 8 0) i nt e gr a l n u mb e r 2 .2 6 2 .1, t he t hir d i nt e gr a l i s

)

−

2 xr

)

2ln(

)

− 1

− r r

1 2

  

  

eq u al to

r 1(2

)

( 1

− r r

1 4

−

   

   

   

 .   

( 2 .1 8 7)

)

1 4

− 1

1 ++ 4

1 4

− r r

r r

1 2 r

− r r

−

   

   

Co mb i ni n g t he t hr ee i n t egr al s g i ve s t h e a v er a ge d no r ma l ized i n te n si t y a s

   

   

2 += 4

1 += 2

+ r 2

1 2

1 2 r

1 r

  . 

  

( 2 .1 8 8)

2.14 Diffuse incidence when the media are different. In terms

of Z,k.

I n t hi s se ct io n t h e ca se wh e n t he t wo me d ia ha v e d i f fe r e nt val u es o f Z a nd k, i s

co n s id er ed . T he i nt e gr a t io n t ha t is p er fo r me d n u me r ic al l y to fi nd t he tr a n s mi tt ed

in te n s it y o v er a ll a n gle s is d er i v ed . Fr o m eq uat i o n ( 2 .1 3 3 )

kZ 2

Re(

)

kZk 1

Zk rx

k 1 k

Re(

kZ 2

k 1 k

Z Z

( 2 .1 8 9)

IZ 2

Re(

)

t 2

Z Z

k 1 k

T hu s

Re(

)

k tx k

2 2

Z Z

k 1 k

( 2 .1 9 0)

Fr o m eq u at io n s ( 2 .1 1 0) to ( 2 .1 1 2) ,

−

cos

sin

θ f

2 f

θ f

( 2 .1 9 1 )

−

cos

sin

θ r

2 k 1

2 f

θ f

( 2 .1 9 2)

and

−

cos

sin

θ t

2 2

2 f

θ f

( 2 .1 9 3 )

−

sin

2 2

θ f

−

sin

2 k 1

2 f

θ f

2 f

θ f

2 f k

   

  Re  

IZ 2

t 2

−

sin

2 k 1

2 2

2 f

θ f

2 f

θ f

Z Z

k 1 k

T hu s

−

cos

sin

2 k 1

θ f

2 f

θ f

2 f 2 2

  Re  

   

−

sin

2 k 1

2 2

2 f

θ f

2 f

θ f

Z Z

k 1 k

( 2 .1 9 4)

He nce ,

−

cos

sin

Re(

sin

)

θ f

2 f 2 k 1

k 1

2 f 2 2

IZ 2

t 2

−

sin

2 2

θ f

2 f

θ f

Z Z

1 k

2 f 2 k 1

( 2 .1 9 5)

No w

−

sin

2 2

2 f

θ f

1 k

2 f 2 k 1

2 k 1 2 k 2

( 2 .1 9 6)

−

sin

θ f

2 f 2 k 1

2 k 1 2 k 2

2 f 2 2

  Re  

   

  Re  

  ,  

( 2 .1 9 7)

and

k f = r . k 1

( 2 .1 9 8)

−

cos

sin

θ f

k 1

2 f 2 k 1

2 k 1 2 k 2

  Re  

   

IZ 2

T hu s

t 2

−

sin

θ f

Z Z

2 f 2 k 1

2 k 1 2 k 2

( 2 .1 9 9)

Using

and defining

gives

Z 1 Z

k 1 k

k 1

−

2 2 α

cos

sin

1Re

sin

θ f

)

(

IZ 2

( 2 .2 0 0)

t 2

−

+ βθ

2 2 α

sin

θ f

( 2 .2 0 1)

Us i n g

sin

−= 1

cos

θ f

( 2 .2 0 2 )

−

2 2 α

cos

1Re

( 1

)

( 1

)

θ f

)

(

IZ 2

t 2

−

2 2 α

cos

1 ( 1

( 1

cos ) + βθ

cos )

θ f

gi v e s

−

cos

1Re

cos

2 2 + αα r

θ f

)

(

−

+ βθ

cos

2 2 + αα r

θ f

( 2 .2 0 3)

IZ 2

sin

θθ d f f

t 2

∫

π 2 0

I n te gr a ti n g o v er a ll a n gl es o f i nc id e nc e gi v e s

π 2

−

2 2 αα + r

cos

1Re

cos

θ f

)

(

sin

θθ d f f

∫

−

βθ +

−

2 2 αα + r

cos

θ f

( 2 .2 0 4)

Us i n g

−=

cos

and

sin

θ f

θθ d f f

( 2 .2 0 5 )

and no ti n g t ha t

cos(

= 1)0

and

cos(

π ) 2

( 2 .2 0 6)

gi v e s, a f ter r e ver si n g t h e li mi ts i n o r d er to r e mo ve t he mi n u s si g n fr o m d x=- sin θ fd θ f

−

2 xr

2 2 αα + r

1Re

)

(

th e i n te gr al as

dx .

∫

−

2 xr

2 2 αα + r

( 2 .2 0 7)

T hi s i s t he a ct u al i nt e gr atio n t ha t i s p er fo r med n u me r ic al l y to fi nd t he r ela ti ve

tr a n s mi tt ed i n te n si t y o v er al l a n g le s .

2.15 Summary

As t he eq ua tio n s i n t h is ch ap te r ar e q ui te co mp l i cated , i n it ia ll y t he ca se o f no r mal

in cid e nce i s co n sid er ed . E v e n i n t he ca se wh er e th e t wo me d ia ei t her s id e o f t h e

j un ct io n ar e t h e sa me , t her e wi l l b e b o t h a tr a ns mi t ted a nd a r e fle ct ed wav e. T h is i s

b eca u se i n t h e fo r c ed wav e ca s e o nl y t he med i u m i n wh ic h t he wa v e i s i nc id e n t wi l l

b e d r i ve n b y t h e fo r c i n g f u n ct io n. Fir s t t h e o ne d i me n sio n al wa ve eq ua ti o n is d er i ved

and so l ved fo r t he ca se wh er e t he wa v e p r o p a g a tio n i n t h e i n cid e n t me d i u m i s fo r c ed .

T hi s i nc l ud e s t he i ner tia eq uat io n wh ic h i s ne ed ed fo r d er i vi n g t he r el at io n s h ip

b et we e n t he p r es s ur e a n d t he p a r t icl e ve lo ci t y.

Ne xt , t he ca se o f a no r m all y i nc id e n t fo r c ed wa v e ( wh e n t he med ia ar e t h e s a me) i s

co n s id er ed . T he tr a n s m itt ed a nd r e fl ec ted so u n d p r es s ur es ar e ca lc ul ate d . T he i ncid e nt

in te n s it y is t he n ca lc u la ted co n s id er i n g b o t h t he i ncid e nt wa v e a nd t he r ef le cted wa v e.

T he tr a ns mi tted i n te n si t y is al so cal c ul ated , a nd fo r f ut ur e us e t h e i n te n si t y o f t he

r ef le cted wa v e i f i t wa s tr a ve li n g alo n e i s a l so c alc u lat ed . T h e n t h e no r ma ll y i nc id e n t

cas e wh e n t h e t wo me d i a ar e d i f fer e nt i s co n sid er ed . T h e sa me ca lc u lat io n s a s

p r ev io u sl y p er fo r med ar e r ep e ated b ut t hi s ti me wi t h d i f f er e n t va l ue s fo r t he me d ia

p r o p er ti e s o n e it h er sid e o f t he j u nc tio n.

T he o b l iq ue i nc id e nc e c ase i s t h e n co n s id er ed . I t i s s ho wn t ha t t he t r a n s mi t ted

ne ar fi eld t ha t i s p r o d u ce d i n t h e ca s e o f to ta l i n t er n al r e fl ec tio n tr a n s mi t s no p o wer .

T he aco u s tic al p ar t ic le ve lo c it y fo r t he o b l iq ue in cid e nt ca se is ca lc u lat ed fo l lo wed b y

th e c alc u la tio n s o f tr a n s mi t ted a nd r e f lec ted p r e s s ur e s. T h e n t h e i n cid e n t a nd r e fle ct ed

in te n s it ie s ar e ca lc u lat e d . T he i ncid e nt i nt e ns it y i ncl ud e s b o t h t he i nc id e nt a nd t he

r ef le cted wa v e.

T he o b l iq ue i nc id e nt val ue s ar e u sed to c al c ula te t he d i f f u se f ield i nc id e nc e ca s e b y

av er a g i n g o ver al l p o ss i b le a n g le s o f i nc id e nc e. W he n t he med i a ar e t h e sa me t h is i s

d o ne b y a na l yt ica l i nte g r atio n . Sep ar at e i n te gr al s need to b e e va l ua ted fo r t he ca se

wh e n t he fo r ced wa ve n u mb e r i s le s s t h a n o r eq ua l to t he fr ee l y p r o p a ga ti n g wa ve

n u mb e r a nd fo r t he ca se wh e n t he fo r ced wa ve n u mb e r i s gr eat er t h a n o r eq ual to t he

fr e el y p r o p a ga ti n g wa v e n u mb er . Fo r t he d i f f us e f ie ld i n cid e n ce ca se wh en t he t wo

med ia ar e d i f fer e nt t he e q ua tio n s wh i c h ha ve to b e e va l uat ed n u me r ic al l y a r e d er i ved .

Chapter 3 The fluid media sound results

3.1 Introduction

I n t hi s c hap t er , t he t heo r eti ca l r e s ul t s o b t ai ned i n t he p r e vio u s c h ap te r a r e p r e se n ted a s

gr ap h s so t ha t t h e co n se q ue n ce s c a n b e ea si l y i n ter p r e ted . I n p ar tic u lar , it is s ho wn

th at t he s u m o f t h e i n te n si ti es p r o p a ga ted i n t h e d ir ec tio n no r mal to t he j u nc tio n b y t he

fo r ced i n cid e nt a nd fr e e l y p r o p a ga ti n g r e f lec ted wa ve s ar e no t eq u al to t he tr a n s mi t ted

in te n s it y. T hi s is b eca u s e t he cr o ss ter ms i n t he in te n s it y cal c ul atio n s d o no t va n is h

u nl e ss t he i nc id e n t wa v e is fr ee l y p r o p a g at i n g. I t is s ho wn t hat t he i nt e n s it y i ncid e nt

o n t he j u nct io n is eq ua l to t he tr a ns mi t ted i nt e n s it y i f t h e cr o s s t er ms ar e i ncl ud ed .

W her e i t is p o s s ib le, t he r es u lt s ar e gr ap hed i n d ecib e l s. T h i s i s no t a l wa ys p o ss ib l e

b eca u se so me o f t h e r e s ul t s ar e ze r o o r c h a n ge s ig n . T he r e s ul ts fo r t he si mp ler ca se o f

no r ma l i nc id e nc e ar e p r ese n ted f ir st, fo l lo we d b y t he o b liq ue i nc id e n ce cas e. T h e

r es u lt s fo r t he wh e n t he med ia o n ea c h sid e o f t h e j u nc tio n ar e t he sa me, ar e p r e se n ted

f ir s t, fo llo wed b y t he ca se wh e n t he me d i a ha ve d i f fer e n t i mp ed a n ce s. Fi na ll y f o r

o b liq ue i nc id e nc e t h e c a se wh e n t he i mp ed a nce s and wa v e n u mb er s ( wa v e sp eed s) ar e

d i f fer e n t is p r e se nt ed .

I t s ho uld b e no t ed t ha t a lt ho u g h t he me d ia ar e t h e s a me o n eac h s id e o f t he j u nc tio n, a

d is co nt i n ui t y s ti ll e xi s ts b eca u se t he fo r ci n g f u n ctio n wh i c h ge n er a te s t h e fo r ced

in cid e nce wa v e o p er ate s o n l y i n t he f ir st med i u m a nd no t i n t he se co nd med i u m, o n t he

o th er sid e o f t he j u nct io n.

3.2 Normal incidence, the same media

T he tr a ns mi tted i n te n si t y is

(

)( r

(

)

+ ri

 + k  

 = 

( 3 .1 )

fr o m eq ua tio n s ( 2 .3 8) a nd ( 2 .4 5 ) , wh e r e

Z ρ=

.0c

( 3 .2 )

P ut

k f= k

( 3 .3 )

Fo r a f o r ced i nc id e nt wa ve t he tr a n s mi t ted i nt e n si t y i s

(

)( r

(

)

+ ri

2  = 

 + 1  2 

( 3 .4 )

fo r a fr e el y p r o p a ga ti n g wa v e ( r =1 ) , t h e t r a n s mi t ted i n te n s it y is

).1(

)1()

(

+ ri

( 3 .5 )

T he l as t eq u al it y o cc ur s b eca u se t he me d i a ar e t h e s a me a nd he nc e t h er e i s no r e fle ct ed

wa v e wh e n t he i nc id e nt wa v e i s fr e el y p r o p a gat i n g. T he no r ma l ized tr a n s mi tt ed

)( r

)

+ ri

in te n s it y is

)1(

( I

)( rI t )1( I

 + 1  2 

2  = 

(

)

+ ri

( 3 .6 )

Eq u at io n ( 3 .6 ) i s s ho wn in F i g ur e 3 .1 i n d ec ib el s . T he fr eel y p r o p a ga ti n g ca se is gi ve n

b y r=1 . Fo r t he r i s l es s th a n o ne ca se, t he tr a n s mi t ted i nte n s it y i s l es s t ha n t he

in te n s it y fo r a fr eel y p r o p ag at i n g i n cid e n t wa ve ( r =1 ) .T h is fi g ur e s ho ws t h at

att e n uat io n o f a no r ma ll y i ncid e nt f o r ced wa v e i s a t mo st 6 d B . T h is i s mo r e t h a n t h e

att e n uat io n o f a no r ma ll y i ncid e nt f r ee l y p r o p a g ati n g wa v e. I t s h o uld a l s o b e no ted t ha t

th e gr ap h i s i nd ep e nd e n t o f f r eq ue n c y. I n t he r i s gr e ate r t h a n o ne ca se t he i nt e n si t y o f

th e t r a n s mi tt ed wa v e is gr e at er t h a n t h at fo r t he fr e el y p r o p a ga ti n g ca se wh e n r= 1 .

-1

-2

) B d ( ) ) 1 ( t I / ) r ( t I ( g 1 0 1

-3

-4

-5

-6 0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 2 1

F i gu r e 3 . 1 : Th e t r a n s m i t t e d i n t e n s i t y d u e t o a n o r m a l l y i n c i d e n t fo r c e d w a v e .

T he r a tio o f t he tr a n s mi t ted ( o r to t al) i nt e ns it y t o t he tr a n s mi tt ed i n te n s i t y wh e n t he

in cid e nce wa v e i s fr ee l y p r o p a gat i n g go e s to i n f i ni t y wh e n t he wa ve n u m b er r at io ( r)

go e s to i n f i ni t y b e ca u se p o wer t hat t he fo r c i n g f u nc tio n h a s to i nj ec t i n t o t he f ir st

med i u m i n o r d er to g e ne r ate t he fo r ced i nc id e n t wa v e o f co n st a nt a mp l it ud e go e s to

in f i n it y as t he wa v e n u mb e r r a tio go es to i n fi n i t y. I n o t her wo r d s, i t is eas ie st to

ge n er a te a wa v e wi t h t h e fr ee l y p r o p a g at i n g wa ve n u mb er a nd b eco me s har d er to

ge n er a te t he wa ve wh e n it s wa v e n u mb er i s v er y d i f fer e n t fr o m t he fr ee l y p r o p a gat i n g

wa v e n u mb er .

T he i n cid e n t i nte n s it y i n t he ab s e nce o f t he r e f le cted wa ve i s ,

)( rI i

f k

pk f f ωρ 0

( 3 .7 )

T he r e f lec ted i nt e ns it y i n t he p o si ti v e x d ir e ct io n i n t h e ab se n ce o f t he i nc id e nc e wa v e

is, fr o m eq u at io n ( 2 .5 2)

−=

)( r

  

 − 1  2 

( 3 .8 )

Fo r t h e fr e el y p r o p a ga ti n g wa ve ( r =1 )

)1(

( 3 .9 )

T he i n te n si ti es ar e no w no r ma l ized b y d i v id i n g t he m b y t h e i n te n si t y o f t he f r ee l y

p r o p ag at i n g i n cid e nt wa ve

= r .

)( rI i )1( I

( 3 .1 0)

Fr o m eq u at io n s ( 3 .8 ) a n d ( 3 .9 ) t h e no r ma li zed r ef le cted i n te n si t y i s

I r I

r )( )1(

 − 1 −=  2 

2  −  r −=  2  

  

( 3 .1 1)

Fr o m eq u at io n s ( 3 .4 ) a n d ( 3 .9 ) t h e no r ma li zed tr an s mi t ted i nt e ns it y i s

rI )( t I )1(

 + 1  2 

  

( 3 .1 2)

T hu s t he s u m o f t he i nc i d en t a nd r e fl ec ted i nte n si ti es ( wh e r e p o si ti v e d e no t e s e ner g y

r )(

−= r

rI )( i I

  

 − r  2 

I )1( 2

b ei n g p r o p a g at ed i n t he p o si ti v e x -a x i s d ir ec tio n ) i s gi v e n b y

i −

−

+ 4 + 6 r 4

( 3 .1 3)

No te t ha t eq u at io n ( 3 .1 3 ) i s no t eq ua l to t he tr a n s mi tt ed i n te n si t y

I I

r )( )1(

2 4

( 3 .1 4)

u nl e ss t he i nc id e n t wa v e is fr ee l y p r o p a g at i n g ( r =1 ) . T he se no r ma liz ed i nt e n si tie s ar e

s ho wn i n Fi g ur e 3 .2 .

T hu s t he p o wer p r o p a ga t ed b y t he i nc id e n t a nd r ef le cted wa v es ca n no t b e ca lc ul at ed

sep a r at el y u n le s s t h e i n c id e nt wa v e i s fr ee l y p r o p a g at i n g. T h e ac t ua l i n c id e nt p o wer o n

th e i n cid e nce s id e o f t he j u nct io n p la ne at x eq u a ls 0 ca n o n l y b e ca lc u lat ed a s I ( i + r ) ( r )

and ca n no t b e cal c ul ated as t he s u m o f I i ( r ) a nd I r (r ) u nle s s t he i ncid e nt wa v e i s fr e el y

p r o p ag at i n g.

2.5 Iiforced/Iifree

Irforced/Iifree 2

(Iiforced+Irforced)/Iifree

1.5 Itforced/Iifree

) 1 ( i I / ) r ( x I

0.5

-0.5 0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 2 1

F i gu r e 3 . 2 : Th e t r a n s m i t t e d , r e fl e c t e d , i n c i d e n t a n d s u m o f i n c i d e n t a n d r e f l e c t e d i n t e n s i t i e s

d u e t o a n o r m a l l y i n c i d e n t fo r c e d w a ve .

I n Fi g u r e 3 .2 I i f o r c e d /I i f r e e i s t h e no r ma l is ed i n cid en t wa ve i nt e ns it y ca lc u lat ed o n i t s

o wn , i n t he ab se nc e o f t he r e fl ec ted wa ve. I r f o r c e d/I i f r e e is t he no r ma li sed r ef le cted wa v e

in te n s it y p r o p a gat ed i n t he p o s it i ve x d ir ect io n. He nce i t i s al wa ys ne ga t iv e o r zer o . I f

th e se t wo ar e ad d ed to g et her to o b tai n ( I i f o r c e d +I r f o r c e d) /I i f r e e , i t i s no ted t ha t t h i s s u m

d oes no t eq u al I t f o r c e d /I i f r e e. T h i s s ho ws t h at t he i nt e n si t y i n t he f ir st med iu m wh i c h

mu s t b e eq u al to t he i nt en s it y i n t h e seco nd me d iu m, c a n no t b e ca lc u la t ed co r r e ct l y b y

co n s id er i n g t he fo r ced i nc id e n t a nd r e fl ect ed wa ve s ep ar a te l y.

T he r a tio s o f t he tr a n s m itt ed a nd r e fl ec ted p r e s s ur e to t he fo r ced i nc id e n t p r e s s ur e ar e

as fo llo ws .

Fr o m eq u at io n ( 2 .2 7)

p t p

1+ 2

( 3 .1 5)

Fr o m eq u at io n ( 2 .2 8)

p p

1− 2

( 3 .1 6)

T he s u m o f t he se t wo eq ua tio n s i s

+= 1

− 2

+ 2

p t p

( 3 .1 7)

T he tr a ns mi tted , r e f lec t ed a nd i n cid e nce p r e ss ur es ar e gr ap hed i n F i g ur e 3 .3 .

1.5

f

0.5

p / x p

pr/pf 0 pt/pf

pf/pf

-0.5 0 0.5 1.5 1 2

F i gu r e 3 . 3 : Th e n o r m a l i s e d t r a n s m i t t e d , r e f l e c t e d a n d i n c i d e n t s o u n d p r e s s u r e s a s a fu n c t i o n o f

t h e r a t i o r o f t h e f o r c e d i n c i d e n t w a v e n u mb e r t o t h e f r e e l y p r o p a g a t i n g w a v e n u mb e r .

Fi g ur e 3 .3 s ho ws t ha t t h e tr a n s mi t ted p r es s ur e i s le s s t h a n o r gr e ate r t h a n t he no r ma l l y

in cid e nt fo r ced p r es s ur e .

3.3 Normal incidence, different media.

Fr o m eq u at io n ( 2 .1 5 6) f o r a no r ma ll y i nc id e n t wav e wh e n t h e t wo med ia ar e d i f f er e n t

eq u at io ns ( 2 .6 7) a nd ( 2 . 7 2) g i ve s

r )(

(

rI )( t

= + I ( ri

)

z 1 +

(

)

z 1

( 3 .1 8)

T hu s

)1(

= + I ( ri

)

z 4 +

)

(

2 z 1

( 3 .1 9)

)( r

+ ri

)

and

)1(

)( rI t )1( I

( I

  

 + r  2 

+ ri

(

)

( 3 .2 0)

Eq u at io n ( 3 .2 0) i s t he s a me a s eq u at io n ( 3 .6 ) . T h u s f i g ur e 3 .1 al so ap p l ie s fo r no r mal

in cid e nce wh e n t he med i a ar e d i f fer e nt.

3.4 Oblique incidence, same media.

Fr o m eq u at io n ( 2 .1 5 6) , t he a ver a ge tr a n s mi tt ed i nt e n si t y fo r t he c as e o f a fr ee l y

p r o p ag at i n g d i f f u s e i n ci d en t f ield i s

)1(

f Z

( 3 .2 1)

wh er e p f is t he r ms so u nd p r es s ur e o f t h e i n cid en t d i f f u s e so u nd fi eld . T he va l ue o f

eq u at io n ( 3 .2 1) wi ll b e u sed to no r ma li ze t he o t her val u es .

Fr o m eq u at io n ( 2 .1 5 0) a nd ( 3 .2 1 )

−

θ )

(

cos

sin

−

sin 2 θ

θ ,( ) rI t )1( I

≥

sin

    0 

( 3 .2 2)

Fr o m eq u at io n ( 2 .1 8 0) a nd ( 2 .1 8 8 ) , t h e no r ma li z ed tr a n s mi t ted i nt e ns it y d ue to

d i f f us el y i nc id e n t fo r c e d so u nd wa ve s wh e n t he med ia ar e t he s a me i s g iv e n b y

+ +

< <

> >

≤ 1r if ≥ r if 1.

rI )( t )1( I

 r 1( 2/)  r 2/)/11( 

( 3 .2 3)

5 0 degrees 4.5 15 degrees 4 30 degrees 3.5 45 degrees 3 60 degrees

2.5 75 degrees

> ) 1 ( t I < / ) r ( t I

90 degrees 2

diffuse 1.5

0.5

0 0 0.5 1.5 1 2

F i gu r e 3 . 4 Th e n o r m a l i s e d t r a n s mi t t e d i n t e n s i t y d u e t o fo r c e d p l a n e w a ve s i n c i d e n t a t a n gl e s

o f i n c i d e n c e fr o m 0 t o 9 0 d e g r e e s i n 1 5 d e gr e e i n c r e m e n t s .

Eq u at io ns ( 3 .2 2) a nd ( 3 . 2 3) ar e gr ap hed i n fi g ur e 3 .4 . Fo r va l ue s o f r gr eate r t h a n o ne,

to ta l i n ter n al r e fl ect io n is s ee n to o cc ur fo r t h e l ar g er a n g le s. T h e z er o v al ue s o f

in te n s it y wh i c h o cc ur b e ca us e o f to ta l i nte r nal r e f lec tio n ar e no t gr ap h ed in fi g ur e 3 .4.

I t i s i nte r e st i n g to no te t ha t t h e c ur v es te nd to i n f i ni t y a s t h e va l ue o f r f o r wh i c h to ta l

in ter n al r e fl ec tio n o c c ur s i s ap p r o ac h ed fr o m t he d ir ec tio n o f r eq ua l s ze r o , e xcep t fo r

th e 9 0 d e g r ee i nc id e n t c ase . T he d i f f u se f ie ld c u r ve i s eq u al to 0 .5 at r= 0 a nd te nd s to

0 .5 as r te nd s to i n f i n it y . T h us fo r t h e d i f f u s e fi eld i n cid e nce ca se, t he f o r ced

tr a n s mi tt ed i n te n si t y is l es s t ha n o r eq ual to t he fr e el y p r o p a ga ti n g tr a n s mi t ted

in te n s it y b y at mo s t 3 d B . I t s ho uld b e no t ed t h at a ll t he c ur ve s a r e eq u al wh e n r eq u al s

zer o . T h i s i s b e ca u se t h e a n gl e o f i ncid e nce ca n no t r ea ll y b e d e f i ned wh en t he i nc id e n t

fo r ced wa ve n u mb er is e q ua l to zer o . I t s ho u ld b e n o ted t ha t t h e no r ma li zat io n u sed i n

f i g ur e 3 .4 i s d i f f er e n t fr o m t ha t u sed i n fi g ur e 3 . 1.

3.5 Oblique incidence, different impedances, same wave

numbers

Fr o m eq u at io n ( 2 .2 0 3) , t he n o r ma liz ed tr a n s mi tt ed i nt e n si t y, fo r a n o b l i q ue l y i n cid e nt

−

2 2 + αα r

cos

1Re

cos

)

(

p la ne wa ve wh e n t he me d ia ar e d i f f er e n t, is gi ve n b y

rIZ )( t 2 2

−

βθ

2 2 + αα r

cos

( 3 .2 4)

k k

Z Z

wh er e

I f t he wa ve n u mb er s o f f r eel y p r o p a ga ti n g wa ve s i n t h e t wo med ia ar e t h e s a me ( α =1 ) ,

th e n eq u at io n ( 3 .2 4) b ec o me s

−

cos

sin

1Re

sin

)

(

)( rIZ 2 t 2

−

sin

( 3 .2 5)

−

cos

)

(

≤

sin

if r

T hu s

)( rIZ 2 t 2

sin 2 θ

−

≥

sin

if r

     0 

( 3 .2 6)

T he r i g h t ha nd s id e o f e q ua tio n ( 3 .2 6) i s t he s a me a s t h e r i g h t sid e o f e q ua tio n ( 3 .2 2) .

≤

He nce a ver a gi n g o v er a d i f f us e f ield i nc id e nc e g iv e s

)( rIZ t 2

≥

2/ ) r

r if r if

( )  r 1  ( + /11 

( 3 .2 7)

rIZβ )( 2

T hu s

( 3 .2 8)

−

cos

)

(

≤

r if

sin

Di v id i n g eq u at io n ( 3 .2 6 ) b y ( 3 .2 1) gi v e s

1 .

sin 2 θ

−

)( rI t )1( I

≥

sin

r if

sin

     0 

( 3 .2 9)

≤

Di v id i n g eq u at io n ( 3 .2 7 ) b y ( 3 .2 8) gi v e s

≥

2/ ) r

r if r if

rI )( t )1( I

( )  1  ( + /11 

( 3 .3 0)

Eq u at io n ( 3 .2 9) i s t he s a me a s eq u at io n ( 3 .2 2) . E q ua tio n ( 3 .3 0) i s t he s a me a s eq u at io n

( 3 .2 3) . H e nc e f i g ur e 3 .4 al so ap p li es wh e n t he wav e n u mb er s o f t he t wo med ia ar e t he

sa me e ve n i f t he c har a ct er i st ic i mp ed a n ce s ar e d i f fer e n t.

3.6 Oblique incidence, different wave numbers

Fo r t h e c as e o f d i f fer e n t fr eel y p r o p a ga ti n g wa v e n u mb er s i n t he t wo me d ia, eq u at io n

( 3 .2 4) ap p l ie s. Ho we v er t hi s eq u at io n wi l l b e m ul tip li ed b y t wo b e fo r e gr ap h i n g it to

−

cos

1Re

cos

2 2 + αα r

)

(

gi v e

)( rIZ 2 t 2

−

+ βθ

cos

2 2 + αα r

( 3 .3 1)

T hi s mea n s t ha t t h e tr a n s mi tt ed i n te n si t y is no r ma li zed b y d i vid i n g it b y h al f t he

in te n s it y o f a p l a ne so u n d wa ve wi t h t he sa me so u nd p r es s ur e a s t h e i n ci d en t p l a ne

so u nd wa v e b u t p r o p a g a ti n g i n t h e seco nd med i u m. T hi s no r ma l iza tio n i s u sed b eca u s e

c a n no t b e ca lc u l ated a nal yt i ca ll y. T he fr eel y p r o p a ga ti n g wa ve n u mb e r s i n t he

t wo me d ia ar e d i f f er e nt , an d t h u s t he i nte gr at io n ne ed s to b e p er fo r med n u me r ic al l y i n

th i s c as e. T h e no r ma li za tio n is gi ve n b y eq ua tio n ( 3 .2 1 ) , e x cep t t ha t i n th i s c as e i t i s

ne ce s sar y to c ho o se wh i ch med i a ’ s i mp ed a nc e to u se b eca u s e t he y ar e no w d i f f er e n t.

2

T he no r ma li zat io n is s i mi la r to t ha t wh i c h ca n b e d ed uced fr o m eq u at io n ( 3 .2 8 ) , e x cep t

1 β+

th at t he f ac to r is no t u se d . T he no r ma l ized tr a n s mi t ted i nte n s it y i s p lo tt ed a s a

f u nc tio n o f t h e r a tio r o f t he fo r ced i nc id e nt wa ve n u mb er to t h e fr e el y p r o p ag at i n g

wa v e n u mb er i n t h e fir s t med i u m f o r a n g le s o f i n cid e nc e i n 1 5 d e gr ee i nc r e me nt s fr o m

0 to 9 0 d e gr ee s. T h e r a ti o o f t he f r ee l y p r o p a g at i n g wa ve n u mb er i n t h e f ir s t med i u m to

th at i n t he se co nd me d i u m ( α ) i s g i ve n t he va l ue s o f ½ o r 2 . T he α eq ua l s 1 c a se is

co v er ed i n t h e p r e v io u s sec tio n . T he r at io o f t he ch ar a cter i st ic i mp ed a n c e o f t h e fir s t

med i u m to t h at o f t h e s e co nd med i u m ( β) i s g i ve n t he va l ue s o f ½ , 1 a nd 2 . T he

r es u lt i n g gr ap h s ar e s ho wn i n f i g ur e 3 .5 t hr o u g h to fi g u r e 3 .1 0.

53

Fr o m eq u at io n ( 2 .2 0 7)

2

2

2

2

2

2

1

+

−

+

−

rx

r

2 xr

r

2 2 αα + r

x

1

Re(

1

)

2

=

dx .

rIZ )( t 2

2

∫

2

2

2

2

2

0

p

f

2 2 βαα

−

+

+

−

+

r

2 xr

r

x

r

1

1

( 3 .3 2)

Eq u at io n ( 3 .3 2) i s gr ap h ed i n f i g ur e 3 .1 1 a nd f i g ur e 3 .1 2 wh e r e it i s no r ma li zed b y

d iv id i n g i t b y whi c h i s t he tr a n s mi t ted i nt e n si t y d ue to a f r ee l y p r o p ag at i n g

in cid e nt d i f f u se f ie ld . T hi s i s t h e s a me no r ma liz atio n a s u sed i n eq ua tio n s ( 3 .3 0) a nd

( 3 .2 3) . T he o n l y d i f fer e nc e i s t ha t i n t hi s ca se t he i nt e gr a tio n s ne ed to b e p er fo r me d

n u me r ic al l y.

T he n u mer ica l i nte g r at i o n s we r e p er fo r med i n Mat lab u si n g t he “ q uad g k” q uad r a t ur e

f u nc tio n to p er fo r m t he in te gr at io ns . W he n to t al i nte r nal r e f lec tio n o cc u r s t he i n te gr al

− /(11

2) αr

54

in eq ua tio n ( 3 .3 2) ne ed s o n l y to b e e v al ua ted f r o m x eq ua l s to x eq u al s 1 .

10

9 15 0

8

45 30 7

6

y t i s n e t n

I

75 60 5

d e s i l

4 90

a m r o N

3

2

1

0 0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 2

F i gu r e 3 . 5 Th e t r a n s m i t t e d i n t e n s i t y d u e t o a p l a n e s o u n d w a v e i n c i d e n t a t a n g l e s r a n gi n g f r o m

0 t o 9 0 d e gr e e s i n 1 5 d e g r e e i n c r e m e n t s . Th e r e s u l t s a r e g r a p h e d a s a fu n c t i o n o f t h e r a t i o r o f

t h e fo r c e d i n c i d e n t w a v e n u mb e r t o t h e w a v e n u mb e r o f a f r e e l y p r o p a g a t i n g w a v e i n t h e fi r s t

m e d i u m. Th e r a t i o o f t h e w a v e n u mb e r o f a fr e e l y p r o p a g a t i n g w a v e i n t h e fi r s t m e d i u m t o t h a t

i n t h e s e c o n d me d i u m ( α ) i s ½ . Th e r a t i o o f t h e i mp e d a n c e o f t h e c h a r a c t e r i s t i c i mp e d a n c e o f

t h e fi r s t m e d i u m t o t h a t o f t h e s e c o n d me d i u m ( β ) i s ½ .

1 r

I n f i g ur e 3 .5 t h e lo cal m in i ma a nd ma x i ma ar e s ho wn i n t he r a n ge fr o m r i s eq ua l to 0

to 2 . T h er e is a mi n i mu m o f zer o at 9 0 d e gr e e c ur ve . T he 4 5 a nd 6 0 d e g r ee c ur v es ha ve

lo ca l ma x i mu ms . T h e 7 5 an d 9 0 d e gr ee c ur v e s h av e lo cal mi n i mu ms . T h e s lo p e s o f t he

55

0 , 1 5 a nd 3 0 d e gr ee c ur ve s i ncr e a se fr o m 0 to 2 .

8

0 7

15 6 30

5 45

y t i s n e t n

I

60 4

d e s i l

75 3

a m r o N

90 2

1

F i gu r e 3 . 6 Th e t r a n s m i t t e d i n t e n s i t y d u e t o a p l a n e s o u n d w a v e i n c i d e n t a t a n g l e s r a n g i n g f r o m

0 t o 9 0 d e gr e e s i n 1 5 d e g r e e i n c r e m e n t s . Th e r e s u l t s a r e gr a p h e d a s a fu n c t i o n o f t h e r a t i o r o f

t h e fo r c e d i n c i d e n t w a ve n u mb e r t o t h e w a v e n u mb e r o f a f r e e l y p r o p a g a t i n g w a v e i n t h e fi r s t

m e d i u m. Th e r a t i o o f t h e w a v e n u mb e r o f a fr e e l y p r o p a g a t i n g w a v e i n t h e fi r s t m e d i u m t o t h a t

i n t h e s e c o n d me d i u m ( α ) i s ½ . Th e r a t i o o f t h e i mp e d a n c e o f t h e c h a r a c t e r i s t i c i mp e d a n c e o f

t h e fi r s t m e d i u m t o t h a t o f t h e s e c o n d me d i u m ( β ) i s 1 .

0 0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 2 1 r

I n f i g ur e 3 .6 , 0 , 1 5 , a nd 3 0 d e gr ee c ur ve s i ncr e as e fr o m 0 to 2 , a n d t he 6 0 , 7 5 , a nd 9 0

56

d egr ee c ur ve s ha v e a lo c al mi n i ma .

2

1.8

1.6

1.4

1.2

y t i s n e t n

I

0 15 30 45 60 75 90 1

d e s i l

0.8

0.6

a m r o N

0.4

0.2

0 0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 2

F i gu r e 3 . 7 Th e t r a n s m i t t e d i n t e n s i t y d u e t o a p l a n e s o u n d w a v e i n c i d e n t a t a n g l e s r a n gi n g f r o m

0 t o 9 0 d e gr e e s i n 1 5 d e g r e e i n c r e m e n t s . Th e r e s u l t s a r e gr a p h e d a s a fu n c t i o n o f t h e r a t i o r o f

t h e fo r c e d i n c i d e n t w a ve n u mb e r t o t h e w a v e n u mb e r o f a f r e e l y p r o p a g a t i n g w a v e i n t h e fi r s t

m e d i u m. Th e r a t i o o f t h e w a v e n u mb e r o f a fr e e l y p r o p a g a t i n g w a v e i n t h e fi r s t m e d i u m t o t h a t

i n t h e s e c o n d me d i u m ( α ) i s ½ . Th e r a t i o o f t h e c h a r a c t e r i s t i c i m p e d a n c e o f t h e fi r s t m e d i u m t o

t h a t o f t h e s e c o n d m e d i u m ( β ) i s 2 .

1 r

I n f i g ur e 3 .7 , t h e 0 , 1 5 a nd 3 0 d e gr ee c ur ve s al l ha v e i n cr ea s i n g slo p e s. T he 4 5 , 6 0 , 7 5 ,

and 9 0 d e gr e e c ur v es al l ha v e a lo c al mi n i ma. T he 7 5 a nd 9 0 d e gr ee c ur ve s d i sp la y a

57

lo ca l ma x i ma b et wee n r =1 a nd r=2 . T he 9 0 d e gr ee c ur v es g o e s to zer o a t r =2 .

5

4.5

4

3.5

3 0 15 30 45 60 75 90

y t i s n e t n

I

2.5

d e s i l

2

a m r o N

1.5

1

0.5

0 0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 2

F i gu r e 3 . 8 Th e t r a n s m i t t e d i n t e n s i t y d u e t o a p l a n e s o u n d w a v e i n c i d e n t a t a n g l e s r a n gi n g f r o m

0 t o 9 0 d e gr e e s i n 1 5 d e g r e e i n c r e m e n t s . Th e r e s u l t s a r e gr a p h e d a s a fu n c t i o n o f t h e r a t i o r o f

t h e fo r c e d i n c i d e n t w a ve n u mb e r t o t h e w a v e n u mb e r o f a f r e e l y p r o p a g a t i n g w a v e i n t h e fi r s t

m e d i u m. Th e r a t i o o f t h e w a v e n u mb e r o f a fr e e l y p r o p a g a t i n g w a v e i n t h e fi r s t m e d i u m t o t h a t

i n t h e s e c o n d me d i u m ( α ) i s 2 . Th e r a t i o o f t h e i mp e d a n c e o f t h e c h a r a c t e r i s t i c i mp e d a n c e o f

t h e fi r s t m e d i u m t o t h a t o f t h e s e c o n d me d i u m ( β ) i s 1 / 2 .

1 r

I n f i g ur e 3 .8 , t h e 0 d e g r ee c ur v e d i sp l a ys a n i nc r ea si n g slo p e. T h e 1 5, 3 0 , 4 5 , 6 0 a nd

7 5 d e gr ee c ur ve s ha v e a lo ca l ma x i mu m a nd t h e n go to z er o . T h e 9 0 d e gr ee c ur v e s lo p e

d ecr ea se s to z er o . T h e 1 5 d e gr ee c ur ve h as i ts p ea k ar o u nd r =1 .7 . T he o th er c ur v e s a ll

ha v e lo wer v al u es o f r f o r t he ir p ea k . T he gr ea t er t he a n gl e, t he lo wer t he r v al u e fo r

58

wh ic h t he p ea k o cc ur s.

4

0 3.5 15 30 3 45 60 75 2.5 90

y t i s n e t n

I

2

d e s i l

1.5

a m r o N

1

0.5

0 0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 2

F i gu r e 3 . 9 Th e t r a n s m i t t e d i n t e n s i t y d u e t o a p l a n e s o u n d w a v e i n c i d e n t a t a n g l e s r a n gi n g f r o m

0 t o 9 0 d e gr e e s i n 1 5 d e g r e e i n c r e m e n t s . Th e r e s u l t s a r e g r a p h e d a s a fu n c t i o n o f t h e r a t i o r o f

t h e fo r c e d i n c i d e n t w a ve n u mb e r t o t h e w a v e n u mb e r o f a f r e e l y p r o p a g a t i n g w a v e i n t h e fi r s t

m e d i u m. Th e r a t i o o f t h e w a v e n u mb e r o f a fr e e l y p r o p a g a t i n g w a v e i n t h e fi r s t m e d i u m t o t h a t

i n t h e s e c o n d me d i u m ( α ) i s 2 . Th e r a t i o o f t h e i mp e d a n c e o f t h e c h a r a c t e r i s t i c i mp e d a n c e o f

t h e fi r s t m e d i u m t o t h a t o f t h e s e c o n d me d i u m ( β ) i s 1 .

1 r

Fi g ur e 3 .9 is ver y si mi la r to fi g u r e 3 .8 i n it s ap p ear a nc e. Ho we v er t h e p e ak s i n t h e 1 5 ,

3 0 , 4 5 , 6 0 a nd 7 5 d e gr e e cu r ve s ha v e a s ma ll er v al ue o f i n te n si t y. T h e 0 d egr ee c ur ve

d isp la ys a n i ncr ea si n g s l o pe. T he 1 5 , 3 0 , 4 5 , 6 0 and 7 5 d e gr e e c ur v es d i sp l a y a

ma x i mu m a nd t h e n t h e y go to ze r o a s r i n cr ea s es . T he 9 0 d e gr ee c ur ve d ecr ea se s to

zer o . T h e 1 5 d e gr ee c ur ve h a s i t s p ea k ar o u nd r =1 .7 8. T h e o t h er c ur v es ar e a ll ha ve

lo wer val u e s o f r fo r t he ir p ea k . T he g r ea ter t he an g le, t he lo we r t he r va lu e fo r wh i c h

59

th e p e a k o cc u r s.

2

1.8

1.6

1.4

1.2

y t i s n e t n

0 15 30 45 60 75 90

I

1

d e s i l

0.8

0.6

a m r o N

0.4

0.2

F i gu r e 3 . 1 0 Th e t r a n s m i t t e d i n t e n s i t y d u e t o a p l a n e s o u n d w a v e i n c i d e n t a t a n gl e s r a n gi n g

f r o m 0 t o 9 0 d e g r e e s i n 1 5 d e g r e e i n c r e m e n t s . Th e r e s u l t s a r e gr a p h e d a s a fu n c t i o n o f t h e

r a t i o r o f t h e fo r c e d i n c i d e n t wa v e n u mb e r t o t h e w a v e n u mb e r o f a fr e e l y p r o p a g a t i n g w a v e i n

t h e fi r s t m e d i u m. Th e r a t i o o f t h e w a ve n u mb e r o f a f r e e l y p r o p a g a t i n g w a v e i n t h e fi r s t

m e d i u m t o t h a t i n t h e s e c o n d m e d i u m ( α ) i s 2 . Th e r a t i o o f t h e i m p e d a n c e o f t h e c h a r a c t e r i s t i c

i mp e d a n c e o f t h e fi r s t m e d i u m t o t h a t o f t h e s e c o n d m e d i u m ( β ) i s 2 .

0 0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 2 1 r

Fi g ur e 3 .1 0 lo o ks ver y mu c h l i ke f i g ur e 3 .9 . Ag ai n t h e d i f f er e n ce is t ha t t he i nte n s it y

is mu c h lo we r . T he 0 d e gr e e c ur v e d i sp la ys a n i ncr ea si n g s lo p e. T he 1 5 , 3 0 , 4 5 , 6 0 a nd

7 5 d e gr ee c ur ve s d isp la y a ma x i mu m a nd t he n g o to ze r o . T he 9 0 d e gr ee cu r ve

d ecr ea se s to z er o . T h e 1 5 d e gr ee c ur ve h as i ts p ea k ar o u nd r =1 .8 2. T he o th er p e a ks al l

ha v e lo wer v al u es o f r f o r t he ir p ea k s. T he gr ea t er t he a n gl e, t he lo wer t he r v al u e fo r

60

wh ic h t he p ea k o cc ur s.

3

(0.5,0.5) 2.5 (0.5,1)

(0.5,2) 2

1.5

> ) 1 ( t I < / > ) r ( t I <

1

0.5

0 0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 2 1

F i gu r e 3 . 1 1 Th e t r a n s m i t t e d i n t e n s i t y d u e t o a fo r c e d d i f f u s e i n c i d e n t s o u n d f i e l d . Th e r e s u l t s

a r e g r a p h e d a s a fu n c t i o n o f t h e r a t i o r o f t h e fo r c e d i n c i d e n t w a v e n u mb e r t o t h e w a ve n u mb e r

o f a f r e e l y p r o p a g a t i n g w a v e i n t h e f i r s t m e d i u m . A l p h a i s e q u a l t o ½ . B e t a i s ½ , 1 o r 2 .

r

Fo r r=0 t he v al u es i n f i g ur e 3 .1 1 ar e c lo s e to 0 . 5 . T he y i ncr e as e a s r i nc r ea se s wi t h a

sl i g ht mi n i ma at r = 1 . Fo r r gr eat er t h a n 1 , t he v a lu e s ar e gr ea ter t ha n 1 a nd h a ve a

61

ma x i ma j u st b elo w r=2 .

3.5

(2,0.5) 3

(2,1) 2.5 (2,2) 2

1.5

> ) 1 ( t I < / > ) r ( t I <

1

0.5

0 0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 2 1

F i gu r e 3 . 1 2 Th e t r a n s m i t t e d i n t e n s i t y d u e t o a fo r c e d d i f f u s e i n c i d e n t s o u n d f i e l d . Th e r e s u l t s

a r e g r a p h e d a s a fu n c t i o n o f t h e r a t i o r o f t h e fo r c e d i n c i d e n t w a v e n u mb e r t o t h e w a ve n u mb e r

o f a f r e e l y p r o p a g a t i n g w a v e i n t h e f i r s t m e d i u m . A l p h a i s e q u a l t o 2 . B e t a i s ½ , 1 , o r 2 .

r

I n f i g ur e 3 .1 2 a ll t he c u r ve s h a ve a ma x i mu m a t ab o ut r = 0 .5 . T h e c ur ve s fo llo w

d i f fer e n t p a t h s b e lo w r = 0 .5 . T he li n es ar e al mo s t t he s a me fo r va l ue s o f r gr e ate r t h a n

0 .5 . T he v al ue s ar e gr ea ter t ha n 1 fo r r le s s t ha n 1 a nd l es s t ha n 1 fo r va lu e s o f r

gr e at er t h a n 1 .

Lo o k i n g b a c k a t t h e ear l ier gr ap h s, na me l y f i g ur e 3 .8 to f i g ur e 3 .1 0 , t he d ecr ea s i n g

va l ue s ab o v e r=0 .5 ar e d ue to to ta l i nte r nal r e fl ect io n o r so me o f t h e a n gl e s o f

in cid e nce. T he b e ha v io r i n fi g u r e 3 .1 2 i s q ua li ta ti ve l y t h e o p p o si te o f wha t hap p e ns

in f i g ur e 3 .1 1. I n f i g ur e 3 .4 , wh i c h is t he alp h a e q ua l s o ne ca se, t he d i f f u se f ie ld va l ue s

62

ha v e a p ea k o f 1 at r = 1 and ar e le s s t h a n o ne wh en r i s l es s t ha n o r gr ea t er t ha n 1 .

3.7 Summary

I n it ial l y t he c as e o f a no r mal l y i n cid e n t wa ve wi th t he s a me med i u m o n eit h er sid e o f t h e

j un ct io n i s co ns id er ed . No te t ha t t r a n s mi s sio n a nd r e fle ct io n o c c ur a t t h e j u nc tio n b eca u s e

th e med i u m i s d r i v e n b y t he fo r ci n g f u n ct io n o n o nl y o ne sid e o f t he j u n ctio n. F i g ur e s

3 .1 -3 .3 d i sp l a y a gr ap h o f i nt e ns it y v er s us t he f o r ced wa v e n u mb er d i v i d ed b y t he fr ee l y

p r o p ag at i n g wa v e n u mb er . Fi g ur e 3 .2 s ho ws t h e r at io o f t he sep ar a tel y calc u la ted

in cid e nt a nd r e f lec ted i n te n si tie s, a nd t h e i n te n s i t y car r ied b y t he co mb i n atio n o f t h e

in cid e nt a nd r e f lec ted wav es . T hi s s ho ws t ha t t h e i nc id e n t i n te n si t y c a n n o t b e c al c ula ted

b y sep ar at el y ca lc ul at i n g t he i nte n s it ie s o f t he f o r ced a nd r e fl ec ted wa v es. F i g ur e 3 .3

s ho ws t he v al ue s o f t he in cid e nt a nd r e f lec ted tr an s mi t ted p r es s ur es . T h en t he ca se o f a

no r ma l l y i n cid e n t wa ve wi t h d i f f er e n t med ia o n eit h er sid e o f t h e j u n ct i o n is co n sid er ed .

Ne xt , t he ca se o f a n o b li q ue i nc id e nc e wa v e i s c o n sid er ed wh e n t he m e d ia o n e it h er sid e

o f t he j u nc tio n ar e t he s a me. A gr ap h o f i n te n si t y v er s u s r is p r o d uced s ho wi n g t he

in cid e nt a n gle s o f 0 , 1 5 , 3 0 , 4 5 , 6 0 , 7 5 , a nd 9 0 d egr ee s. T h e o b l iq ue i nc i d en ce wa ve i s

ag ai n co ns id er ed b ut wi t h d i f fer e nt i mp ed a nce s o n eit h er sid e o f t he j u n ctio n. T he sa me

wa v e n u mb er s ar e o n e it her s id e o f t he j u nc tio n . Aga i n t he o b l iq ue i ncid en ce wa ve ca se i s

co n s id er ed b ut t hi s ti me wi t h d i f f er e n t wa v e n u mb e r s o n ei t her s id e o f t he j u nc tio n.

Fi g ur es 3 .5 -3 .1 2 a r e p r o d uc ed . T h e y s ho w t he n o r mal is ed i n te n si t y ver s u s r. T h e gr ap h s

d isp la y li n es s ho wi n g t h e i nc id e n t a n g le s o f 0 , 1 5 , 3 0 , 4 5 , 6 0 , 7 5 , a nd 9 0 d e gr ee s. T he

gr ap h s s ho w e ve r y p o ss i b le co mb i na tio n o f ( α = ( k 1 / k 2) ) eq ua l to 0 .5 , 1 o r 2 , a nd ( β =

63

( Z 1/ Z 2) ) eq ual to 0 .5 , 1 o r 2 .

Chapter 4 The transmission of bending waves between two panels at a pinned joint.

4.1 Introduction

I n t hi s c hap t er , t he b e nd in g wa v e i n te n s it y tr a n s mi t ted a t t he p i n ned li ne j u nct io n o f t wo

in f i n ite hal f p la te s, wh e n a fo r ced b e nd i n g wa v e is i nc id e n t o n t he li ne j u nc tio n, i s

calc u la ted . T he no r ma l l y i ncid e nt ca se i s co n sid er ed fi r s t, fo llo wed b y t he o b l iq u e

in cid e nt ca se. T he n t he cas e o f a d i f f u se i nc id e n t vib r at io n fi eld i s co n si d er ed . Fi n al l y,

th e c as e wh e n t h e i n cid e nt vib r at io n fi eld i s e xc i ted b y a d i f f u se aco u st ic f ie ld i s st ud ied .

4.2 The transmitted and reflected wave equations for normal

incidence with a freely propagating incident wave.

T wo ha l f i n f i n ite fla t p l ate s l yi n g i n t h e y =0 p l a ne, ar e r i g id l y co n nec te d at a p i n ned

j o int alo n g t he z - a xi s ( s ee fi g ur e 4 . 1 ) . A fr ee l y p r o p ag at i n g tr a n s ver se b end i n g wa ve i n

p lat e o ne mo v i n g i n t h e p o si ti v e x d ir ec tio n i s i n cid e n t no r ma ll y o n t h e z ax i s. Fo ur

th i n g s ne ed to b e fo u nd . W ha t a r e t he r a tio s o f t he a mp l it ud e s o f t h e p r o p ag at i n g

r ef le cted wa v e ( r) , t he n ear fi eld r e fl ect ed wa v e ( r j) , t he p r o p a ga ti n g tr a n s mi tt ed wa v e

( t) a nd t he n ear f ield tr a n s mi tt e d wa v e ( t j) to t he a mp l it ud e o f t h e fr e el y p r o p ag at i n g

in cid e nt wa ve . T he se fo ur u n k no wn s wi l l no w b e fo u nd u si n g mu c h s ub s tit u tio n ,

d i f fer e n ti at io n a nd s i mu lta n eo us eq ua tio n s.

T wo ha l f i n f i n ite p la te s j o ined r i g id l y at a j u n cti o n ar e b e i n g co n s id er ed . T he y ar e

r i gid l y co n ne ct ed a t a p i n j o i n t. T h e p l ate s ma y r o tat e ab o u t t h e p i n j o i n t . Ho we v er

th e y ar e co n st r ai n ed b y th e p i n to h a ve z er o tr a n s ver se v elo ci t y a t t h e j o in t ( t he y

64

ca n no t mo v e up a nd d o wn a t t he j o i n t) . T h e t wo p lat es ar e i n li ne wi t h e ac h o t h er .

P late 1 i s i n t h e ha l f i n f in it e p l a ne y=0 , a nd x i s le s s t h a n o r eq u al to z e r o. P lat e 2 is i n

th e ha l f i n fi n ite p la n e y =0 a nd x i s gr eat er t h a n o r eq ual to zer o . T he t wo p lat e s ar e

y

x

Plate 1

Plate 2

z

F i gu r e 4 . 1 P l a t e 1 a n d P l a t e 2 .

j o ined at x=0 a nd y =0 . P lea se see t he d ia g r a m b elo w.

t

I t i s a s s u me d t h at t he o s cil la tio n s o f t he p la te va r iab l es ha ve a n a n g u lar fr e q ue nc y o f

je ω . T his fac to r wi ll b e o mi tted

ω . T h u s t h eir v ar ia tio n wi t h t i me t i s p r o p o r t io n al to

+1v is

fr o m t he eq ua tio n s. A p l an e b e nd i n g wa v e wi t h t r an s v er s e ve lo ci t y a mp l i tud e

in cid e nt no r mal l y o n t he j u nct io n b e t we e n t he t wo p lat e s ( z -a xi s) fr o m p l ate 1 ( x is le s s

th a n o r eq ual to zer o ) .

T he r a tio s o f t he a mp l it ud e s o f t he p r o p a ga ti n g r ef le cted wa v e, t he ne ar f ield r e f lec ted

wa v e, t he p r o p a ga ti n g tr an s mi t ted wa ve a nd t he ne ar fi eld tr a n s mi t ted wa ve s to t h e

a mp li t ud e o f t h e fo r c ed in cid e nt wa ve s, ar e r, r j, t, t j r e sp e ct i vel y. Also k 1 a nd k 2 ar e

th e fr e el y b e nd i n g p r o p a ga ti n g wa v e n u mb er s o f th e t wo p lat e s. T he to tal ve lo c it ie s v y 1

x

and v y 2 i n p lat e s 1 a nd 2 ar e g i ve n b y eq uat io n 6 .1 4 o f Cr e me r et a l. ( 2 0 0 5) as

-jk 1

xjk 1

xk 1

+

+

≤

v

re

x

x )(

,0

( e

)

y

er j

1

= + v 1

( 4 .1 )

−

−

jk

x

2

xk 2

and

=

+

≥

v

x )(

x

.0

( te

)

+

y

et j

2

v 1

65

( 4 .2 )

At t he j u nct io n o f t he p l ate s ( y =0 , x=0 ) , x is zer o i n eq ua tio n ( 4 .1 ) a nd ( 4 .2 ) a nd t h e

ve lo c it ie s o f t he t wo p la te s b eco me

≤

v

++ r

r

x

)0(

1(

)

0,

y

j

1

= + v 1

( 4 .3 )

and

+

≥

v

t

x

)0(

t (

)

.0

2

= + v 1

y

j

( 4 .4 )

B eca u se t he b o u nd a r y i s p i n ned t he ve lo c it ie s ( v y 1( 0 ) a nd v y 2( 0 ) ) ar e eq u al to zer o at

x=0 . T h u s

1

++ r

r

=+= t

t

.0

j

j

( 4 .5 )

Fr o m Cr e mer e t a l. ( 2 0 0 5) eq uat io n 3 .6 9

=

w

,

z

∂ v y ∂ x

( 4 .6 )

wh er e w z i s t he a n g ula r ve lo c it y o f t he p la te ab o ut t he z a xi s. B y ap p l yi n g e q uat io n

−

xjk 1

xjk 1

xk 1

( 4 .6 ) to eq uat io n s ( 4 .1 ) and ( 4 .2 ) we ha v e t h e fo llo wi n g eq u at io n s

=

−

+

+

≤

w

(

jk

re

)

x

0,

z

v + 1

ejk 1

1

erk j 1

( 4 .7 )

−

−

jk

x

2

xk 2

and

=

−

−

≥

(

)

0.

w

jk

te

x

z

v + 1

2

etk j 2

( 4 .8 )

At x=0 , eq uat io n s ( 4 .7 ) and ( 4 .8 ) b e co me

−

+

+

(

),

w

jk

jk

r

z

= + v 1

1

1

rk 1

j

( 4 .9 )

and

−

−

w

(

jk

t

).

z

= + v 1

2

tk 2

j

( 4 .1 0 )

B eca u se t he t wo p la te s a r e r i g id l y co n nec ted at x =0 t h e a n g u lar ve lo c it y gi v e n b y

eq u at io ns ( 4 .9 ) a nd ( 4 .1 0 ) mu s t b e eq ual . T her e f o r e

−

+

+

−=

−

jk

t

.

jk 1

rjk 1

rk 1

j

2

tk 2

j

( 4 .1 1)

T he a n g u lar mo me n t p er u n it le n g t h ab o u t t h e z ax i s i s g i ve n b y Cr e mer et a l . ( 2 0 0 5)

66

eq u at io n 3 .7 7

∂

z

−=

B

.

M ∂ t

∂ w z ∂ x

var ie s wi t h t i m e t a s e j ω t, eq ua tio n ( 4 .1 2 ) g i ve s t he fo llo wi n g e q ua tio n f o r t h e

( 4 .1 2)

Si n ce M z

an g u lar mo me n t

−=

M

.

z

B ω j

∂ w z ∂ x

( 4 .1 3)

.

−=

M

z

B ω j

∂ w z ∂ x

T he p l ate 1 mo me n t is g iv e n b y

−

xjk 1

xjk 1

xk 1

−=

−

−

+

≤

k

re

x

(

)

when

.0

2 ek 1

2 1

2 erk 1 j

B v + 1 ω j

( 4 .1 4)

−

xk 1

xjk 1

xjk 1

T hu s

+

−

≤

.0

x

= vM + 1 z

2 kB re 11 ω j

2 erkB 11 j ω j

2 ekB 11 ω j

   

  when  

( 4 .1 5)

T he p l ate 2 mo me n t is g iv e n b y

−

−

jk

x

2

xk 2

−=

−=

−

+

≤

(

) w

hen

.0

M

x

2 tek 2

z

2 etk 2 j

B ω j

∂ w z ∂ x

B v + 1 ω j

( 4 .1 6)

−

−

xk 2

jk

x

2

2

2

T hu s

−

≥

when

x

.0

= vM + 1 z

2 tekB 2 ω j

2 etkB 2 j ω j

   

   

( 4 .1 7)

At t he j u nct io n ( x=0 ) , t h e mo me n ts o f p l ate 1 a n d 2 , gi ve n b y eq ua tio n s ( 4 .1 5) a nd

j

j

( 4 .1 7) ar e eq ua l. H e nce

+

−

=

−

v + 1

v + 1

2 rkB 11 ω j

2 tkB 22 ω j

2 tkB 22 ω j

2 kB 11 ω j

2 rkB 11 ω j

   

  ,  

   

   

67

( 4 .1 8)

−

−=

2 2 − rkBrkB 11 11

2 + tkBtkB 22

2 22

.2 kB 11

j

j

( 4 .1 9 )

−−= 1(

r

)

t and

−= t

.

r

j

j

( 4 .2 0)

No w u si n g eq ua tio n ( 4 .2 0 ) , r j a nd t j ar e r ep la ced i n eq ua tio n ( 4 .1 9) to gi ve

+

−

−=

2

2

.

2 kB 11

2 rkB 11

2 tkB 2 2

2 kB 11

( 4 .2 1 )

2 11kB

Di v id e eq u at io n ( 4 .2 1) b y .

−

−=

+ 21

r

2

t

.1

2 kB 22 2 kB 11

( 4 .2 2)

2 2

De fi n e

=ψ

.2

kB 2 kB 11

( 4 .2 3)

.

Ca n cel i n g o u t t h e 2 ’ s gi ve s

.1−=

r ψ − t

( 4 .2 4 )

T he ab o ve eq ua tio n wi ll b e us ed a s o n e o f t h e si mu l ta n eo u s eq ua tio n s.

Re ar r a n g i n g eq ua tio n ( 4 .1 1 ) g i ve s

+

+

+

=

jk

t

rjk 1

rk 1

2

tk 2

jk .1

j

j

( 4 .2 5 )

S ub s ti t ut i n g eq u at io n ( 4 .2 0 ) i nto t he ab o ve eq ua tio n ( 4 .2 5 ) gi ve s

−

−

+

−

=

jk

t

jk

rjk 1

k 1

rk 1

2

tk 2

.1

( 4 .2 6 )

.

Di v id i n g b y k 1 gi v e s

−

+

−

j

r

j

+= j

(

)1

(

t )

.1

k 2 k 1

k 2 k 1

( 4 .2 7)

κ i s d e f i ned to b e k 2 /k 1

68

T hu s eq uat io n ( 4 .2 7) b e co me s

+− 1(

rj )

+−+ 1(

j

κ ) t

+= 1

j

.

( 4 .2 8 )

j+−1

Di v id i n g ( 4 .2 8 ) b y and mu l t ip l yi n g t h e d e no mi n ato r a nd t he n u mer a to r b y t he

j−−1

j

co mp le x co nj u g ate gi ve s

=

r

+κ t

.

+ 1 +− 1

j

−− 1 −− 1

j j

( 4 .2 9)

and

−=

r

+κ t

.j

( 4 .3 0 )

No w t her e ar e fo ur u n k n o wn s ( t, t j, r , r j) a nd fo u r eq uat io n s to so l ve t he m. T he y wi ll

b e so l ved u s i n g fo ur si m ul ta n eo us eq ua tio n s.

Eq u at io n ( 4 .3 0) mi n u s ( 4 .2 4) i s

t ψκ + t

.1+−= j

( 4 .3 1 )

.

So l v i n g fo r t, t he t r a n s m itt ed wa v e i s

=

,

t

− 1 j ψκ+

( 4 .3 2)

and t he tr a n s mi t ted n ear f ield i s

=

.

t j

+− 1 j ψκ+

( 4 .3 3)

Re ar r a n g i n g eq ua tio n ( 4 .3 0 ) g i ve s u s t he r e fle ct ed wa v e

r

κ−−= j .t

( 4 .3 4 )

S ub s ti t ut i n g t he va l ue o f t t h at h as j u st b ee n fo u nd i n eq ua tio n ( 4 .3 2) i nt o eq uat io n

−

( 4 .3 4) , gi v es

=

−

.

r

+ ψκ ( ) j + ψκ

− κ 1( ) j + ψκ

( 4 .3 5)

.

69

and

−=

.

r

+ ψκ j ψκ +

( 4 .3 6)

T hu s t he r e fle ct ed nea r f ield i s

=

.

rj

+− ψ 1( ) j + ψκ

( 4 .3 7)

4.3 Derivation of wave numbers for the obliquely incident

forced wave case.

I n t hi s se ct io n, t h e wa v e n u mb er eq u at io ns ( ( 6 .1 4 2 b ) , ( 6 .1 42 c) , ( 6 .1 4 2 d ) ) fr o m Cr e me r

et a l . ( 2 0 0 5) a r e d er i ved . E q uat io n ( 6 .1 4 2 a) i s al so p r e se n ted .

A s i n gl e fr eq u e nc y, f r ee l y p r o p a ga ti n g b e nd i n g wa v e i n a p la te sa ti s f ie s t he t wo

d i me n sio n al ho mo g e neo u s b e nd i n g wa v e eq uat io n. Fr o m eq ua tio n ( 3 .1 8 4 a) o f Cr e me r e t

a l. ( 2 0 0 5) , t hi s i s

4

∇

−

=

v

v

,0

y

y

2 ω m B

( 4 .3 8)

wh er e t he eq ua tio n h as b ee n d i f f er e n ti ated wi t h r esp ect to t he ti me t . v y is t he

tr a n s ver se v elo c it y o f t h e p la te i n t h e d i r ec tio n o f t he y a x is i n f i g ur e 4 . 1 . ω i s t he

an g u lar f r eq ue n c y, m i s th e ma s s p e r u ni t ar ea a nd B i s t h e b e nd i n g st i f f ne s s o f t he

p lat e.

As s u me t h at t her e i s a p la ne b e nd i n g wa ve i n t h e p la te. Ro ta te t he x a nd t he z a x e s so

th at t he x -a x is p o i n ts i n th e d i r ec tio n o f p r o p a ga tio n o f t he p la te a nd so t ha t t h e

p r o p er ti e s o f t h e p l a ne b end i n g wa ve d o no t v ar y i n t h e z -a xi s d ir e ct io n. T he n eq ua tio n

4

( 4 .3 8) b eco me s

−

=

v

v

.0

y

y

4

∂ ∂ x

2 ω m B

( 4 .3 9)

70

Let

−

jkx

=

,

xv )( y

ev y

( 4 .4 0 )

wh er e k i s t he wa v e n u m b er o f t h e fr e el y p r o p a g ati n g p la n e b e nd i n g wa v e i n t he p lat e.

2 ω

m

T he n p ut ti n g eq ua tio n ( 4 .4 0) i nto eq ua tio n ( 4 .3 9 ) g i ve s

4

=

k

.

B

( 4 .4 1)

T hu s eq uat io n ( 4 .3 8) ca n b e wr it te n as

=

4 vk

.0

4 −∇ v y

y

( 4 .4 2 )

2

2

Si n ce, i n Car te si a n co -o r d i nat es

2 ≡∇

+

2

,2

∂ ∂ x

∂ ∂ z

( 4 .4 3)

2

2

2

eq u at io n ( 4 .4 2) ca n b e wr i tte n a s

+

−

=

v

4 vk

,0

y

y

2

2

∂ ∂ x

∂ ∂ z

  

  

( 4 .4 4)

4

4

4

∂

∂

∂

o r

+

+

−

=

2

.0

4 vk

y

v y 4

2

v y 4

∂ x

v y 2 ∂∂ x z

∂ z

( 4 .4 5)

−

j

xk.

Eq u at io n ( 4 .4 0) b eco me

=

)x(

,

v

y

ev y

( 4 .4 6 )

wh er e,

k

),

( x kk= ,

z

( 4 .4 7 )

and

( zx=

,

).

x

( 4 .4 8 )

−

( + zkxkj z

x

Eq u at io n ( 4 .4 6) ca n b e wr i tte n a s

=

).

),( zxv y

ev y

( 4 .4 9 )

4

P ut ti n g eq uat io n ( 4 .4 9) i nto eq ua tio n ( 4 .4 5) gi v e s

+

+

−

=

k

2

k

k

,0

4 x

2 kk x

2 z

4 z

( 4 .5 0 )

71

o r

2

2

+

=

z

.4

k

k

( k

)

2 x

( 4 .5 1)

T her e fo r e

+

±=

k

k

k

,2

2 x

2 z

( 4 .5 2 )

T hu s

2

±±=

−

k

k

k

x

.2 z

( 4 .5 3)

T her e a r e fo ur p o s sib le va l ue s o f k x.

2

±=

−

k

k

k

x

,2 z

( 4 .5 4)

and

2

±=

+

k

kj

k

x

.2 z

( 4 .5 5)

Fo r a p la n e wa v e wh ic h is o b liq u el y i nc id e n t o n th e j u n ct io n o f t he t wo p lat es at a n

an g le o f θ i r el at i ve to t h e p o s it i ve d ir ec tio n o f t he p o s it i ve x -a xi s f r o m p lat e 1 , wi t h

fo r ced wa ve n u mb er k i,

=

k

k

.

zi

i

θ sin i

( 4 .5 6 )

Al so

=

cos

cos

,

k

k

xi

i

χθ = a

i

k 1

θ i

( 4 .5 7 )

wh er e

.

a =χ

ki 1k

( 4 .5 8)

I n o r d er to ma i nta i n co n ti n ui t y o f a n g ul ar ve lo ci t y a nd a n g u lar mo me n t u m at t he

j un ct io n b e t we e n t he t wo p lat e s, a ll r e fl ect ed a n d tr a n s mi tt ed wa v e s mu s t ha ve t he

−

jk

zi

sa me var iab i li t y i n t h e z -a x i s d ir ec tio n as t he i n cid e nc e wa v e, n a me l y

−

jk

i

θ sin i

=

=

.

( ) zE

e

e

( 4 .5 9 )

T hu s

=

=

=

sin

,

k

k

k

k

z

2

z

z

i

θ i

1

( 4 .6 0 )

72

wh er e 2 , 1 a nd i d e no te th e t r a n s mi tt ed , r e f le cte d a nd i nc id e n ce wa ve s.

Si n ce t h e r e f le cted wa v e s p r o p a g ate o r d e ca y i n th e ne g at i ve x d ir ect io n, t he

2

2

p r o p ag at i n g r e f le cted wav e ha s a k x va l ue g i ve n b y

−

=

+=

−

=

−

k

k

k

k

sin

k

1

sin

.

x

x 1

2 1

2 i

θ i

1

2 χ a

θ i

( 4 .6 1 )

No te t ha t t h i s i s o n l y a p r o p ag at i n g wa v e i f

>

sin

.

k 1

ik

θ i

( 4 .6 2 )

I f

<

sin

,

k 1

ik

θ i

( 4 .6 3 )

th e r e f le cted wa v e b e co me s a no n -p r o p a g at i n g n ear fi eld wa v e.

I f t he i nc id e nc e wa v e is a fr ee l y p r o p a g at i n g wa ve

=

=

k

k

.

i

k 1

r

( 4 .6 4 )

An d θ i wi l l b e d e n o ted b y θ 1.

Eq u at io n ( 4 .6 0) g i ve s

=

=

=

,

k

k

k

2

1

θk sin 1 1

z

z

z

( 4 .6 5 )

and eq ua tio n ( 4 .6 1 ) gi v e s

2

−

=

+=

−

+=

k

k

k

k

1

sin

cos

.

x

x 1

1

θ 1

1

θ 1

( 4 .6 6)

T hu s

−

=

=

.

jk

jk

jk

1

θ cos 1

x

1 x

( 4 .6 7 )

De no t i n g t h e a n g le o f r e f lec tio n r el at i ve to t h e d ir ec tio n o f t he p o s it i ve x- a xi s b y θ r

gi v e s

=

−=

cos

cos

,

θ 1

θ r

k x k 1

( 4 .6 8)

1

and

=

=

sin

sin

.

θ r

θ 1

k z k 1

( 4 .6 9)

He nce

−=r .1θπθ

73

( 4 .7 0 )

I n c i d e n t Wa v e

No t ice t ha t t h e a n g le is in r ad ia n s fo r t he c a se o f a fr e el y p r o p a gat i n g i n cid e n t wa v e .

x

θ i = θ 1 = π - θ r

θ i=θ 1

θ i=θ 1

R e fl e c t e d Wa v e

z

F i gu r e 4 . 2 Th e a n gl e o f r e fl e c t i o n e q u a l s t h e a n g l e o f i n c i d e n c e fo r a f r e e l y p r o p a g a t i n g w a v e . Th e p o s i t i v e y- a xi s p o i n t s ve r t i c a l l y o u t o f t h e p a ge .

θ r=π - θ 1

Fi g ur e 4 .2 s ho ws t ha t t h e r e fl ec ted a n g le d e fi n e d i n t h e u s ual wa y i s eq ua l to t he

in cid e nt a n gle i f t he i nc id e nt wa v e i s a fr e el y p r o p ag at i n g o ne.

T he r e f lec ted d eca yi n g ne ar fi eld so u nd f ie ld ha s

=

+

k

kj

k

x

2 1

,2 zi

( 4 .7 1 )

−=

k

jk

1

N

x

and

.

2

=

1

sin

,

k 1

θ i

2 χ+ a

( 4 .7 2)

a =χ

ki 1k

wh er e . I f t he i nc id e nt wa v e i s fr e el y p r o p a gat i n g

2

+

1

sin

,

1

= k 1

θ 1

kN

( 4 .7 3)

74

T hi s i s eq u at io n ( 6 .1 4 2 c ) o f C r e mer et a l. ( 2 0 0 5 ) .

=

=

−

k

k

k

k

x

x

2

2 2

2 zt

T he p r o p a ga ti n g tr a ns mi tted wa v e ha s

2

=

sin 2

,

k 1

2 χκ − a

θ i

=κ

( 4 .7 4)

k 2 k 1

. T hi s i s eq u atio n ( 6 .1 4 2 b ) o f Cr e me r e t a l. ( 2 0 05 ) . I f t h e i nc id e n t wa ve wh er e

is f r ee l y p r o p a g at i n g,

2

=

=

2 κ

−

,1

and

k

sin

.

χ a

= θθ 1

i

k x

2

1

θ 1

( 4 .7 5)

Si mi l ar l y to t he r e f le cte d wa ve , t h is tr a n s mi t ted wa ve i s o nl y a p r o p a gat in g wa v e i f

>

k

sin

.

2

ik

θ i

( 4 .7 6 )

I f

<

k

sin

,

2

ik

θ i

( 4 .7 7 )

to ta l i n ter n al r e fl ect io n o cc ur s a nd t he tr a n s mi tt ed wa v e b eco me s a no n - p r o p ag at i n g

ne ar fi eld wa v e.

I f t he i nc id e nt wa v e i s a fr eel y p r o p a ga ti n g wa v e a nd i f t h e tr a ns mi tted fr e el y

p r o p ag at i n g wa v e tr a ve l s a t a n a n gl e o f θ 2 , r e la t iv e to t he d ir ect io n o f t h e p o s it i ve x -

ax i s,

=

=

=

k

k

sin

k

k

sin

.

z

2

2

θ 2

z

1

1

θ 1

( 4 .7 8 )

T hi s i s S ne ll s la w a nd e q ua tio n ( 6 .1 4 2 a) o f Cr e me r e t a l. ( 2 0 0 5) . T h e tr an s mi t ted

d eca yi n g near f ie ld wa v e ha s

−=

+

k

kj

k

2 2

x

,2 2z

( 4 .7 9)

=

k

jk

N

2

x

and

2

=

k

sin 2

.

1

2 χκ + a

θ i

( 4 .8 0)

75

I f t he i nc id e nt wa v e i s f r eel y p r o p a ga ti n g,

2

2 κ +

sin

,

kN

2

= k 1

θ 1

( 4 .8 1)

T hi s i s eq u at io n ( 6 .1 4 2 d ) o f C r e mer et a l. ( 2 0 0 5 ) .

4.4 The angular velocity and the torsional moment for the

obliquely incident wave case.

−

jk

x

jk

x

k

x

1

1

1

x

x

N

T he tr a ns v er se b e nd i n g wa v e v elo c it ie s i n p l at e s 1 a nd 2 ar e

=

+

+

≤

,0

v

re

x

)

+

y

er j

1

( )( ezEv 1

( 4 .8 2 )

−

−

jk

x

k

x

x

N

2

2

and

=

+

v

te

( te

),

y

2

zEv )( + 1

( 4 .8 3 )

wh er e v 1 + i s t he r ms v el o cit y o f t h e fo r c ed i n cid en t wa ve. At x=0 , eq ua ti o n s ( 4 .8 2) a nd

( 4 .8 3) ar e zer o b eca u se o f t he p i n ni n g.

=

1

,0

++ r

jr

( 4 .8 4 )

and

t

.0=+ jt

( 4 .8 5 )

T he a n g u lar ve lo c it ie s a b o u t t he z - a xi s i n p l at es 1 a nd 2 ar e

=

,

w z

∂ v y ∂ x

( 4 .8 6)

as g i ve n i n eq u at io n ( 4 . 6) a nd i n eq ua tio n ( 3 .6 9 ) o f C r e mer et a l. ( 2 0 0 5 ) .

−

jk

x

jk

x

k

x

x

x

N

1

1

1

T hu s

=

+

+

≤

jk

e

jk

re

x

,0

( −

)

w z

xi

xr

erk jNr

1

zEv )( + 1

( 4 .8 7 )

−

−

jk

x

k

x

x

N

2

2

and

=

−

≥

jk

te

x

.0

( −

)

w z

x

2

zEv )( + 1

2

etk N j 2

( 4 .8 8 )

76

At x=0 , eq uat io n s ( 4 .8 7) an d ( 4 .8 8) ar e eq u al to eac h o t her b eca u se o f co nt i n ui t y.

T hu s

−

+

+

−=

−

jk

jk

jk

t

.

x 1

r 1 x

rk jN 1

x

2

tk 2 N

j

( 4 .8 9 )

T hi s i s eq u at io n ( 4 .1 1) , wi t h t he fir s t k 1 r ep lac e d wi t h k x 1 , t he s eco nd k 1 r ep l aced wi t h

k x 1, t he t hi r d k 1 r ep laced wi t h k N 1, t he f ir st k 2 r e p laced wi t h k x 2 a nd t he s eco nd k 2

r ep la ced wi t h k N 2.

T he a n g u lar ve lo c it y ab o ut t he x -a x i s i s

−=

w

.

x

∂ v y ∂ z

( 4 .9 0)

Cr e me r e t a l. ( 2 0 0 5) p o i nt s o u t i n sec tio n 6 .7 .2 . 2 t hat t h is gi ve s

vjk=w z

x

y

( 4 .9 1 )

Eq u at io n ( 4 .9 1) me a n s t ha t t h e a n g u lar v elo cit y ab o u t t h e x - a xi s is co mp let el y

d eter mi n ed b y t he val u e s o f t he o t her v ar iab le s.

Si n ce a ll t he k z ’ s ar e t he sa me o n b o t h sid e s o f t he z - a xi s j u nct io n a nd t h e v elo ci tie s v y

o n eac h s id e o f t he j u nc t io n ar e eq ua l a t t h e j u n c tio n, t he w x ’ s ar e a l so e q ua l o n b o t h

sid es o f t h e j u n ct io n ( wher e x =0 a nd y=0 ) .

2

∂

y

Eq u at io n ( 4 .1 3) o f t he p r ev io u s sec tio n ca n b e wr it te n a s

−=

M

z

v .2

B ω j

∂ x

( 4 .9 2)

Fo r a n o b liq u el y i nc id e n t wa v e t h i s ha s to be mo d i fied to ( se e Cr e me r e t a l. ( 2 0 0 5)

2

∂

eq u at io n ( 6 .1 4 4 b ) ) .

−=

+

µ

M

xz

y 2

v y 2

∂ v ∂ x

∂ z

 B   ω j 

  ,  

( 4 .9 3)

wh er e µi s P o i sso n ’ s r at io . T he t er m co n ta i ni n g it a p p ear s b ec a us e t h e P o i s s o n

exp a ns io n a nd co nt r ac ti o n i n t h e x - a xi s d ir ec ti o n d u e to wa v e p r o p a ga t io n i n t he z -a x i s

d ir ec tio n g e ner a te s a mo me n t ab o u t t he z -a x i s.

2

∂

Eq u at io n ( 4 .9 3) b eco me s

µ

−=

−

M

xz

2 vk z

y

v y 2

∂ x

  .  

 B   ω j 

77

( 4 .9 4 )

Fo r t h e p i n n ed j o i nt ca s e b ei n g co n sid er ed her e, wh e n x=0 , v y=0 . T h u s e q ua tio n ( 4 .9 4)

2

∂

r ed u ce s to

−=

−=

M

)0(

( ) 0

( ).0

xz

v y 2

∂ x

B ω j

∂ w z ∂ x

B ω j

( 4 .9 5)

2

1

B eca u se o f co nt i n ui t y o f M x z a t x =0 ,

=

( ) 0

( ).0

B 2

B 1

∂ w z ∂ x

∂ w z ∂ x

( 4 .9 6)

−

jk

x

jk

x

k

x

1

xi

x

1

N

1

Di f f er e nt ia ti n g eq ua tio n s ( 4 .8 7) a nd ( 4 .8 8) wi t h r esp ect to x gi ve s

=

−

+

≤

k

x

,0

( −

)

zEvB )( +

B 1

11

2 ek xi

2 re 1 x

2 erk 1 jN

∂ w z ∂ x

( 4 .9 7)

−

−

jk

t

k

x

2

x

N

2

2

and

=

+

≥

k

te

x

.0

( −

)

zEvB )( +

B 2

12

2 2 x

2 etk 2 N j

∂ w z ∂ x

( 4 .9 8)

Co mb i ni n g eq uat io n s ( 4 . 9 7) to ( 4 .9 8) g i ve s

−

+

=

+

k

( −

)

( −

).

B 1

2 xi

2 rk x 1

2 rk jN 1

B 2

2 tk x 2

tk 2 N

j

( 4 .9 9 )

2

) µ

) µ

) µ

B

B

B

Cr e me r e t a l. ( 2 0 0 5) eq u atio n ( 6 .1 4 4 a) g i ve s t he to r s io na l mo me n t p e r u n it le n gt h a s

=

−=

−=

M

k

k

.

xx

z

ω z z

( − 1 ω

( − 1 ω

( − 1 ω j

∂ v ∂∂ zx

∂ v y ∂ x

( 4 .1 0 0)

T hi s s ho ws t h at t he to r si o na l mo me n t p er u ni t le n gt h i s co mp le te l y d e ter mi n ed b y t he

va l ue s o f t he o t her v ar i a b le s.

Alt h o u g h no t u sed her e b eca u se t he j o i n t i s p i n n ed , i t i s wo r t h wh i le no ti n g t hat

eq u at io ns , ( 3 .1 48 e) a nd ( 6 .1 4 4 c) o f Cr e mer et a l . ( 2 0 0 5) gi v e t h e s h ear f o r ce p er u n it

zz

−=

−

Q

x

∂ M ∂ z

le n gt h ac ti n g o n a p l a ne no r ma l to t he x - a xi s as

∂

−

=

2 wk z

z

∂ M xz ∂ x 3 v y 3

∂ x

   

 B   ω j 

78

( 4 .1 0 1)

B eca u se o f t he ap p ear a n ce o f t he t hir d p ar t ia l d e r i vat i ve o f v y wi t h r esp e ct to x , Q x is

ind ep e nd e n t o f t he o t her va r iab le s.

Fr o m eq u at io n ( 6 .1 4 5 ) o f Cr e mer et a l. ( 2 0 0 5) , t he e xt er na l s up p o r t i n g o r r eac tio n

3

3

∂

∂

+

=

+

−

) µ

= QF

( 2

x

v y 3

2

∂ M xx ∂ z

∂ x

v y ∂∂ zx

 B   ω j 

   

fo r ce p er u ni t l e n gt h at a b o u nd ar y is

3

∂

=

−

−

( 2

) 2 µ wk z z

v y 3

∂ x

 B   ω j 

  .  

( 4 .1 0 2)

No te t ha t Cr e me r e t a l. ( 2 0 0 5) h a ve t he wr o n g si g n i n fr o n t o f t h e p ar ti al d er i va ti v e o f

th e to r s io na l mo me n t M x x a nd d i f fer e n ti at e i t wi t h r esp ect to x, r a t her t ha n z a s i n t heir

ear l ier ed i tio n s.

4.5 Derivation of the transmitted and reflected waves for

obliquely incident freely propagating waves.

I f t he i nc id e nt wa v e i s f r eel y p r o p a ga ti n g, k x i=k x 1, a nd θ i =θ 1. E q ua tio n s ( 4 .8 9) a nd

( 4 .9 9) b eco me

−

−=

−

jk

( 1

) +− r

jk

t

,

x 1

rk 1 jN

x

2

tk 2 N

j

( 4 .1 0 3 )

and

=

+

( 1

) ++ r

]

[ −

].

[ − kB 1

2 x 1

2 rk jN 1

B 2

2 tk x 2

2 tk N 2

j

( 4 .1 0 4 )

Eq u at io ns ( 4 .8 4) , ( 4 .8 5 ) , ( 4 .1 0 3) a nd ( 4 .1 0 4) a g r ee wi t h eq ua tio n ( 6 .1 4 6 ) o f C r e mer et

a l. ( 2 0 0 5) .

Us i n g eq u at io ns ( 4 .8 4) , ( 4 .8 5) i n eq uat io n s ( 4 .1 0 3) a nd ( 4 .1 0 4) g i ve s

−

+

−

=

+

(

(

jk

k

) r

jk

k

) t

jk

k

x 1

N

1

N

2

x

2

x 1

,1

N

( 4 .1 0 5 )

and

+

−

+

−=

+

k

k

k

) t

).

( kB 1

2 x 1

2 N

1

( ) kBr 2

2 x 2

2 N

2

( kB 1

2 x 1

2 N

1

79

( 4 .1 0 6)

2

2

Fr o m eq u at io n s ( 4 .6 6) a nd ( 4 .7 3 ) .

+

=

=

k

k

++ 1

sin

( cos

)

1

2 k 1

θ 1

θ 1

2 k .2 1

2 1 x

2 N

( 4 .1 0 7 )

2

2

Fr o m eq u at io n s ( 4 .7 5) a nd ( 4 .8 1 )

+

=

−

+

=

k

k

sin

sin

2 κ 2

.

( 2 κ

)

2

2 k 1

2 + κθ 1

θ 1

2 k 1

2 2 x

2 N

( 4 .1 0 8 )

I n ser ti n g eq ua tio n s ( 4 .1 0 7) a nd ( 4 .1 0 8) i nto eq u atio n ( 4 .1 0 6) gi v e s

r ψ − t

,1−=

( 4 .1 0 9 )

wh er e

=

ψ

.2

B 2 = κ 2 B 1

2 kB 22 kB 11

( 4 .1 1 0)

No te t ha t eq u at io n ( 4 .1 0 9) i s t h e s a me a s eq ua ti o n ( 4 .2 4 ) .

Us i n g eq u at io n ( 4 .1 0 9) t o r ep la ce r i n eq ua tio n ( 4 .1 0 5) g i ve s

=

t

2 jk ) −

).

( kj

+ ψ k

1 x ( k

+ ψ k

x

N

N

2

x 1

1

2

( 4 .1 1 1)

x

2

N

2

N

1

Al so

=

r

) ) .

( ψ j k ( kj

− k 1 x ψ + k

) ( + k ( ) − k

+ ψ k ψ + k

2

x 1

N

2

N

1

( 4 .1 1 2)

S ub s ti t ut i n g eq u at io n ( 4 .1 0 9) i nto eq ua tio n ( 4 .8 4 ) g i ve s

rj ψ−= .t

( 4 .1 1 3 )

Co mb i ni n g eq uat io n s ( 4 . 1 1 1) a nd ( 4 .1 1 3) g i ve s

=

r

j

2 )

).

− ψ + k

ψ kj x 1 ( − k

( kj

ψ + k

N

x

N

x 1

2

2

1

( 4 .1 1 4)

Co mb i ni n g eq uat io n s ( 4 . 8 5) a nd ( 4 .1 1 1) g i ve s

=

t

j

+

+

).

− ψ k

jk 2 1 x ( ) − k

( kj

ψ k

x 1

x

2

N

1

N

2

2

( 4 .1 1 5)

=

t

2

2

2

+

+

−

+

2 − κθ

2 κ

j

j cos − ψθψθ cos

θ 1 1

sin

sin

sin

(

1

1

θ 1

) 1 T hi s eq ua tio n i s t h e sa m e a s eq ua tio n ( 6 .1 4 7 a) o f Cr e mer et a l. ( 2 0 0 5 ) a p ar t fr o m a

( 4 .1 1 6)

80

d i f fer e n ce i n si g n wh i c h ar i se s fr o m t he fac t t h a t i n p la te 2 , t he p o s it i ve y -a xi s

d ir ec tio n o f t h is t he s is i s t r a n s fo r med to t he ne g ati v e x - a xi s d i r ec tio n i n Cr e me r e t a l.

( 2 0 0 5 ) b eca u se t h is t he s is co n sid er s a str ai g h t p i n ned j u nc ti o n wh i le Cr e me r e t a l.

2

2

2

−

−

+

+

2 κ

+ ψθ

2 + κθ

sin

1

sin

sin

j

1

( 2 0 0 5 ) co ns id er a r i g ht an g le p i n ned j u nc tio n.

=

.

r

j 2

2

2

+

+

−

θψ cos 1 2 − κ

1 2 − κθ

ψθψθ cos

sin

sin

1

)

(

j

θ 1 θ 1

1

1

1

+ sin Ag ai n t hi s i s t h e sa me a s eq u at io n ( 6 .1 4 7 b ) o f C r e mer e t a l. ( 2 0 0 5 ) .

−

( 4 .1 1 7)

=

.

rj

2

2

2

−

+

+

θψ 2 cos 1 +

j −

2 κ

2 − κθ

j

(

sin

ψθψθ cos

)

sin

sin

1

1

1

θ 1

1

−

2

( 4 .1 1 8)

=

.

t j

2

2

2

−

j −

+

+

2 κ

2 − κθ

j

(

sin

cos ψθψθ cos

)

θ 1 + 1

sin

sin

1

1

1

θ 1

( 4 .1 1 9)

4.6 Obliquely incident forced waves

S ub s ti t ut i n g eq u at io ns ( 4 .6 6) , ( 4 .7 3) , ( 4 .7 5 ) a nd ( 4 .8 1) i n to eq u atio n s ( 4 .1 1 1) , ( 4 .1 1 2) ,

and ( 4 .1 1 4) , g i ve s t h e fo llo wi n g eq u at io n s. P ut ti n g θ 1 eq ua l to zer o r ed u ces t he se

eq u at io ns to t he eq uat io n s ( 4 .3 2) , ( 4 .3 3) , ( 4 .3 6 ) and ( 4 .3 7) .

T he o b l iq ue l y i n cid e n t wa v e i s a s s u med to b e a fo r ced wa ve wi t h a wa v e n u mb er o f k i.

Us i n g eq u at io ns ( 4 .8 4) a nd ( 4 .8 5 ) i n eq ua tio n s ( 4 .8 9) g i ve s

−

+

−

=

+

(

(

jk

k

) r

jk

k

) t

jk

k

x 1

N

1

x

2

N

2

xi

,1

N

( 4 .1 2 0 )

and

+

−

+

−=

+

k

k

k

( ) kBr

) t

).

( kB 1

2 x 1

2 N

1

2

2 x 2

2 N

2

( kB 1

2 xi

2 N

1

( 4 .1 2 1 )

81

Fr o m eq u at io n s ( 4 .6 1) a nd ( 4 .7 2 )

2

2

+

=

−

+

+

=

k

k

k

sin

k

sin

1

2 k 1

2 k 1

2 k .2 1

2 1 x

2 N

2 i

θ i

2 i

θ i

( 4 .1 2 2 )

Fr o m eq u at io n s ( 4 .7 4) a nd ( 4 .8 0 )

+

=

−

+

+

=

k

k

k

k

k

k

2

k

2 κ= 2

,

2

2 2

2 2

2 2

2 k 1

2 2 x

2 N

2 2 z

2 2 z

=κ

( 4 .1 2 3 )

k 2 k 1

wh er e .

2

=

+

+

cos

sin

k

k

k

2 1

2 i

2 θ i

2 i

θ i

Fr o m eq u at io n s ( 4 .5 7) a nd ( 4 .7 2 )

2 xi =

2 1 N +

k

+ k ( 1

k ).

2 1

2 χ a

( 4 .1 2 4)

( +− 1

)

Us i n g eq u at io ns ( 4 .1 2 2) , ( 4 .1 2 3) a nd ( 4 .1 2 4) i n eq u at io n ( 4 .1 2 1) g i ve s

2 χ a

=

.

r

ψ − t

2

( 4 .1 2 5)

,1=aχ

an d eq uat io n ( 4 .1 2 5) a gr e e s wi t h W he n t he i nc id e n t wa v e is f r ee l y p r o p a g at i n g

eq u at io n ( 4 .1 0 9) .

(

jk

2/

)

1

1

S ub s ti t ut i n g eq u at io n ( 4 .1 2 5) i nto eq ua tio n ( 4 .1 2 0 ) g i ve s

=

t

.

1 x +

)( 1 −

+ xi ( ψ

jk )

− (

2 χ a )

k N jk

) ( + − k

k N jk

+ k

x 1

1

N

x

2

2

N

( 4 .1 2 6)

( ψ

(

2/

)

N

1

N

S ub s ti t ut i n g eq u at io n ( 4 .1 2 6) i nto eq ua tio n ( 4 .1 2 5 ) g i ve s

=

r

.

2 x +

jk )

2 χ a )

+ jk k xi ( ψ jk

) − − k

− k ( jk

)( + 1 2 − k

x 1

x

2

N

1

N

2

( 4 .1 2 7)

(

jk

2/

)

1

1

Fr o m eq u at io n s ( 4 .8 5) a nd ( 4 .1 2 6 )

−=

t

.

j

1 x +

)( 1 −

+ xi ( ψ

jk )

− (

2 χ a )

k N jk

) ( + − k

k N jk

+ k

x 1

1

N

x

2

2

N

( 4 .1 2 8)

−

+

−

) 1

x

Fr o m eq u at io n s ( 4 .8 4) a nd ( 4 .1 2 7 )

=

.

r

j

k N −

( ψ j k ( ψ

( )

jk ( +

+ xi jk

) k x 1 − k

− 2 jk

)( 2 χ 2 a ) k

N

x

N

x 1

1

2

2

( 4 .1 2 9)

S ub s ti t ut i n g eq u at io ns ( 4 .5 6) , ( 4 .6 1) , ( 4 .7 2 ) , ( 4 . 7 4) , a nd ( 4 .8 0) i nto eq u atio n s ( 4 .1 2 6) ,

82

( 4 .1 2 7) , ( 4 .1 2 8) , a nd ( 4 . 1 2 9) g i ve s

2

+

+

1

sin

θχ cos j i

a

2 χ a

θ i

)

(

2

2

+

−

−

+

+

1

sin

1

sin

j

) 2/1

2 χ a

θ i

2 χ a

2 χθ a i

)(

(

=

t

,

2

2

ψ

−

−

+

1

sin

1

sin

j

2 χ a

θ i

2 χ a

θ i

)

2

2

2

2

+

+

−

sin

sin

j

2 − χκ a

2 χκθ a

i

θ i

)

( (

( 4 .1 3 0)

2

+

+

cos

1

sin

j

2 χ a

θ i

)

2

2

2

2

and

−

−

+

+

sin

sin

j

) 2/1

2 χκ − a

2 χκθ a

i

2 χθ a i

( (

   

 θχψ i a   

=

r

,

2

2

−

−

+

ψ

1

sin

1

sin

j

2 χ a

θ i

2 χ a

θ i

)( )

2

2

2

2

−

+

+

sin

sin

j

2 − χκ a

2 χκθ a

i

θ i

)

( (

( 4 .1 3 1)

2

+

+

1

sin

θχ cos j i

a

2 χ a

θ i

and

)

(

2

2

−

−

+

+

+

1

sin

1

sin

j

) 2/1

2 χ a

θ i

2 χ a

2 χθ a i

)(

(

−=

t

,

j

2

2

−

−

+

ψ

1

sin

1

sin

j

2 χ a

θ i

2 χ a

θ i

( 4 .1 3 2)

)

2

2

2

2

+

−

+

sin

sin

j

2 − χκ a

2 χκθ a

i

θ i

)

( (

2

+

−

−

cos

1

sin

θχψ j a i

2 χ a

θ i

)

(

2

2

2

2

and

+

−

+

−

j

sin

sin

) 2/1

2 χκ − a

2 χκθ a

i

2 χθ a i

(

   

   

=

r

,

j

2

2

ψ

−

−

+

sin

1

sin

j

1

2 χ a

θ i

2 χ a

θ i

)( )

2

2

2

2

+

+

−

sin

sin

j

2 χκ − a

2 χκθ a

i

θ i

)

( (

( 4 .1 3 3)

2

2

2

2

Let

=

=

2 = κθ

s

sin

sin

sin

.

2 χ a

θ i

1

θ 2

( 4 .1 3 4 )

2

2

2

2

−

+

+

−

−

+

+

+

j

s

s

s

s

j

1

1

T he n eq ua tio n s ( 4 .1 3 0) , ( 4 .1 3 1) , ( 4 .1 3 2) , a nd ( 4 . 1 3 3) ca n b e wr it te n a s

) 2/1

2 χ a

2 χ a

)(

)

(

(

=

t

,

1 ∆

83

( 4 .1 3 5)

2

2

2

2

2 κ

−

+

+

−

−

+

+

+

1

s

s

s

s

j

) 2/1

2 χψ j a

2 χ a

)

)(

(

(

=

,

r

2 κ ∆

2

2

2

2

−

+

+

−

−

+

+

+

1

1

s

s

s

s

j

j

( 4 .1 3 6)

) 2/1

2 χ a

2 χ a

)(

)

(

(

−=

,

t

j

1 ∆

( 4 .1 37 )

2

2

2

2

2 κ

2 κ

+

−

−

+

+

−

−

+

−

j

s

s

s

s

1

) 2/1

2 χψ j a

2 χ a

)

)(

(

(

and

=

r

,

j

∆

( 4 .1 3 8)

wh er e

2

2

2

2

ψ

2 κ

2 κ

+

=∆

−

−

+

−

−

+

j

j

s

s

s

s

1

1

( 4 .1 3 9)

)

) .

(

(

Eq u at io ns ( 4 .1 2 6) , ( 4 .1 3 0) a nd ( 4 .1 3 5) a gr ee wi t h e q uat io n ( 9 ) o f V il lo t and G u i go u -

Car ter ( 2 0 0 0 ) ap ar t fr o m d i f fer i n g i n s i g n. T h i s d i f fer e n ce i n si g n i s d ue to t h e fa ct

th at t he t wo p la te s ar e i n t he sa me p l a ne i n t h i s th e si s, w h i le i n V il lo t a nd G ui go u -

Car ter ( 2 0 0 0 ) t h e seco n d p lat e ha s b ee n r o tat ed th r o u g h +9 0 d e gr ee s i n o r de r to fo r m a

r i g ht a n gled j u nc tio n . T h u s t h e ve lo c it y i n t h e p o si ti v e y -a x i s d ir ec tio n i n t he se co nd

p lat e b eco me s a v elo cit y i n t h e ne ga ti v e x - a xi s d ir ec tio n i n t he Vi llo t a n d G ui go u -

Car ter ( 2 0 0 0 ) p ap e r . No te t h at t he p o s it i ve d ir ec tio n o f t he x -a x is i n Vi ll o t a nd

G ui go u - Car ter ( 2 0 0 0) f i g ur e 6 a is i nco r r ec t. I t d o es no t co r r e sp o nd to t h e s i g n s o f t h e

exp o n e nt ial e xp o ne n t s i n t hei r eq ua tio n ( 4 ) . Al s o no te t hat o nce t he p o s i ti ve d ir e ct io n

o f t he x -a xi s i s c ha n g ed in t he ir f i g ur e 6 , t h e p o s iti v e d i r ec tio n o f t he z - a xi s al so h as to

b e c ha n ged i n o r d er to ma i nt ai n a r i g ht h a nd ed co -o r d i na te s ys t e m.

2

2

2

−

+

−

+

−

+

−

+

s

j

s

s

1

Eq u at io n ( 4 .1 3 5) ca n b e wr i tte n a s

[

]

=

t

.

2

) 2/1 2

1 2

2

−

ψ

( 2 χ a +

) 2/1 2 κ

+

2 χ a +

( 2 χ a 2 κ

−

+

ψ

−

+

s

s

s

s

j

1

1

)

)

(

(

( 4 .1 4 0)

4.7 Transmitted bending wave power.

84

As wa s s ho wn i n c hap te r s 2 a nd 3 fo r t he aco u st ica l ca s e, i t is no t p o s si b le to ca lc u lat e

th e ne t i n te n s it y i nc id e n t o n a j u nc tio n b y s ep ar ate l y ca lc u la ti n g t he i nt en s it y o f t he

in cid e nt wa ve a nd t he i n te n si t y o f t h e r e f le cted wa v e, wh e n t he i nc id e n t wa ve i s no t

fr e el y p r o p a ga ti n g. T hi s is b ec a us e t h e cr o s s t er ms i n t h e i n te n si t y c alc u lat io n d o no t

ca nce l u nl e ss t he i nc id e nt wa v e i s f r ee l y p r o p a g ati n g. T h u s o nl y t he to t a l i n te n si t y

in cid e nt o n t he j u nc tio n ca n b e cal c ul ated . B ec a u se t hi s is eq ua l to t he t r an s mi t ted

in te n s it y b y e ner g y co ns er v at io n, o n l y t h e tr a ns mi t ted i nte n s it y wi ll b e calc u la ted i n

th i s c h ap ter . O n e o f t he co n seq u e nce s o f no t b ei n g ab le to c al c ula te a m ea ni n g f u l

in cid e nt i nt e ns it y d ue to a fo r ced i nc id e n t wa v e alo ne, i s t h at it i s no t p o s sib le t o

me a ni n g f u ll y d e fi n e a tr an s mi s s io n e f f ic ie nc y a s Vi llo t a nd G ui go u - C ar t er ( 2 0 0 0 ) ha v e

att e mp t ed to d o i n t he ir eq u at io n ( 1 0 ) . T hi s i s o ne o f t h eir t wo er r o r s whi c h ar e

co r r ec ted i n t h i s t h es i s.

T he tr a n s mi t ted b e nd i n g wa ve p o wer p er u n it le n gt h o f t h e p l ate j u nc t io n I i s no w

−

−

jk

x

k

x

2

2

x

N

d er i ved . Fr o m eq ua tio n ( 4 .8 8) t he a n g ul ar ve lo c it y ab o u t t he z -a x i s i n p l ate 2 i s

=

−

jk

te

).

w z

2

( )( − zEv + 1

x

2

etk 2 j N

( 4 .1 4 1 )

−= t

t j

No w

−

−

jk

x

k

x

T hu s

x

N

2

=

−

jk

e

).

+

w z

2

( ) ( − tzEv 1

x

2

ek 2 N

( 4 .1 4 2 )

At t he j u nct io n, x=0 , t h u s,

=

=

+

jk

k

)0

).

xw ( 2 z

( ) ( − tzEv + 1

x

2

N

2

( 4 .1 4 3 )

Fr o m eq u at io n ( 4 .9 4) , t h e mo me n t p er u n it le n g t h a b o ut t he z -a x i s, M x z 2, wh i c h i s

2

∂

ex er t ed acr o s s a l i ne no r ma l to t h e x - a xi s i n p la t e 2 i s

−

−=

µ

M

xz

2 vk z

y

2

v y 2

∂ x

 B  2  ω j 

  .  

( 4 .1 4 4)

2

2

B ut t he tr a n s v er s e ve lo c it y v y (x =0 ) i s zer o b ec a u se o f t he p i n ned j o in t. T hu s

=

−=

=

=

(

)

)

x

j

x

0

0

(

).0

( xM xz 2

∂ w z ∂ x

∂ wB z 2 ω ∂ x

 = 

 B 2  ω j 

( 4 .1 4 5)

85

Di f f er e nt ia ti n g eq ua tio n ( 4 .1 4 2) , set ti n g x=0 a n d i ns er t i n g i n to eq u at io n ( 4 .1 4 5) , g i ve s

=

=

−

−=

+

)

0

j

k

k

j

k

).

[ −

]

( xM 2 xz

tzEv )( + 1

2 x 2

2 N

2

2 N

( ktzEv )( + 1

2 x 2

2

B 2 ω

B 2 ω

( 4 .1 4 6)

T he tr a ns mi tted b e nd i n g wa ve p o wer ( I ) p er u ni t le n gt h o f t h e p l at e j u n c tio n is t he r ea l

p ar t o f t he p r o d u ct o f t h e a n g ul ar ve lo c it y a nd t he mo me n t b ec a u se t he t r an s v er s e

=

=

=

(

I

Re

x

0

0

2

( ω z

2

2

2

2

2

ve lo c it y o f t he j u nc tio n is zer o .

=

+

+

)

)

t

Re

k

k

k

k

) ( xM 2 xz { ( k

) ) }

( ( k

2

2

2

2

2

v + 1

* 2 x

x

2 N

x

N

2 N

B B 2 − j ωω

  

) .   

2

=

−

( 4 .1 4 7)

sin

,

k

k

k

k

2 2

2 2 x

2 i

θ i

2 2 x

2

Si n ce is al wa ys r e al. T h u s

+

].

I

k

[ Re k

[ 2 kt

]

= + v 1

2 x 2

2 N

2

x

2

B 2 ω

( 4 .1 4 8)

No w

=

−

k

k

k

sin 2

,

2 x 2

2 i

θ i

2 2

( 4 .1 4 9 )

and

=

+

k

k

k

sin 2

.

2 N

2

2 2

2 i

θ i

( 4 .1 5 0 )

T hu s

+

=

k

k

2

.2 2 k 2

2 2 x

2 N

( 4 .1 5 1 )

2

2

i

=

−

k

k

1

sin

x

θ i

2

2

k k

2

  

  

Fr o m eq u at io n ( 4 .7 4)

2

2

=

−

k

1

sin

θ i

2

χ a κ

  

  

( 4 .1 5 2)

2

2

2

2

I n ser ti n g eq ua tio n s ( 4 .1 5 1) a nd ( 4 .1 5 2) i nto eq u atio n ( 4 .1 4 8) gi v e s

=

−

2

1

sin

I

t

v + 1

θ i

3 kB 22 ω

χ a κ

  

  

  Re  

  .  

( 4 .1 5 3)

86

No w fr o m t h e ho mo ge n e o u s b e nd i n g wa v e eq uat i o n ( se e eq u atio n ( 4 .4 1) ) ,

4

kB .2 22 ω= m 2

( 4 .1 5 4)

T her e fo r e

=

3 kB 22 ω

ω m 2 . k

2

( 4 .1 5 5)

Si n ce

=

,

c 2

ω k

2

( 4 .1 5 6)

th e fo llo wi n g eq uat io n i s o b t ai n ed .

=

cm .22

3 kB 22 ω

( 4 .1 5 7)

2

2

2

P ut ti n g eq uat io n ( 4 .1 5 7) i nto ( 4 .1 5 3) gi v es

=

−

I

2

1

sin

θ i

v + 1

2 cmt 22

χ a κ

  

  

  Re  

  .  

( 4 .1 5 8)

2

2

He nce

=

−

2

1

I

v + 1

2 cmt 22

s 2 κ

  ,  

  Re  

( 4 .1 5 9)

2

2

and

=

−

1

t

s 2 κ

2

I 2 cmv + 1 22

  ,  

  Re  

( 4 .1 6 0)

T he tr a ns mi tted b e nd i n g wa ve p o wer ( I ) p er u ni t le n gt h o f t h e p l at e j u n c tio n c a n a l so

2

2

=

)

2

Re

( cos

I

t

vcm + 1 22

θ 2

2

2

2

b e d er i v ed fr o m eq ua tio n s ( 3 .8 3 ) , ( 3 .8 4 ) a nd ( 3 . 9 2 ) o f C r e mer et a l. ( 2 0 0 5) .

=

−

2

1

t

vcm + 1 22

s κ

  Re  

   

( 4 .1 6 1)

+1v

is t h e wh er e m 2 i s t h e ma s s o f p lat e 2 , c 2 is t he b e nd i n g wa ve sp eed i n p l ate 2 and

87

r o o t me a n s q uar e v elo ci t y o f t he f o r ced i nc id e n t b end i n g wa ve. No te t hat t he f acto r o f

2 d o es no t ap p ear i n Cr e me r e t a l.( 2 0 0 5 ) b ec a us e t he y ar e u s i n g p ea k wa ve a mp l it ud e

in s tead o f r o o t mea n sq u ar e ( r ms ) a mp l it ud e a s i s b e i n g u sed h er e.

4.8 The power per unit length transmitted by a diffuse bending

wave field.

T he p o we r p er u n it le n g th tr a ns mi t ted to p lat e 2 b y a d i f f u s e b e nd i n g wa ve f ie ld i n

p lat e 1 wi l l no w b e cal c ul at ed . T h is p o we r i s p r o p o r tio na l to t h e i n te gr a l o f eq u at io n

( 4 .1 6 1) o ver t he i ncid e n t a n gl e θ i fr o m 0 r ad i a ns to π /2 r ad ia n s o r t he mi ni mu m a n g le

at wh ic h to t al i nt er na l r ef le ct io n o cc ur s wh e n t h e r ea l f u nc tio n i n eq u at i o n ( 4 .1 6 1 ) i s

2

eq u al to z er o . T h i s o cc u r s wh e n

−

≤

1

.0

s 2 κ

( 4 .1 6 2)

2

2

T hi s mea n s t ha t

≥

2 κ

sin

= s

.

2 χ a

θ i

( 4 .1 6 3 )

Re ar r a n g i n g gi v es,

sin

,

κ θ ≥ i χ a

( 4 .1 6 4)

and

≥

arcsin

θ i

κ χ a

  

  . 

( 4 .1 6 5)

if

≥ χκ a

T hu s t he up p er l i mi t o f t he i nt e gr a l i s

=

.

θ u

if

χκ < a

κ χ a

  

  

π   2    arcsin  

( 4 .1 6 6)

He nce t he b e nd i n g wa v e i nte n s it y tr a n s mi t ted b y a n i nc id e n t d i f f u s e fo r c ed b e nd i n g

88

wa v e f ie ld i n p la te 1 i s p r o p o r tio na l to

2

θ

2

−

tu∫

0

s 2 κ

  θ d .  

  1 Re  

89

( 4 .1 6 7)

4.9 The transmitted power when plate 1 is excited by a diffuse acoustic field.

φ

y

θ

Plate 1

Plate 2

x

z

F i gu r e 4 . 3 An i n c i d e n t a c o u s t i c s o u n d wa v e .

Fr o m eq u at io n ( 1 2 ) o f V illo t a nd G u i go u - Car ter ( 2 0 0 0) , t he wa v e i mp ed a nc e o f p lat e 1

is

=

−

+

z

)

4 k 1

.4 η kB 111

2 i

B ( 1 k ω j

( 4 .1 6 8)

wh er e B 1 i s t he b e nd i n g st i f f ne s s o f p la te 1 , k 1 i s t he wa ve n u mb er o f p la te 1 fo r a n

k

k = i

φsina

an g u lar f r eq ue n c y o f ω , η 1 i s t h e i n s it u d a mp i n g lo s s f ac to r o f p lat e 1 a n d

is t he wa v e n u mb er o f t he p la n e wa v e fo r ced b y an ai r b o r ne p la ne wa v e o f wa ve

to t he no r mal to p lat e 1 . T h e n u mb e r k a wh i c h is i nc i d en t o n p lat e 1 a t a n a n g le o f

ϕ d a mp i n g lo s s ter m h a s b ee n ad d ed to t he Vi llo t and G u i go u - Car t er ( 2 0 0 0 ) eq uat io n.

I f t he r ms a mp l it ud e o f t he i nc id e n t a ir b o r ne d i f f u se so u nd f ie ld a t t h e p l ate s ur f ace i s

90

1 , t he n f r o m eq ua tio n ( 1 1 ) o f V il lo t a nd G u i go u - Car ter ( 2 0 0 0 ) ,

2 ω

2 =+ v 1

2

−

+

k

k

)

24 1

η 1

4 i

8 i

{ ( 2 kB 1

}

1

=

2

2

−

+

) 1

2 χη 1 a

{ ( 2 4 χω m 1 a

}8

( 4 .1 6 9)

2

2

−

t

1

T hu s

s 2 κ

=

,

4

8

2 ω

2

    −

+

φ

φ

2 IB 1 cm 22

sin

sin

    2 η k

( k

)

24 k 1

4 a

8 a

( 4 .1 7 0)

2

2

=

s

2 χ a

and

sin 2

2

sin

k

2 a

=

.

θ i θφ sin i 2 k 1

( 4 .1 7 1)

Fo r a d i f f u se f ie ld a co us tic i nc id e n t wa v e eq u at i o n ( 4 .1 7 1 ) ha s to b e a v e r ag ed o ver

2

2

π π

−

φ

1

sin

t

ϕ and θ i.

2

2

s 2 κ

Av

2

4

8

∫ ∫

φθ . dd i φ

  Re   φ −

+

2

sin

sin

    2 η k

( k

)

2 IB 1 cm 22

4 k 1

4 a

8 a

0

0

 2 = πω 

  

91

( 4 .1 7 2)

y

φ=0

φ

θ i

x

φ=0,

z

φ=0 θ i=0

F i gu r e 4 . 4 Th e o c t a n t t h a t e q u a t i o n 4 . 1 8 7 i s a v e r a g e d o ve r .

d b li nt . m c alc u la te s t h e v al ue s o f t he i nt e gr a l t h at is p r o p o r t io na l to t he p o wer p e r u ni t

le n gt h tr a n s mi t ted t hr o u g h t he j u nct io n wh e n p l ate 1 i s e x ci ted b y a d i f f u se aco u st ic

so u nd f ield ( s ee ap p e nd i x 1 ) . T he i nt e gr a l i s co n d uc ted o n l y fr o m 0 to π/ 2 wh i c h i s t he

so l id a n g le o f t he o ct a nt o v er wh i c h t he i nt e gr at i o n is mad e . No te t ha t t h e a n g ul ar

we i g h ti n g i s d i f f er e n t fr o m t ha t u sed i n eq ua tio n ( 1 3 ) b y V il lo t a nd G u i g o u- Car ter

( 2 0 0 0 ) . I n t h e no t at io n o f t hi s t he si s, Vi llo t a nd G ui go u - Car ter e va l uat e an i nt e gr a l o f

π

k

a

th e fo r m.

F

θ ,(

dk

.

dkk ) i

i

θ i

i

∫

∫

0

2 0

( 4 .1 7 3)

Si n ce

=

φ

=

k

k

dk

k

sin

,

cos

φ .

i

a

a

i

( 4 .1 7 4 )

ππ

2

2

T hu s t hei r i n te gr a l b e co me s

φθφφ

F

k

k

θ ,(

sin

φ )

sin

cos

a

2 a

dd i

∫∫

0

0

92

( 4 .1 7 5)

T he k a sq ua r ed a nd t he 2 /π ar e j u s t co n st a nt s wh i ch ca n b e co r r ect ed fo r b ut t he co sφ

is i nco r r ec t b ec a us e t h e fo r ced b e nd i n g wa ve v el o cit y d ep e nd s o n t he i nc id e nt

aco us ti c p o we r .

4.10 Summary

T he r e f lec ted wa v e a mp l it ud e s ( r) , t he r e fl ec ted ne ar fi eld wa ve a mp li t ud es ( r j ) , t h e

tr a n s mo t ted wa ve a mp li t ud e s ( t) , a nd t h e tr a ns mi tted n ear f ie ld ( t j) wa ve s ar e d er i ved .

T hes e wa v e a mp l it ud e s ar e c alc u la ted fo r t he s it ua tio n o f no r ma l i n cid e n ce. T h e n t h e

wa v e n u mb er eq u at io ns fr o m C r eme r et a l . ( 2 0 0 5 ) ar e d e r i ved fo r t h e o b liq ue l y

in cid e nt fr ee l y p r o p a g at in g wa v e c as e. Ne xt , t h e an g u lar velo ci t y a nd t h e to r sio n al

mo me n t i n t h e x d ir e ct io n a r e d e ter mi n ed b y v al ue s i n t h e y a nd z d ir ec ti o n s. T h i s i s

s ho wn f o r t h e o b l iq ue l y in cid e nt wa ve ca se . T he n t he d e r i va tio n o f t h e t r an s mi t ted

and r e fl ec ted wa ve a mp l it ud e f o r t h e o b l iq ue l y i nc id e n t fo r c ed wa v e is g iv e n. T h es e

a mp li t ud e s ar e u sed to c alc u lat e t h e tr a n s mi tt ed b end i n g wa ve i nt e ns it y. T he n t h e

in te n s it y tr a n s mi t ted b y a d i f f u se b e nd i n g wa v e f ield i s c alc u la ted . F i nal l y, t h e

tr a n s mi tt ed i n te n si t y wh en p la te 1 i s e x ci ted b y a d i f f u se aco u st ic f ie ld i s d er i ved .

93

T hi s fi n al cal c ul atio n was t he ma i n ai m o f t h i s t he s is .

Chapter 5 The results for pinned plates.

I n t hi s c hap t er , gr ap h s o f t he r ela ti v e t r a n s mi tt e d i nte n s it y at t he j u nc tio n o f t wo s e mi -

f i ni te i n t h e p l a ne p lat es wi ll b e p r e se n ted . As e xp l ai n ed i n c hap ter s 2 a nd 3 , a nd i n

sec tio n 4 .6 i t is no t p o s s ib le to c al c ula te mea n i n g f u l i n te n si ti es fo r t h e i nc id e n t a nd

r ef le cted wa v e s u nl es s t he i nc id e n t wa v e is fr ee l y p r o p a gat i n g. O nl y t he to ta l i n te n s it y,

wh ic h i s i n cid e nt o n t he j u nct io n c a n b e mea n i n g f u ll y cal c ul ated a nd t h i s i s eq ua l to

th e t r a n s mi tt ed i n te n si t y . T h us o nl y t he tr a n s mi t ted i n te n s it y is co n sid er ed i n t hi s

ch ap te r .

As i n t he p r e vio u s c h ap t er t he p la te s wi ll b e a s s u med to h a ve b e nd i n g st i f f ne s se s B i

and fr ee l y p r o p a g at i n g wa v e n u mb er s k i wh e r e i =1 o r 2 . T h e r a tio o f t he fr eel y

p r o p ag at i n g wa v e n u mb er s i s

=κ

.

k 2 k 1

( 5 .1 )

T he var iab l e ψ is d e f i ne d to b e

ψ

=

=

.2 κ

2 kB 22 2 kB 11

B 2 B 1

( 5 .2 )

T he i n te n si t y o f a fr e el y p r o p a gat i n g p la ne b e nd in g wa v e wi t h tr a ns v er s e v elo ci t y v i i n

2

th e i t h p lat e i s ( s ee Cr e me r e t a l. ( 2 0 0 5) a nd se ctio n 4 .7 )

2

,

I = i

vcm i i i

( 5 .3 )

wh er e m i is t he ma s s p er u n it ar ea a nd c i i s t he f r eel y p r o p a ga ti n g b e nd i n g wa ve

94

ve lo c it y o f t he i t h p l ate. T hu s

Z

2=

i

cm i i

( 5 .4 )

ca n b e i nt er p r e ted a s t he i mp ed a n ce e xp er ie nc ed b y a fr ee l y p r o p a gat i n g b end i n g wa ve.

Fo r a f r ee l y p r o p a g a t i n g b e nd i n g wa ve C r e mer e t a l. ( 2 0 0 5)

4 ω=i ,2

B i k m i

( 5 .5 )

and

=

c ,i

ω k

i

( 5 .6 )

wh er e ω i s t he a n g ular f r eq u e nc y.

Eq u at io ns ( 5 .4 ) , ( 5 .5 ) a n d ( 5 .6 ) a nd g i ve

=

=

.

kB i

2 i

2 cm i i

cZ i i 2

( 5 .7 )

2

2

He nce ψ c a n b e wr i tt e n as

ψ

=

=

=

1 κ

Z Z

k 1 k

Z Z

cZ 22 cZ 11

1

2

1

( 5 .8 )

T hu s ψ ca n b e i n ter p r ete d as t he r at io o f t he “f r e el y p r o p a g at i n g b e nd i n g wa ve

i mp ed a n ce s” d i vid ed t h e r at io o f t h e fr e el y p r o p ag at i n g wa v e n u mb er s .

T he fo r c ed i n cid e n t wa v e wi ll b e i n p l ate 1 a nd ha v e a fo r ced wa ve n u m b er o f k i. I t

wi l l b e i nc id e n t a t a n a n gl e o f θ i to t h e no r ma l t o t he li ne j u nc tio n b et wee n t h e t wo

se mi - f i n it e i n p la n e p l at es. T he va r iab le χ a i s t h e r at io o f t he fo r ced b e n d in g wa v e

n u mb e r o f t h e i n cid e nt wa v e to t h e fr e el y p r o p a ga ti n g wa v e n u mb er i n p lat e 1 .

.

a =χ

ki 1k

95

( 5 .9 )

T he fr e el y p r o p a gat i n g wa v e ca se i s gi ve n b y χ a eq u al s 1 . F ir st gr ap h s o f t he r ela ti v e

tr a n s mi tt ed i n te n si t y ( r e lat i ve p o wer p er u ni t le n gt h o f t h e j u n ct io n) wi l l b e gi ve n .

Eq u at io n ( 4 .1 6 0) wi ll b e gr ap hed .

2

2

2

T hi s eq ua tio n i s

=

−

t

1

sin

θ i

χ a κ

  

  

2

I 2 cmv + 22 1

  Re  

  .  

( 5 .1 0)

T hu s it ca n b e s ee n t ha t th e t r a n s mi tt ed i n te n si t y is ei t her no r ma liz ed b y o r i s r e lat i ve

to t he i n te n si t y o f a fr ee l y p r o p a ga ti n g p la n e b e nd i n g wa v e . T h i s r e f er e nc e wa v e ha s

th e sa me tr a ns v er se b e n d in g wa v e ve l o c it y as t h e i nc id e n t wa v e b u t i s p r o p ag at i n g i n

p lat e 2 a t a tr a n s mi t ted an g le o f ο0 r elat i ve to t h e no r ma l o f t he li n e j u n ct i o n.

T he tr a ns mi tted i n te n si t y h as b ee n f ur t her no r m ali zed b y d i vid i n g i t b y th e t r a n s mi tt ed

in te n s it y fo r t he fr ee l y p r o p ag at i n g i n cid e nt wa v e ca se ( χ a=1 ) . T hi s i s no t p o s s ib l e i n

all t he se gr ap h s b e ca u se so me ti me s t h e tr a ns mi tt ed i nt e n si t y fo r t he fr eel y p r o p a gat i n g

in cid e nt wa ve ca se i s z e r o . I n t h es e ca s es t he tr a n s mi tt ed i n te n si t y is no r ma li zed b y

d iv id i n g i t b y i t s ma x i m u m v al ue . T h is f u r t he r n o r mal iza tio n ma ke s t he cur v e s i n eac h

f i g ur e ne ar l y o ver la y e a ch o t her .

I t ca n b e see n i m me d i at el y t h at t he r el at i ve tr a n s mi tt ed i n te n si t y is zer o wh e n

.

χ ≥ a

κ θ sin i

( 5 .1 1)

I n t hi s ca se to ta l i nter n a l r e f lec tio n o cc ur s. T he mi n i mu m v al u e o f χ a fo r wh i c h to tal

iθ ar e gi v e n i n t he fo l lo wi n g tab l e 5 .1 .

96

in ter n al r e fl ec tio n o c c ur s a s a f u nc tio n o f κ a nd

κ=1/2

κ=1

κ=2

ο

∞

θ 0=i

ο

1.93

3.86

7.73

θ 15=i

ο

1

2

4

θ 30=i

ο

0.707

1.41

2.83

θ 45=i

ο

0.577

1.15

2.31

θ 60=i

ο

0.518

1.04

2.07

θ 75=i

ο

0.5

1

2

θ 90=i

T ab le 5 .1 T he mi n i mu m va l ue o f χ a fo r wh i c h to t al i n ter n al r e fl ect io n o c cur s fo r

d i f fer e n t va l ue s o f κ a nd θ i.

T he r o o t mea n sq u ar e co mp le x a mp l it ud e o f t h e fr e el y p r o p a ga ti n g tr a n s mi t ted wa ve i n

p lat e 2 d i v id ed b y t he r o o t mea n sq u ar e co mp l e x a mp l it ud e o f t h e fo r c ed i ncid e nt wa v e

2

−

1

sin

j

2 χ a

θ i

2

+

+

+

+

1

sin

) 2/1

θχ cos j i

a

2 χ a

θ i

( 2 χ a

)

(

2

−

+

1

sin

2 χ a

θ i

   

   

in p la te 1 i s t. Fr o m eq u atio n ( 4 .1 3 0)

=

t

.

2

2

−

−

+

ψ

1

sin

1

sin

j

2 χ a

θ i

2 χ a

θ i

)

2

2

2

2

+

−

+

sin

sin

j

2 − χκ a

2 χκθ a

i

θ i

)

( (

( 5 .1 2)

T he o nl y o n e o f t he ter ms i n t h e n u mer ato r o f t he r i g h t ha nd si d e o f eq ua tio n ( 5 .1 2)

97

th at ca n p o s sib l y b e co m e zer o i s

j

1

sin

.

2 θχ− a i

( 5 .1 3 )

Eq u at io n ( 5 .1 3) i s zer o wh e n

.

χ = a

1 θ sin i

( 5 .1 4)

I t ca n b e see n fr o m eq u a tio n s ( 4 .6 2) a nd ( 4 .6 3) , th at t he v al u e o f χ a gi ve n b y eq uat io n

( 5 .1 4) i s t h e ma x i mu m v al ue o f χ a fo r wh i c h t he r ef le cted p r o p a ga ti n g wav e i s ac t u al l y

a wa v e. Fo r va l ue s o f χ a gr eat er t h a n t h at g i ve n b y eq uat io n ( 5 .1 4) t he p r o p ag at i n g

r ef le cted wa v e b e co me s a n o n -p r o p a ga ti n g n ear f ield wa v e.

2

At t he va l ue o f χ a g i ve n b y eq uat io n ( 5 .1 4) it i s l ik el y t ha t t h e gr ap h s wi l l e x hib it a

.

t

lo ca l mi n i mu m b eca u se th e gr ap h s ar e p r o p o r tio na l to Ho we v er t h i s lo c al mi n i mu m

wi l l no t b e o b se r ved i f v al ue s o f χ a gi ve n b y eq u atio n ( 5 .1 4) ar e gr e ate r th a n o r eq ual

to t he val u e o f χ a gi v e n b y t he eq u al s si g n i n eq ua tio n ( 5 .1 1) . T h i s wi l l hap p e n wh e n κ

is eq ua l to o r le s s t ha n o ne. T hi s i s b ec a u se t h e r eal f u n ct io n i n eq ua tio n ( 5 .1 0) ma ke s

th e r e la ti v e tr a ns mit ted in te n s it y zer o i n t h e r e g io n o f t h e va l ue o f χ a wher e a lo c al

mi n i mu m wo u ld p r o b ab l y h a ve o c c ur r ed . T he v a lu e s o f χ a f o r wh i c h t he lo ca l mi n i mu m

98

wi l l p r o b ab l y o cc ur a r e gi v e n i n tab le 5 . 2 .

κ>1

ο

θ 0=i

ο

3.86

θ 15=i

ο

2

θ 30=i

ο

1.41

θ 45=i

ο

1.15

θ 60=i

ο

1.04

θ 75=i

ο

1

θ 90=i

T ab le 5 .2 Val u es o f χ a f o r wh i c h a lo ca l mi n i mu m i n t h e t r a n s mi tt ed i n te n si t y wi l l

p r o b ab l y o c c ur fo r d i f fe r en t va l ue s o f θ i.

Ap ar t fr o m t he mo d u li o f t he ter m g i ve n b y eq ua tio n ( 5 .1 3 ) , a ll t he mo d u li o f t he ter ms

in t he n u mer ato r o f eq ua tio n ( 5 .1 2 ) i nc r ea se a s χ a i ncr ea se s. T h u s ap ar t f r o m t he

p o s sib le lo ca l mi n i mu m and t he u lt i ma te d r i vi n g to ze r o ( e xc ep t f o r θ i =0 ) b y t he r ea l

f u nc tio n i n eq uat io n ( 5 . 1 0) , t he g r ap h s ar e e xp e cted to b e i ncr ea si n g f u nc tio n s o f χ a.

Fi g ur es 5 .1 to 5 . 3 s ho w th e r e la ti v e tr a ns mit ted in te n s it y at 0 ˚ f o r κ eq u al s ½, 1 a nd 2

r esp ect i ve l y. E ac h ne x t se t o f t h r ee f i g ur e s s ho ws t he s a me gr ap hs fo r t he ca se wh e n θ i

ha s b ee n i ncr e as ed b y 1 5 ˚ . T hi s tr e nd is co nt i n u ed u nti l θ i eq u al s 9 0 ˚ .

Fi g ur es 5 .1 to 5 . 2 1 s ho w a wid e var iet y o f b e ha vio u r . E xc ep t f o r t h e no r mal i nc id e nc e

cas e, to ta l i nter n al r e fl e ctio n o cc ur s fo r s ma l ler va l ue s o f χ a a s κ d ecr e a s es a nd θ

in cr e as es . O n mo s t o f t h e g r ap h s t h e c ur v es fo r d i f fer e n t va l ue s o f ψ ar e al mo st

id e nt ica l. T he se c u r ve s ar e o nl y s i g ni f ic a nt l y d i f fe r e nt o n t h e κ eq ual s 2 gr ap hs , fo r t he

lar ger v al u es o f χ a a nd θ . T he vo l u me o f χ a fo r whi c h a mi n i mu m o c c ur s i n so me o f t h e

99

gr ap h s i s we l l p r ed ict ed b y v al ue s g i ve n i n t ab l e 5 .2 .

T he r e lat i ve b e nd i n g i nt en s it y ( p o wer p er u n it l e n gt h o f t h e j u n ct io n) t r a n s mi tt ed b y a

fo r ced d i f f u s e f ield i n p lat e 1 wi l l no w b e cal c u lat ed . T h is i s d o ne b y i n te gr a ti n g

eq u at io n ( 5 .1 0) o v er a n g le s o f i n cid e nce θ i r a n g i n g fr o m 0 r ad ia n s up to th e a n g le at

wh ic h to t al i nt er na l r e f l e ct io n o cc u r s o r π /2 r ad i an s ( 9 0 ˚ ) i f to t al i nt er na l r e f lec tio n

if

≥ χκ a

d o es no t o c c ur . T h u s t he up p er l i mi t o f i nt e gr a ti o n θ u is g i ve n b y eq ua ti o n ( 4 .1 6 6 )

=

.

θ u

if

χκ < a

κ χ a

  

  

π   2    arcsin  

( 5 .1 5)

2

θ

2

2

He nce t he r e lat i ve i nt e n si t y i s p r o p o r t io nal to e q ua tio n ( 4 .1 6 5)

−

tu

Re

1

sin

.

i

∫

0

χ a κ

  

  

  θθ d i  

   

( 5 .1 6)

Eq u at io n ( 5 .1 6) wi ll al wa ys g i ve no n - zer o r es u lt s. T h u s t he r es u lt o f eq u atio n ( 5 .1 6)

wi l l b e f ur t h er no r ma li z ed b y d i v id i n g it b y t he r es u lt fo r a fr eel y p r o p a ga ti n g i nc id e n t

wa v e ( χ a=1 ) . T he se r e s u lt s wi l l t he n b e co n ver te d to d ec ib e l s a nd gr ap he d i n fi g u r e s

5 .2 2 to 5 . 2 4 . T he no r ma liz ed c ur v es i n t he se f i g ur e s o v er l a y ea c h o t h er ex cep t wh e n χ a

is g r ea ter t ha n o n e i n t h e κ eq u al s t wo ca se, s ho wn i n f i g ur e 5 . 2 4 . T h i s o ut co me ca n b e

p r ed ic ted fr o m t h e r e s ul ts s ho wn i n fi g ur es 5 . 1 t o 5 . 2 1 .

Ex cep t fo r t h e lo ca l mi n i mu m i n t he κ=1 /2 r e s u lt s s ho wn i n f i g ur e 5 .2 2 , t he c ur ve s a r e

in cr e as i n g f u nc tio n s o f χ a. T he lo c al mi n i mu m i n t he κ =1 /2 ca se ap p ear s to b e a n

in ter ac tio n b et wee n t he in cr e as i n g na t ur e o f t he r ela ti v e tr a ns mi s sio n as a f u nct io n o f

χ a fo r a n g le s o f i nc id e nc e clo se to no r ma l i nc id e nc e a nd t he d ecr e as e i n th e a n g le at

wh ic h to t al i nt er na l r e f l ec t io n o cc u r s a s a f u n ct i o n o f χ a.

Ne xt it i s a s s u med t ha t t he b e nd i n g wa v e fi eld i n p lat e 1 is fo r ced b y a n i ncid e nce

d i f f us e a co us ti c fi eld . T he f o r ced b e nd i n g wa v e n u mb e r k i wi l l var y f r o m k a, wh er e k a

100

is t he v al u e o f t h e wa v e n u mb e r o f t h e d i f f u s e so u nd f ie ld . T he var iab l e r is d e fi n ed to

b e t he r at io o f t he air b o r ne wa v e n u mb e r k a to t he f r ee l y p r o p a g at i n g wav e n u mb er k 1

in p la te 1 .

r

.

k a= 1k

( 5 .1 7)

I n t hi s ca se t he r e la ti v e tr a n s mi tt ed b e nd i n g wa v e i nt e n si t y i s p r o p o r tio n al to eq ua tio n

2

2

2

π π

φ

−

t

1

sin

sin

θ i

( 4 .1 7 2) .

2

2

χ a κ

  

φθ dd .

4

8

∫ ∫

φ

η

φ

−

+

  Re   sin

    sin

( k

   )

4 a

82 k a

4 k 1

0

0

( 5 .1 8)

T he 2 / π fr o m eq uat io n ( 4 .1 7 2) h a s b ee n d r o p p ed b eca u se eq ua tio n ( 5 .2 4 ) wi ll b e

no r ma l ized b y d i v id i n g i t b y i t s va l u e fo r t he f r e el y p r o p a g at i n g i n cid e n ce wa v e c a se

wh e n χ a eq ua l s o ne. No t e t ha t t h e a n g u lar we i g h ti n g is d i f f er e nt fr o m t h at u sed i n

eq u at io n ( 1 3) o f b y V il l o t a nd G ui go u - C ar te r ( 2 0 0 0 ) .

Af te r no r ma li zat io n, t he va l ue o f eq ua tio n ( 5 .1 8 ) wi ll b e co n v er ted to d e cib e ls b e fo r e

b ei n g gr ap h ed i n fi g ur es 5 .2 5 to 5 . 2 7 wh er e t h e cur v e s i n eac h f i g ur e o v er la y eac h

o th er . T h i s i s b eca u se t h e c ur ve s i n fi g u r e s 5 .1 t o 5 .2 1 o ver l a y e ac h o t he r , ap ar t fr o m

th e κ =2 ca s e wh e n t h e v al ue s o f χ a ar e gr e ater t h an o ne. Fi g ur e s 5 .2 5 to 5 . 2 7 d i f fer

fr o m f i g ur e s 5 .2 2 to 5 . 2 4 b eca u se a we i g h t ed a v er a ge o f t he tr a n s mi t ted in te n s it y h as

b ee n t a ke n o ve r b y va l u es o f χ a fr o m zer o to r.

Fo r va l ue s o f r gr eat er t ha n o ne t h i s a v er a ge i s d o mi nat ed b y va l ue s o f χ a clo se to o n e.

I n t hi s ca se:

≈φ

sin

.

ka

1k

( 5 .1 9 )

B eca u se t he d a mp i n g lo s s fa cto r η i s s mal l ( 0 .0 3 ) , t he d e no mi n ato r o f t h e i nt e gr a nd i n

eq u at io n ( 5 .1 8) i s c lo s e to zer o wh e n eq ua tio n ( 5 .2 4 ) ap p l ie s. T h i s me a n s t h at t he

101

in te gr ad ha s a s h a r p ma xi mu m a t χ a =1 . T h e e f f e ct i s t ha t t h e i n te gr al is r ela ti v el y

co n s ta nt fo r r gr e ater t h an o r eq ua l to o ne. B eca u se t he i nt e gr al i s no r m ali zed b y

d iv id i n g b y i t s va l ue wh en r eq ua ls o ne, t he no r ma li zed i nt e gr al i s c lo se to o ne wh i c h

is 0 d B wh e n r i s gr e at e r t ha n o n e.

W he n r i s l es s t ha n o n e, t he c u r ve s fo l lo w t he g en er a l b e ha v io r o f f i g ur es 5 . 2 2 to 5 . 2 4 .

T hat is , t h e y ar e i n cr ea s in g f u nc tio n s o f r e xcep t fo r a lo c al mi n i mu m i n t he κ =1 /2

102

cas e.

6

5

ψ=1/2

ψ=1

y t i s n e t n

I

4

ψ=2

d e t t i

3

2

l

1

m s n a r T e v i t a e R

0

0

0.5

1.5

2

1 χa

F i gu r e 5 . 1 Th e r e l a t i v e t r a n s mi t t e d i n t e n s i t y a t t h e j u n c t i o n o f t wo i n f i n i t e p a n e l s d u e t o a

fo r c e d w a ve i n t h e f i r s t p a n e l i n c i d e n t a t a n a n gl e o f i n c i d e n c e t o t h e n o r m a l o f 0 ˚ . Th e r a t i o κ

o f t h e w a v e n u mb e r i n t h e s e c o n d p a n e l t o t h a t i n t h e f i r s t p a n e l i s ½ . C u r ve s a r e gi ve n f o r t h e

r a t i o ψ e q u a l s ½ , 1 a n d 2 .

103

6

5

ψ=1/2

y t i s n e t n

I

4

ψ=1

d e t t i

3

2

ψ=2

l

1

m s n a r T e v i t a e R

0

0

0.5

1.5

2

1 χa

F i gu r e 5 . 2 . Th e r e l a t i ve t r a n s m i t t e d i n t e n s i t y a t t h e j u n c t i o n o f t wo i n fi n i t e p a n e l s d u e t o a

fo r c e d w a ve i n t h e f i r s t p a n e l i n c i d e n t a t a n a n gl e o f i n c i d e n c e t o t h e n o r m a l o f 0 ˚ . Th e r a t i o κ

o f t h e w a v e n u mb e r i n t h e s e c o n d p a n e l t o t h a t i n t h e f i r s t p a n e l i s 1 . C u r v e s a r e gi ve n fo r t h e

r a t i o ψ e q u a l s ½ , 1 a n d 2 .

104

6

5

ψ=1/2

ψ=1

y t i s n e t n

I

4

ψ=2

d e t t i

3

2

l

1

m s n a r T e v i t a e R

0

0

0.5

1.5

1

2

χa

F i gu r e 5 . 3 Th e r e l a t i v e t r a n s mi t t e d i n t e n s i t y a t t h e j u n c t i o n o f t wo i n f i n i t e p a n e l s d u e t o a

fo r c e d w a ve i n t h e f i r s t p a n e l i n c i d e n t a t a n a n gl e o f i n c i d e n c e t o t h e n o r m a l o f 0 ˚ . Th e r a t i o κ

o f t h e w a v e n u mb e r i n t h e s e c o n d p a n e l t o t h a t i n t h e f i r s t p a n e l i s 2 . C u r v e s a r e gi ve n fo r t h e

r a t i o ψ e q u a l s ½ , 1 a n d 2 .

105

2.5

2

ψ=1/2 ψ=1 ψ=2

y t i s n e t n

I

1.5

d e t t i

1

0.5

l

m s n a r T e v i t a e R

0

0

0.5

1.5

2

χa 1

Fi g ur e 5 .4 T h e r e la ti ve t r an s mi t ted i nt e ns it y at t he j u nc tio n o f t wo i n f i ni te p a n el s d ue

to a fo r ced wa ve i n t he f ir s t p a n el i nc id e nt at a n an g le o f i nc id e n ce to t h e n o r ma l o f

1 5 ˚ . T he r at io κ o f t he wav e n u mb er i n t he se co n d p a nel to t hat i n t he f ir st p a ne l i s 1 /2 .

106

C ur ve s ar e g i ve n fo r t he r at io ψ eq u al s ½, 1 a nd 2 .

5

4.5 ψ=1/2 4

y t i s n e t n

3.5 ψ=1

I

3

d e t t i

2.5 ψ=2

2

1.5

1

l

m s n a r T e v i t a e R

0.5

0 0 0.5 1.5 1 2

χa

Fi g ur e 5 .5 T h e r e la ti ve t r an s mi t ted i nt e ns it y at t he j u nc tio n o f t wo i n f i ni te p a n el s d ue

to a fo r ced wa ve i n t he f ir s t p a n el i nc id e nt at a n an g le o f i nc id e n ce to t h e n o r ma l o f

1 5 ˚ .T he r a tio κ o f t he wav e n u mb er i n t he se co n d p a nel to t hat i n t he f ir st p a ne l i s 1 .

107

C ur ve s ar e g i ve n fo r t he r at io ψ eq u al s ½, 1 a nd 2 .

6

5

ψ=1/2

ψ=1

y t i s n e t n

4

I

ψ=2

d e t t i

3

2

1

l

m s n a r T e v i t a e R

0

0

0.5

1

1.5

2

χa

Fi g ur e 5 .6 T h e r e la ti ve t r an s mi t ted i nt e ns it y at t he j u nc tio n o f t wo i n f i ni te p a n el s d ue

to a fo r ced wa ve i n t he f ir s t p a n el i nc id e nt at a n an g le o f i nc id e n ce to t h e n o r ma l o f

1 5 ˚ . T he r at io κ o f t he wav e n u mb er i n t he se co n d p a nel to t hat i n t he f ir st p a ne l i s 2 .

108

C ur ve s ar e g i ve n fo r t he r at io ψ eq u al s ½, 1 a nd 2 .

1.2

1

ψ=1/2

y t i s n e t n

ψ=1

I

0.8

ψ=2

d e t t i

0.6

0.4

l

0.2

m s n a r T e v i t a e R

0

0

0.5

1.5

1

2

χa

F i gu r e 5 . 7 Th e r e l a t i v e t r a n s mi t t e d i n t e n s i t y a t t h e j u n c t i o n o f t wo i n f i n i t e p a n e l s d u e t o a

fo r c e d w a ve i n t h e f i r s t p a n e l i n c i d e n t a t a n a n gl e o f i n c i d e n c e t o t h e n o r m a l o f 3 0 ˚ . Th e r a t i o

κ o f t h e w a v e n u mb e r i n t h e s e c o n d p a n e l t o t h a t i n t h e fi r s t p a n e l i s 1 / 2 . C u r v e s a r e gi ve n fo r

t h e r a t i o ψ e q u a l s ½ , 1 a n d 2 .

109

1.8

1.6

ψ=1/2

1.4

ψ=1

y t i s n e t n

ψ=2

I

1.2

d e t t i

1

0.8

0.6

l

0.4

m s n a r T e v i t a e R

0.2

0

0

0.5

1.5

1

2

χa

F i gu r e 5 . 8 Th e r e l a t i v e t r a n s mi t t e d i n t e n s i t y a t t h e j u n c t i o n o f t wo i n f i n i t e p a n e l s d u e t o a

fo r c e d w a ve i n t h e f i r s t p a n e l i n c i d e n t a t a n a n gl e o f i n c i d e n c e t o t h e n o r m a l o f 3 0 ˚ . Th e r a t i o

κ o f t h e w a v e n u mb e r i n t h e s e c o n d p a n e l t o t h a t i n t h e fi r s t p a n e l i s 1 . C u r v e s a r e g i v e n fo r

t h e r a t i o ψ e q u a l s ½ , 1 a n d 2 .

110

3

2.5

ψ=1/2

ψ=1

y t i s n e t n

I

2

ψ=2

d e t t i

1.5

1

l

0.5

m s n a r T e v i t a e R

0

0

0.5

1.5

2

1

χa

Fi g ur e 5 .9 T h e r e la ti ve t r an s mi t ted i nt e ns it y at t he j u nc tio n o f t wo i n f i ni te p a n el s d ue

to a fo r ced wa ve i n t he f ir s t p a n el i nc id e nt at a n an g le o f i nc id e n ce to t h e n o r ma l o f

3 0 ˚ . T he r at io κ o f t he wav e n u mb er i n t he se co n d p a nel to t hat i n t he f ir st p a ne l i s 2 .

111

C ur ve s ar e g i ve n fo r t he r at io ψ eq u al s ½, 1 a nd 2 .

1.2

1

y t i s n e t n

I

0.8

ψ=1/2

d e t t i

0.6

ψ=1

0.4

ψ=2

l

0.2

m s n a r T e v i t a e R

0

0

0.5

1.5

2

1 χa

Fi g ur e 5 .1 0 T h e r e la ti ve tr a n s mi tt ed i n te n s it y at th e j u n ct io n o f t wo i n fi n ite p a ne ls d ue

to a fo r ced wa ve i n t he f ir s t p a n el i nc id e nt at a n an g le o f i nc id e n ce to t h e n o r ma l o f

4 5 ˚ . T he r at io κ o f t he wa v e n u mb er i n t h e seco nd p a nel to t hat i n t he f i r st p a ne l i s

112

1 /2 . C ur ve s a r e g i ve n fo r t he r at io ψ eq u al s ½, 1 an d 2 .

1.2

1

ψ=1/2

y t i s n e t n

I

0.8

ψ=1

d e t t i

0.6

0.4

ψ=2

l

0.2

m s n a r T e v i t a e R

0

0

0.5

1.5

2

1 χa

Fi g ur e 5 .1 1 . T he r ela ti v e tr a n s mi t ted i nt e ns it y a t t he j u nc tio n o f t wo i n f in it e p a n el s

d ue to a fo r ced wa v e i n th e f ir s t p a n el i nc id e n t a t a n a n g le o f i n cid e n ce t o t he no r ma l

o f 4 5 ˚ . T he r at io κ o f t h e wa v e n u mb er i n t he se co nd p a ne l to t ha t i n t he f ir st p a nel i s

113

1 . C ur v es ar e gi ve n fo r t he r at io ψ eq ual s ½, 1 a nd 2 .

9

8

7

ψ=1/2

ψ=1

y t i s n e t n

6

I

5

ψ=2

d e t t i

4

3

2

l

m s n a r T e v i t a e R

1

0

0

0.5

1.5

1

2

χa

F i gu r e 5 . 1 2 Th e r e l a t i v e t r a n s m i t t e d i n t e n s i t y a t t h e j u n c t i o n o f t wo i n fi n i t e p a n e l s d u e t o a

fo r c e d w a ve i n t h e f i r s t p a n e l i n c i d e n t a t a n a n gl e o f i n c i d e n c e t o t h e n o r m a l o f 4 5 ˚ . Th e r a t i o

κ o f t h e w a v e n u mb e r i n t h e s e c o n d p a n e l t o t h a t i n t h e fi r s t p a n e l i s 2 . C u r v e s a r e g i v e n fo r

t h e r a t i o ψ e q u a l s ½ , 1 a n d 2 .

114

1.2

1

ψ=1/2

0.8

ψ=1

y t i s n e t n

I

ψ=2

d e t t i

0.6

0.4

0.2

l

m s n a r T e v i t a e R

0

0

0.5

1.5

2

1 χa

F i gu r e 5 . 1 3 Th e r e l a t i v e t r a n s m i t t e d i n t e n s i t y a t t h e j u n c t i o n o f t wo i n fi n i t e p a n e l s d u e t o a

fo r c e d w a ve i n t h e f i r s t p a n e l i n c i d e n t a t a n a n gl e o f i n c i d e n c e t o t h e n o r m a l o f 6 0 ˚ . Th e r a t i o

κ o f t h e w a v e n u mb e r i n t h e s e c o n d p a n e l t o t h a t i n t h e fi r s t p a n e l i s 1 / 2 . C u r v e s a r e gi ve n fo r

t h e r a t i o ψ e q u a l s ½ , 1 a n d 2 .

115

1.8

1.6

1.4

ψ=1/2

1.2

ψ=1

y t i s n e t n

I

1

ψ=2

d e t t i

0.8

0.6

0.4

0.2

l

m s n a r T e v i t a e R

0

0

0.5

1.5

1

2

χa

Fi g ur e 5 .1 4 T h e r e la ti ve tr a n s mi tt ed i n te n s it y at th e j u n ct io n o f t wo i n fi n ite p a ne ls d ue

to a fo r ced wa ve i n t he f ir s t p a n el i nc id e nt at a n an g le o f i nc id e n ce to t h e n o r ma l o f

6 0 ˚ . T he r at io κ o f t he wav e n u mb er i n t he se co n d p a nel to t hat i n t he f ir st p a ne l i s 1 .

116

C ur ve s ar e g i ve n fo r t he r at io ψ eq u al s ½, 1 a nd 2 .

16

14

12

ψ=1/2 ψ=1 ψ=2

y t i s n e t n

I

10

d e t t i

8

6

4

l

m s n a r T e v i t a e R

2

0

0

0.5

1.5

2

1 χa

F i gu r e 5 . 1 5 Th e r e l a t i v e t r a n s m i t t e d i n t e n s i t y a t t h e j u n c t i o n o f t wo i n fi n i t e p a n e l s d u e t o a

fo r c e d w a ve i n t h e f i r s t p a n e l i n c i d e n t a t a n a n gl e o f i n c i d e n c e t o t h e n o r m a l o f 6 0 ˚ . Th e r a t i o

κ o f t h e w a v e n u mb e r i n t h e s e c o n d p a n e l t o t h a t i n t h e fi r s t p a n e l i s 2 . C u r v e s a r e g i v e n fo r

t h e r a t i o ψ e q u a l s ½ , 1 a n d 2 .

117

1.2

1

ψ=1/2

y t i s n e t n

0.8

I

ψ=1

d e t t i

0.6

0.4

ψ=2

l

0.2

m s n a r T e v i t a e R

0

0

0.5

1.5

1

2

χa

F i gu r e 5 . 1 6 . Th e r e l a t i v e t r a n s m i t t e d i n t e n s i t y a t t h e j u n c t i o n o f t wo i n f i n i t e p a n e l s d u e t o a

fo r c e d w a ve i n t h e f i r s t p a n e l i n c i d e n t a t a n a n gl e o f i n c i d e n c e t o t h e n o r m a l o f 7 5 ˚ . Th e r a t i o

κ o f t h e w a v e n u mb e r i n t h e s e c o n d p a n e l t o t h a t i n t h e fi r s t p a n e l i s 1 / 2 . C u r v e s a r e gi ve n fo r

t h e r a t i o ψ e q u a l s ½ , 1 a n d 2 .

118

9

8

7

ψ=1/2 ψ=1 ψ=2

y t i s n e t n

6

I

5

d e t t i

4

3

2

l

1

m s n a r T e v i t a e R

0

0

0.5

1.5

1

2

χa

F i gu r e 5 . 1 7 . Th e r e l a t i v e t r a n s m i t t e d i n t e n s i t y a t t h e j u n c t i o n o f t wo i n f i n i t e p a n e l s d u e t o a

fo r c e d w a ve i n t h e f i r s t p a n e l i n c i d e n t a t a n a n gl e o f i n c i d e n c e t o t h e n o r m a l o f 7 5 ˚ . Th e r a t i o

κ o f t h e w a v e n u mb e r i n t h e s e c o n d p a n e l t o t h a t i n t h e fi r s t p a n e l i s 1 . C u r v e s a r e g i v e n fo r

t h e r a t i o ψ e q u a l s ½ , 1 a n d 2 .

119

45

40

35

30

ψ=1/2

ψ=1

y t i s n e t n

I

25

ψ=2

d e t t i

20

15

10

5

l

m s n a r T e v i t a e R

0

0

0.5

1.5

2

1 χa

F i gu r e 5 . 1 8 Th e r e l a t i v e t r a n s m i t t e d i n t e n s i t y a t t h e j u n c t i o n o f t wo i n fi n i t e p a n e l s d u e t o a

fo r c e d w a ve i n t h e f i r s t p a n e l i n c i d e n t a t a n a n gl e o f i n c i d e n c e t o t h e n o r m a l o f 7 5 ˚ . Th e r a t i o

κ o f t h e w a v e n u mb e r i n t h e s e c o n d p a n e l t o t h a t i n t h e fi r s t p a n e l i s 2 . C u r v e s a r e g i v e n fo r

t h e r a t i o ψ e q u a l s ½ , 1 a n d 2 .

120

1.2

1

ψ=1/2

y t i s n e t n

I

0.8

ψ=1

ψ=2

d e t t i

0.6

0.4

l

0.2

m s n a r T e v i t a e R

0

0

0.5

1.5

1

2

χa

F i gu r e 5 . 1 9 . Th e r e l a t i v e t r a n s m i t t e d i n t e n s i t y a t t h e j u n c t i o n o f t wo i n f i n i t e p a n e l s d u e t o a

fo r c e d w a ve i n t h e f i r s t p a n e l i n c i d e n t a t a n a n gl e o f i n c i d e n c e t o t h e n o r m a l o f 9 0 ˚ . Th e r a t i o

κ o f t h e w a v e n u mb e r i n t h e s e c o n d p a n e l t o t h a t i n t h e fi r s t p a n e l i s 1 / 2 . C u r v e s a r e gi ve n fo r

t h e r a t i o ψ e q u a l s ½ , 1 a n d 2 .

121

1.2

1

ψ=1/2

y t i s n e t n

ψ=1

I

0.8

ψ=2

d e t t i

0.6

0.4

l

0.2

m s n a r T e v i t a e R

0

0

0.5

1.5

2

1 χa

F i gu r e 5 . 2 0 Th e r e l a t i v e t r a n s m i t t e d i n t e n s i t y a t t h e j u n c t i o n o f t wo i n fi n i t e p a n e l s d u e t o a

fo r c e d w a ve i n t h e f i r s t p a n e l i n c i d e n t a t a n a n gl e o f i n c i d e n c e t o t h e n o r m a l o f 9 0 ˚ . Th e r a t i o

κ o f t h e w a v e n u mb e r i n t h e s e c o n d p a n e l t o t h a t i n t h e fi r s t p a n e l i s 1 . C u r v e s a r e g i v e n fo r

t h e r a t i o ψ e q u a l s ½ , 1 a n d 2 .

122

1.2

1

0.8

ψ=1/2

ψ=1

y t i s n e t n

I

0.6

ψ=2

d e t t i

0.4

0.2

l

0

m s n a r T e v i t a e R

0

0.5

1.5

2

1 χa

F i gu r e 5 . 2 1 Th e r e l a t i v e t r a n s m i t t e d i n t e n s i t y a t t h e j u n c t i o n o f t wo i n fi n i t e p a n e l s d u e t o a

fo r c e d w a ve i n t h e f i r s t p a n e l i n c i d e n t a t a n a n gl e o f i n c i d e n c e t o t h e n o r m a l o f 9 0 ˚ . Th e r a t i o

κ o f t h e w a v e n u mb e r i n t h e s e c o n d p a n e l t o t h a t i n t h e fi r s t p a n e l i s 2 . C u r v e s a r e g i v e n fo r

t h e r a t i o ψ e q u a l s ½ , 1 a n d 2 .

123

5

ψ=1/2

4

ψ=1

3

) B d ( y t i s n e t n

ψ=2

I

2

d e t t i

1

0

-1

l

-2

m s n a r T e v i t a e R

-3

0

0.5

1.5

2

1 χa

F i gu r e 5 . 2 2 Th e i n c i d e n t f i e l d i s a d i f f u s e vi b r a t i o n a l fi e l d . Th e i n t e gr a t i o n i s d o n e o v e r a l l

t h e p o s s i b l e a n gl e s o f i n c i d e n c e . B e c a u s e o f s ym m e t r y, t h e i n t e g r a t i o n i s o n l y d o n e fr o m 0 t o

9 0 d e gr e e s . Th e r a t i o κ o f t h e w a ve n u m b e r i n t h e s e c o n d p a n e l t o t h a t i n t h e f i r s t p a n e l i s 1 / 2 .

C u r v e s a r e gi v e n fo r t h e r a t i o ψ e q u a l s ½ , 1 a n d 2 .

124

6

4

ψ=1/2

ψ=1

2

ψ=2

) B d ( y t i s n e t n

I

d e t t i

0

-2

l

-4

m s n a r T e v i t a e R

-6

0

0.5

1.5

2

1

2.5

χa

F i gu r e 5 . 2 3 Th e i n c i d e n t f i e l d i s a d i f f u s e vi b r a t i o n a l fi e l d . Th e i n t e gr a t i o n i s d o n e o v e r a l l

t h e p o s s i b l e a n gl e s o f i n c i d e n c e . B e c a u s e o f s ym m e t r y, t h e i n t e g r a t i o n i s o n l y d o n e fr o m 0 t o

9 0 d e gr e e s . Th e r a t i o κ o f t h e w a ve n u m b e r i n t h e s e c o n d p a n e l t o t h a t i n t h e f i r s t p a n e l i s 1 .

C u r v e s a r e gi v e n fo r t h e r a t i o ψ e q u a l s ½ , 1 a n d 2 .

125

8

ψ=1/2

6

ψ=1

4

) B d ( y t i s n e t n

I

ψ=2

2

d e t t i

0

-2

l

-4

m s n a r T e v i t a e R

-6

0

0.5

1.5

2

1 χa

F i gu r e 5 . 2 4 Th e i n c i d e n t f i e l d i s a d i f f u s e vi b r a t i o n a l fi e l d . Th e i n t e g r a t i o n i s d o n e o v e r a l l

t h e p o s s i b l e a n gl e s o f i n c i d e n c e . B e c a u s e o f s ym m e t r y, t h e i n t e g r a t i o n i s o n l y d o n e fr o m 0 t o

9 0 d e gr e e s . Th e r a t i o κ o f t h e w a ve n u m b e r i n t h e s e c o n d p a n e l t o t h a t i n t he f i r s t p a n e l i s 2 .

C u r v e s a r e gi v e n fo r t h e r a t i o ψ e q u a l s ½ , 1 a n d 2 .

126

0.5

ψ = 1/2

0

ψ = 1

ψ = 2

-0.5

) B d ( y t i s n e t n

I

-1

d e t t i

-1.5

-2

l

-2.5

m s n a r T e v i t a e R

-3

0

0.5

1

1.5

2

r

F i gu r e 5 . 2 5 . Th e vi b r a t i o n a l fi e l d i n t h e 1 s t p a n e l i s e x c i t e d b y a d i f f u s e i n c i d e n t a c o u s t i c

fi e l d . Th e r a t i o κ o f t h e w a v e n u mb e r i n t h e s e c o n d p a n e l t o t h a t i n t h e fi r s t p a n e l i s 1 / 2 .

C u r v e s a r e gi v e n fo r t h e r a t i o ψ e q u a l s ½ , 1 a n d 2 .

127

0.5

-0.5

-1.5

) B d ( y t i s n e t n

I

ψ=1/2

d e t t i

-2.5

ψ=1

-3.5

ψ=2

-4.5

l

m s n a r T e v i t a e R

-5.5

0

0.5

1

1.5

2

r

F i gu r e 5 . 2 6 . Th e vi b r a t i o n a l fi e l d i n t h e 1 s t p a n e l i s e x c i t e d b y a d i f f u s e i n c i d e n t a c o u s t i c

fi e l d . Th e r a t i o κ o f t h e w a v e n u mb e r i n t h e f i r s t p a n e l t o t h a t i n t h e s e c o n d p a n e l i s 1 . C u r v e s

a r e g i v e n fo r t h e r a t i o ψ e q u a l s ½ , 1 a n d 2 .

128

1

ψ=1/2

0

ψ=1

-1

ψ=2

) B d ( y t i s n e t n

I

-2

d e t t i

-3

-4

l

-5

m s n a r T e v i t a e R

-6

0

0.5

1.5

2

1 r

F i gu r e 5 . 2 7 . Th e vi b r a t i o n a l fi e l d i n t h e 1 s t p a n e l i s e x c i t e d b y a d i f f u s e i n c i d e n t a c o u s t i c

fi e l d . Th e r a t i o κ o f t h e w a v e n u mb e r i n t h e s e c o n d p a n e l t o t h a t i n t h e fi r s t p a n e l i s 2 . C u r v e s

a r e g i v e n fo r t h e r a t i o ψ e q u a l s ½ , 1 a n d 2 .

129

2

0

-2

-4

) B d ( y t i s n e t n

I

-6

d e t t i

-8

Villot & Guigou-Carter

-10

-12

l

m s n a r T e v i t a e R

Irwin

-14

F i gu r e 5 . 2 8 Th e r e l a t i v e i n t e n s i t y t r a n s m i t t e d a t a p i n n e d j o i n t b e t w e e n t wo p a n e l s wh e n t h e

t wo p a n e l s h a ve t h e s a m e m a t e r i a l p r o p e r t i e s . Th e fi r s t p a n e l i s e xc i t e d o n o n e o f i t s s i d e s b y a

d i f fu s e s o u n d fi e l d . Th e I r w i n c u r v e s h o w s t h e c a l c u l a t i o n s m a d e i n t h i s t h e s i s , wh i l e t h e

V i l l o t & Gu i go u - C a r t e r c u r v e s h o w s t h e c a l c u l a t i o n s ma d e b y V i l l o t a n d G u i go u - C a r t e r

( 2 0 0 0 ) . Th e x - a xi s va r i a b l e , r i s t h e r a t i o o f t h e w a v e n u mb e r o f t h e d i f fu s e s o u n d fi e l d t o t h e

f r e e b e n d i n g w a v e n u mb e r o f t h e t w o i d e n t i c a l p a n e l s . Th e c r i t i c a l f r e q u e n c y o c c u r s wh e n r

e q u a l s o n e .

-16 0 0.2 0.4 0.6 1 1.2 1.4 1.6 1.8 2 0.8 r

T he cr it ica l fr eq u e nc y o cc ur s wh e n r i s eq u al to o ne . Ab o v e t h e cr it ica l f r eq u e nc y,

wh er e r i s gr e ater t ha n o r eq ual to o n e, t h e t wo c ur ve s a gr ee we ll wi t h ea ch o t her .

B elo w t he cr i ti cal fr eq u en c y t h e r e la ti ve tr a n s m itt ed i n te n s it y is mu c h g r eat er t h a n t h at

calc u la ted b y Vi llo t a nd G ui go u - C ar te r ( 2 0 0 0 ) . He nce i t i s no t a s o b v io u s a s cla i med

b y Vi llo t a nd G ui go u - C a r ter ( 2 0 0 0 ) t ha t t h e i n te n si t y tr a n s mi tt ed b y t h e fo r ced wa ve

ca n b e i g no r ed r el at i ve t o t hat tr a ns mi t ted b y r e s o na n t wa ve s.

T he p r o d uc tio n o f fi g ur e 5 .2 8 wa s t he ma i n ai m o f t hi s t he si s. T he id e nt i f ica tio n o f t wo

er r o r s i n Vi llo t a nd G ui go u - Car ter ’ s ( 2 0 0 0 ) ca lc ul at io n s me a nt t ha t i t was i mp o r t a nt to

130

co r r ec t t h eir ca lc ul at io n s. T h i s ha s b e e n a c hie v e d i n fi g u r e 5 .2 8 a nd l ar g e d i f fe r e nc es

b et we e n t he co r r ec ted c alc u lat io n s a nd Vi llo t a nd G ui go u - C ar te r ’s ( 2 0 0 0 ) cal c ul at io ns

ha v e b ee n id e n ti f ied b el o w t h e cr it ica l fr eq u e nc y.

Fr o m eq u at io n ( 4 .4 1) , t h e b e nd i n g wa v e n u mb er o f p la te 1 is

=

.

2 k 1

2 ω m 1 B 1

( 5 .2 0)

T he a ir b o r ne wa ve n u mb er i s

=

ka

ω . c

( 5 .2 1)

T he cr it ica l a n g u lar fr eq ue n c y o f p lat e 1 ω c 1 o r t he cr i ti ca l fr eq u e nc y f c 1 o f p la te 1

o cc ur s wh e n

ka =

.1k

( 5 .2 2 )

a

T her e fo r e , a f ter mu c h m an ip ul at i uo n i t i s fo u nd th at,

=

=

=

.

r

k k

f f

1

ω ω c 1

c 1

( 5 .2 3)

T hu s r fo r Vi llo t a nd G u igo u - C ar t er ’ s ( 2 0 0 0 ) f i g ur e 7 i s cal c ula ted b y ta ki n g t he

sq uar e r o o t o f t h e r a tio o f t he fr eq ue nc y d i v id ed b y t he cr i ti cal fr eq ue n c y. I n V il lo t

and G u i go u - Car t er ’ s ( 2 0 0 0 ) f i g ur e 7 , t h e cr it ica l fr eq ue nc y i s wh e r e t he i r fo r ced c ur ve

sto p s d ecr e as i n g wi t h i n cr ea si n g f r eq ue n c y a nd b eco me s a l mo st c o n sta n t ab o ve 3 .4

k Hz.

2

c

2

2

I t s ho uld al so b e no ted t ha t

κ

=

=

=

.

f f

k k 1

ω c ω c 1

c 1

131

( 5 .2 4)

Chapter 6 Conclusion

I t ha s b e e n s ho wn i n t h i s t h e si s t h at t he tr a n s mi s sio n o f f o r ced wa v es at an i n ter fa ce

b et we e n t wo me d i a i s d i f fe r e nt fr o m t he tr a n s mi s sio n o f f r ee l y p r o p a g at i n g wa ve s. T he

Vil lo t a nd G ui go u - C ar te r ( 2 0 0 0 ) eq ua tio n f o r t h e a mp l it ud e o f t he tr a n s mi t ted b e nd i n g

wa v e at a p i n n ed j u n ct io n b et wee n t wo p a ne ls d u e to a fo r ced i nc id e n t wav e i s co r r ec t.

Ho we v er , t hei r e xp r e s si o n fo r t he tr a n s mi t te d b end i n g wa ve i nt e ns it y i s no t co r r ect .

T hi s i s b e ca u se it i s no t p o s sib le to d e f i ne a tr a n s mi s sio n fac to r fo r f o r c ed i nc id e n t

wa v e s as Vi llo t a nd G u i go u - Car ter ( 2 0 0 0) ha v e att e mp t ed to d o .

T her e ap p ear to b e t wo er r o r s i n Vi llo t a nd G ui go u - Car ter ’ s ( 2 0 0 0 ) p ap er . T he f ir st

er r o r i s b eca u se it i s no t p o s sib le to ca lc u la te t h e e ner g y i nc id e n t o n a j u nc tio n b y j u st

co n s id er i n g t he i nc id e n t wa ve a nd i g n o r i n g t he r ef le cted wa v e. T h e i n ter act io n o f t he

in cid e nt a nd r e f lec ted wav es mu s t b e co n sid er ed . T hi s i s b eca u s e t he cr o s s t er ms

b et we e n t he tr a n s ver se v elo c it y o f o ne o f t he wa ve s wi t h t h e tr a n s ver se f o r ce o f t he

o th er wa ve i s no t zer o . T hi s i s d i f fer e n t fr o m t h e ca se wh e n t he i nc id e n t wa ve s ar e

fr e el y p r o p a ga ti n g a nd t he se cr o s s t er ms ar e zer o .

T he seco nd er r o r is d ue to t he we i g ht i n g o ve r t h e d ir e ct io n o f t he i ncid e nt wa v e u sed

b y Vi llo t a nd G ui go u - C a r ter ( 2 0 0 0 ) . T hi s i s i n co r r ect b eca u s e i t ha s a s a co s θ ter m t h at

s ho uld no t b e t he r e. T h e co s θ t er m co me s fr o m t he d i f fe r e nt ia l o f t he fo r ce d wa v e

n u mb e r i n t he p a ne l, wh ic h ap p ear s i n t he ir i nt e gr a l s.

B eca u se o f t he se t wo er r o r s, Vi llo t a nd G ui go u - Car ter ’ s ( 2 0 0 0) n u mer ic all y ca lc ul at ed

gr ap h o f t h e tr a n s mi s sio n lo s s i s i n co r r ec t. I t s h o ws t ha t t h e a tt e n uat io n o f t he fo r ced

wa v e ca n b e 1 6 d B o r mo r e t ha n t he at te n u a tio n o f t he fr ee l y p r o p a ga ti n g wa v e. T he

calc u la tio n s i n t h i s t h es i s s ho w t ha t t h e ma x i mu m e xtr a a tte n u at io n fo r t he s a me

132

si t ua tio n is le s s t ha n 6 d B .

B eca u se o f t he co mp l e x na t ur e o f t he p r o b l e m b e in g co n sid er ed i n t h i s t h es is , i n it ial l y

th e si t ua tio n o f fo r ced aco us ti ca l wa v e s p r o p a g ati n g f r o m a h al f i n f i ni t e fl u id med i u m

to a no t h er ha l f i n fi n it e f lu id me d i u m wa s co ns id er ed . T h e no r ma ll y i nc i d en t c as e wa s

co n s id er ed f ir st . T he n t h e o b liq u e i n cid e nce ca se wa s co n s id er ed . F i na ll y t he d i f f u se

f ield i nc id e n c e c as e wa s co ns id er ed . I f t he wa ve sp e ed i n t he t wo med ia is t he s a me, a

ver y s i mp l e fo r mu l a fo r th e r a tio o f t h e e ne r g y t r an s mi t ted b y t he f o r ced d i f f u se f ield

wa v e s to t h at tr a n s mi t te d b y t he f r ee l y p r o p a g at in g wa v e s wa s ca lc u lat e d . U si n g

1

r

an al yt ic al i nt e gr a tio n, t hi s e ner g y r at io wa s fo u nd to b e ( eq ua tio n ( 2 .1 8 0) a nd ( 2 .1 8 8) ) :

≤

r

,1

+ 2

( 6 .1 )

and

≥

+

1

.1

r

1 2

1 r

  

  

( 6 .2 )

I n t he se fo r mu la e, r is t he r at io o f t he wa ve n u mb e r o f t h e fo r c ed i n cid en t a co us ti ca l

wa v e to t h e wa ve n u mb e r o f a fr eel y p r o p a ga ti n g wa ve .

W he n t he wa v e sp eed was d i f fer e nt ( i n t he t wo d i f fer e n t i n fi n ite hal f m ed ia) , i t wa s

ne ce s sar y to u se n u me r i cal i nt e gr a tio n o v er t h e an g le o f i nc id e n ce to ca l cu la te t he

d i f f us e f ield tr a n s mi t ted en er g y . T h i s t h e si s p r o vid e s gr ap h s s ho wi n g t h e r at io o f t he

tr a n s mi tt ed i n te n si t y fo r a fo r ced i nc id e n t wa v e to t ha t fo r a fr e el y p r o p ag at i n g

in cid e nt wa ve .

T he n, t h e c as e o f t wo h a l f i n f i ni te p la te s co n n ec ted b y a p i n ned j o i nt wa s co n s id er ed.

T he p i n n ed j o i nt wa s co n sid er ed b e ca u se t hi s i s a r ea so nab le ap p r o xi ma t io n fo r t wo

ha l f i n fi n ite p la te s co n n ected at r i g h t a n g le s a t l o w fr eq u e nci es . B ec a us e, fo r t h is

p in n ed j o i nt ca se , t h e a n gl e b e t we e n t he p lat e s ma k e s no d i f f er e n ce, t hi s t h e si s

co n s id er s t h e c as e wh e n t he p lat e s ar e i n t he sa me p la n e. Ag ai n b ec a us e o f t h e

133

co mp le x it y, no r ma l i n ci d e n ce wa s co n s id er ed f i r st , fo l lo we d b y t h e o b l i q ue i nc id e nc e

cas e. T wo d i f fer e n t d i f f u se f ie ld ca s es wer e co n sid er ed . F ir s tl y, ca lc u lat io n s wer e

mad e fo r t he ca se o f a si n gl e fo r c ed wa ve n u mb e r d i f f u se i ncid e nt b e nd i n g wa ve fie ld

in t he fir s t p l ate . I n t h e seco nd ca se , t he d i f f u se in cid e nt b e nd i n g wa ve f ield i n t he

f ir s t p lat e i s e xci ted b y a d i f f u se air b o r n e so u nd f ie ld o n o ne s id e o f t he f ir st p la te .

T he fo r c ed wa v e n u mb er i n t h e fir s t p l at e d ep e n d s o n t he a n g le o f i nc id en ce o f t he

fo r ci n g a co u st ic wa ve to t he no r ma l to t he p lat e.

Fo r t h e c as e wh e n t h e wav e i n t he f ir st p la te i s e xc it ed b y a d i f f u se air b o r ne so u nd

wa v e as s ta ted ab o v e, t h e tr a n s mi s s io n o f t he fo r ced wa v e c a n b e up to 6 d B le ss t ha n

th e t r a n s mi s sio n o f t h e f r eel y p r o p a ga ti n g wa ve . T hu s t he as s u mp tio n t h at i s o f te n

mad e, wh e n p r ed ic ti n g f la n ki n g tr a n s mi s s io n u si n g t he me t ho d s i n t h e E N1 2 3 5 4

st a nd ar d , t ha t t h e e ne r g y tr a n s mi tt ed b y t he fo r ced wa v e c a n b e i g no r ed , i s no t

ne ce s sar i l y co r r ec t. T h i s is b ec a us e t h e e x tr a at t en u at io n o f t he fo r ced b end i n g wa ve

co mp ar ed to t hat o f t h e fr e el y p r o p a ga ti n g i nc id en t wa ve s is no t as lar g e as t ha t

p r ed ic ted b y Vi llo t a nd G ui go u - Car ter ( 2 0 0 0) . O f co ur s e, i n ma n y ca se s th e e n er g y o f

th e fo r c ed b e nd i n g wa ve s wi l l b e mu c h le s s t h a n t he e n er g y o f t he fr ee l y p r o p a gat i n g

wa v e s, b u t t h at i ss u e i s no t co n sid er ed i n t h i s t h es is .

T he r e se ar c h co nd u ct ed in t h is t he s is co uld b e e xt e nd ed b y co n sid e r i n g j u nc tio n t yp e s

o th er t ha n p i n ned j u nct i o n s. U n fo r t u n ate l y t he e q ua tio n s wi ll b eco me e v en mo r e

134

co mp li ca ted t ha n t ho s e p r es e nted i n t hi s t he si s.

Appendix 1

Matlab code function z = squareroot( x, r, alpha) ralpha=r.*alpha; r2alpha2=ralpha.*ralpha; z=sqrt(1-r2alpha2.*(1-x.*x)); end T hi s f u nc tio n cal c ul ate s t he sq uar er o o t ter ms u s ed i n t he i nte g r al f u nct i o n func tio n y = In te gral (r, alp ha, be ta ) ralp ha= r.* alp ha ; if r alp ha< =1 low er= 0; else low er= sqr t( 1- 1./( ral pha .*r al ph a)); end y=qu adg k(@ p,l ow er ,1); fun cti on z= p( x) te mp1 =s qu arer oot (x, r,1 ); te mp2 =s qu arer oot (x, r,a lp ha ); z= abs (r .* x+te mp1 ).^ 2.* re al (tem p2) ; z= z./ ab s( temp 1+t emp 2.* be ta ).^2 ; end end T hi s f u nc tio n cal c ul ate s t he r ela ti v e tr a n s mi tted i nte n s it y fo r a d i f f u se i nc id e n t so u nd f ield i n t he aco u st ic c a s e a s a f u n ct io n o f t he r a t io r o f t h e fo r c ed wa v e n u mb e r to t he fr e el y p r o p a ga ti n g wa v e n u mb er a nd t he r at io s a lp h a a nd b et a. Alp ha i s t he wa v e n u mb e r i n med i u m o ne d iv id ed b y t he wa v e n u m b er i n me d i u m t wo . B e ta is t he i mp ed a n ce o f med i u m o ne d i vid ed b y t he i mp ed an ce o f med i u m t wo . func tio n r = tr an smit (al pha ,be ta ) lowe r=0 ; uppe r=2 ; numb er= 101 ; step =(u ppe r- l ow er )./( num ber -1) ; r=ze ros (nu mbe r, 3) ; r(1: num ber ,1) =l ow er:s tep :up per ; for m=1 :nu mbe r r(m ,2) =In te gr al(r (m, 1), alp ha ,b eta) ; if r(m ,1) <= 1 r( m,3 )= (1 +r(m ,1) )./ 2; els e r( m,3 )= (1 +1./ r(m ,1) )./ 2; end end r(1: num ber ,2) =r (1 :num ber ,2) ./I nt eg ral( 1,a lph a,b et a) ; plot (r( 1:n umb er ,1 ),r( 1:n umb er, 2) ,r (1:n umb er, 1), r( 1: numb er, 3)) end T hi s f u nc tio n cal c ul ate s an d gr ap h s t he o u tp ut o f t he i nt e gr al f u nc tio n at 1 0 1 eq ual l y sp ac ed va l ue s o f r fr o m 0 to 2 . T h e va l ue s ar e n o r mal ized b y t he val u e o f t he i nt e gr al f u nc tio n fo r r eq u al s 1 . T he r i s eq u al to t he fo r ced wa v e n u mb er d i v id e d b y t he wa v e n u mb e r i n med i u m o ne. T he va l ue s ar e ca lc u lat e d fo r t he gi ve n v al u es o f alp ha a nd b eta. Alp h a i s t he wa v e n u mb e r i n med i u m o ne d iv id ed b y t he wa v e n u m b er i n me d i u m t wo . B eta i s t h e i mp ed a nc e o f me d i u m o n e d i v i d ed b y t he i mp ed a nc e o f med i u m t wo .

135

136

func tio n t = ta u ( ps i, kap pa, c hi a, t het a) sin2 =si n(t het a) .^ 2; cost het a=s qrt (1 -s in2) ; chia 2=c hia .*c hi a; chia 2si n2= chi a2 .* sin2 ; kapp a2= kap pa. *k ap pa; x=ch ia2 sin 2./ ka pp a2; y1=s qrt (1- chi a2 si n2); y2=s qrt (1+ chi a2 si n2); y3=s qrt (1- x); y4=s qrt (1+ x); t=1i .*c hia .*c os th eta+ y2+ (1i .*y 1- y2 ).*( 1+c hia 2). /2 ; b=ps i.* (1i .*y 1- y2 )+ka ppa .*( 1i. *y 3- y4); t=t. /b; t=ab s(t ).^ 2.* re al (y3) ; end T hi s f u nc tio n cal c ul ate s t he r ela ti v e tr a n s mi tted i nte n s it y fo r t h e ca se o f a p i n ned j un ct io n b e t we e n t wo p a ne l s fo r a wa v e i n cid e nt at a n a n gl e o f t he ta to t he n o r ma l to th e j u n ct io n. T h e c alc u l atio n i s p er fo r med fo r t h e g i v e n v al u es o f p si, k a p p a a nd c h i a. W her e p si i s t h e r a tio o f t he b e nd i n g st i f f ne s s o f p lat e t wo mu l t ip l ied b y t he sq uar e o f th e wa ve n u mb er i n p l at e t wo , to t he b e nd i n g s ti f f n es s o f p la te o n e b y t h e sq u ar e o f t he wa v e n u mb er o f p la te o n e. T he k ap p a ter m i s t h e wa ve n u mb er i n p l at e t wo d i v i d ed b y th e wa ve n u mb er i n p l at e o ne. T h e t er m c h ia i s t he wa v e n u mb er o f a fo r ced i n cid e n t wa v e d i vid ed b y t h e wa ve n u mb er o f a fr e el y p r o p ag at i n g wa v e n u mb er in p la te o n e. func tio n c hia = p anel (ps i, kap pa , thet a) lowe r=0 ; uppe r=2 ; numb er= 101 ; step =(u ppe r-l ow er )./( num ber -1) ; chia =ze ros (nu mb er ,2); chia (1: num ber ,1 )= lowe r:s tep :up pe r; for m=1 :nu mbe r chi a(m ,2) =t au ( p si, ka ppa , ch ia(m ,1) , t het a) ; end chia (1: num ber ,2 )= chia (1: num ber ,2 ). /tau ( psi , k ap pa , 1, thet a); plot (ch ia( 1:n um be r,1) ,ch ia( 1:n um be r,2) ) end T hi s f u nc tio n gr ap hs t he no r ma l ized o utp u t va l u es o f ta u a s a f u n ct io n o f c hia . T he ter m c hia i s t h e wa v e n u mb e r o f a f o r ced i nc id e n t wa v e d i v id ed b y t he wav e n u mb er o f a fr ee l y p r o p a g at i n g wa ve n u mb er i n p l at e o ne. T he va l ue s ar e no r ma li z ed b y d i v id i n g b y t he o u tp ut o f ta u fo r th e fr e el y p r o p a ga ti n g c ase wh e n c hi a eq u al s 1 . func tio n y = in tt au(p si, ka ppa , ch ia) [m,n ]=s ize (ch ia ); for i= 1:m fo r j =1: n i f k app a>= ch ia (i,j ); upp er= pi. /2 ; e lse upp er= asi n( ka ppa. /ch ia( i,j )) ; e nd y =qu adg k(@ p, 0, uppe r); en d end fun cti on z=p (t he ta) z= tau ( psi , ka ppa, ch ia( i,j ), t heta ); end end

137

T hi s f u nc tio n cal c ul ate s t he r ela ti v e tr a n s mi tted i nte n s it y fo r t h e ca se o f a d i f f u se b end i n g wa ve f ie ld i n t h e fir st p a ne l b y i nte gr at i n g ta u o ve r t h e a n g le o f in cid e nce. func tio n c hia = i ntpa nel (ps i, ka pp a) lowe r=0 ; uppe r=2 ; numb er= 101 ; step =(u ppe r- l ow er )./( num ber -1) ; chia =ze ros (nu mb er ,2); chia (1: num ber ,1 )= lowe r:s tep :up pe r; for m=1 :nu mbe r chi a(m ,2) =i nt tau( psi , k app a, c hia( m,1 )); end chia (1: num ber ,2 )= 10*l og1 0(c hia (1 :n umbe r,2 )./ int ta u( psi, ka ppa , 1)); plot (ch ia( 1:n um be r,1) ,ch ia( 1:n um be r,2) ) end T hi s f u nc tio n gr ap hs t he no r ma l ized o utp u t o f i n tta u a s a f u n ct io n o f c h i a fo r 1 0 1 eq u al l y sp ac ed va l ue s fr o m 0 to 2 . T he no r ma l iz atio n i s p er for med b y d i vid i n g b y t he va l ue o f i n ta u fo r t he f r eel y p r o p a ga ti n g ca se whe n c hi a eq ua l s o ne. T h e ter m c h ia is th e wa ve n u mb er o f a fo r ced i n cid e nt wa ve d i v id ed b y t he wa v e n u mb e r o f a fr e el y p r o p ag at i n g wa v e n u mb er i n p la te o ne . function y = dblint( psi, kappa, r, neta ) a=quadgk(@p,0,pi/2); function z=p(phi) sinphi=sin(phi); chia=r.*sinphi; temp=(chia.^4-1).^2+neta.^2.*chia.^8; z=inttau(psi, kappa, chia).*sinphi./temp; end b=quadgk(@q,0,pi/2); function z=q(phi) sinphi=sin(phi); chia=r.*sinphi; temp=(chia.^4-1).^2+neta.^2.*chia.^8; z=(pi./2).*sinphi./temp; end y=a/b; end T hi s f u nc tio n cal c ul ate s t he r ela ti v e tr a n s mi tted i nte n s it y fo r t h e ca se whe n t he in cid e nt b e nd i n g wa ve f ield i s fo r c ed b y a d i f f u se f ie ld aco u s ti c wa v e. I t p er fo r ms a d o ub le i n te gr al ( o v er a n gl e o f i nc id e nc e a nd azi mu t h al a n gl e) b y i nt e gr a ti n g t he o u tp ut o f i nt ta u. T he va l ue s ar e cal c ula ted f o r t h e gi v e n an g le s o f t he r at io s p si, ka p p a, r a nd ne ta ( t he d a mp i n g lo s s f acto r o f t he fir s t p a ne l) . W he r e p si i s t h e r a tio o f t he b e nd i n g st i f f ne s s o f p la te t wo m ul tip li ed b y t he sq uar e o f t he wa ve n u mb er i n p l ate t wo , to t he b end i n g st i f f ne s s o f p la t e o ne b y t he sq uar e o f t he wa v e n u mb er o f p l ate o n e. T he kap p a t er m is t he wa v e n u mb e r i n p la te t wo d i vi d ed b y t he wa v e n u mb e r i n p l ate o ne. T he r i s eq u al to t he fo r ced wa v e n u mb er d i v id e d b y t he wa v e n u mb er i n med i u m o n e. T he ne ta ter m i s t h e i n s it u d a mp i n g lo ss f ac to r o f p la te o ne , i t ha s t he v al ue 0 .0 0 3 . func tio n f = gr ap hdbl int (ps i, ka pp a, n eta ) lowe r=0 ; uppe r=2 ; numb er= 101 ; step =(u ppe r- l ow er )./( num ber -1) ; f=ze ros (nu mbe r, 2) ; f(1: num ber ,1) =l ow er:s tep :up per ; for m=1 :nu mbe r

138

f(m ,2) =db li nt (psi , k app a, sq rt (f(m ,1) ), net a) ; end f(1: num ber ,2) =1 0. *log 10( f(1 :nu mb er ,2). /db lin t(p si , kapp a, 1, neta )); plot (f( 1:n umb er ,1 ),f( 1:n umb er, 2) ) end T hi s f u nc tio n no r ma l ize s t h e o utp u t o f d b li n t b y d i vid i n g i t b y t he o u tp u t o f d b l i nt fo r th e c as e wh e n r eq u al s o ne. T he r i s eq ua l to t h e fo r ced wa ve n u mb er d i vid ed b y t h e wa v e n u mb er i n med i u m o n e.

References

Brunskog, J., & Chung, H. (2011). "Non-diffuseness of vibration fields in

ribbed plates". Journal of the Acoustical Society of America, 129,

1336-1343.

CEN. (2000a). "CEN EN12354-1:2000(E)-Building acoustics-Estimation of

acoustic performance in buildings from the performance of elements-

Part 1: Airborne sound between rooms" (pp. 1- 20). Brussels, Belgium:

European Committee for Standardization.

CEN. (2000b). "CEN EN12354 -2:2000(E)-Building acoustics-Estimation of

acoustic performance in buildings from the performance of elements-

Part 2: Impact sound insulation between rooms" (pp. 1-20). Brussels,

Belgium: European Committee for Standardization.

CEN. (2000c). "CEN EN12354-3:2000(E)-Building acoustics-Estimation of

acoustic performance in buildings from the performance of elements-

Part 3: Airborne sound insulation against outdoor sound" (pp. 1- 20).

Brussels, Belgium: European Committee for Standardization.

CEN. (2000d). "CEN EN12354-4:2000(E)-Building acoustics-Estimation of

acoustic performance in buildings from the performance of elements-

Part 4: Transmission of indoor sound to the outside" (pp. 1-20).

Brussels, Belgium: European Committee for Standardization.

CEN. (2000e). "CEN EN12354-5:2000(E)-Building acoustics-Estimation of

acoustic performance in buildings from the performance of elements-

Part 5: Sound levels due to the service equipment" (pp. 1-20). Brussels,

Belgium: European Committiee for Standardization.

139

CEN. (2000f). "CEN EN12354-6:2000(E)-Building acoustics-Estimation of

acoustic performance in buildings from the performance of elements-

Part 6: Sound absoption in enclosed spaces" (pp. 1- 20). Brussels,

Belgium: European Committee for Standardization.

Cremer, L, Heckl, M, & Petersson, B.A.T. (2005). Structure -Borne Sound.

Berlin, Germany: Springer.

Crispin, C., Ingelaere, B., Van Damme, M., & Wuyts, D. (2006). "The

vibration reduction index Kij: Laboratory measurements for rigid

junctions and for junctions with flexible interlayers,". Journal of

Building Acoustics, 13, 99 -112.

CSTC. (2008). Détails de construction pour améliorer l’isolationaux bruits

aériens pour les constructions légères: Laboratoire acoustique, Centre

Scientifique et Technique de la Construction (CSTC), Limelette,

Belgium.

Dav y, John. (2009). The directivit y of the sound radiation from panels and

openings. Journal of the Acoustical Society of America, 125, 3795 -

3805.

Davy, John.L, Mahn, J.P, Guigou- Carter, Catherine, & Villot, Michel. (2012).

The prediction of flanking sound transmission below the critical

frequency. Journal of the Acoustical Society of America, 132(4), 12.

doi: 10.1121/1.4746945

Gerretsen, E. (1979). Calculation of the sound transmission between

dwellings by partitions and flanking structures. Applied Acoustics , 413-

433.

140

Gerretsen, E. (1986). Calculation of Airborne and Impact Sound Insulation

Between Dwellings. Applied Acoustics, 19, 245-264.

Gerretsen, Edd y. (2007). Some aspects to improve sound insulation

prediction models for lightweight elements. Paper presented at the

Inter-noise 2007

Istanbul, Turkey.

Gradshteyn, I.S, & Ryzhik, I.M. (1980). Table of Integrals, Series, and

Products (A. Jeffrey Ed. 4th edition ed.). United States of America:

Academic Press.

Guigou-Carter, Catherine, Villot, Michel, & Roland, Wetta. (2006).

Prediction Method Adapted to Wood Frame Lightweight Constructions.

Journal of Building Acoustics, 13, 173 -188.

ISO. (2005a). "ISO 15712-1:2005(E) Building acoustics-Estimation of

acoustic performance of buildings from the performance of elements-

Part 1: Airborne sound insulation between rooms" (pp. 1-59). Geneva,

Switzerland: International Organization for standardization.

ISO. (2005b). "ISO 15712-2:2005(E) Building acoustics-Estimation of

acoustic performance of buildings from the performance of elements-

Part 2: Impact sound insulation between rooms" (pp. 30). Geneva,

Switzerland: International Organization for standardization.

ISO. (2005c). "ISO 15712-3:2005(E) Building acoustics-Estimation of the

acoustic performance of buildings from the performance of elements-

Part 3: Airborne sound insulation against outdoor sound" (pp. 1- 31).

Geneva, Switzerland: International Organization for Standardization.

141

ISO. (2005d). "ISO 15712-4:2005(E) Building acoustics-Estimation of

acoustic performance of buildings from the performance of elements-

Part 4: Transmission of indoor sound to the outside (pp. 1-28). Geneva,

Switzerland: International organization for standardization

ISO. (2006). "ISO 10848-2:2006(E)-Acoustics-Laboratory measurement of

the flanking transmission of airborne and impact sound between

adjoining rooms-Part2: Application to light elements when the junction

has a substantial influence- Technical Corrigendum 1". Geneva,

Switzerland: International Organization for Standardization.

ISO. (2006a). "ISO 10848-1:2006(E)-Acoustics-Laboratory measurement of

the flanking transmission of airborne and impact sound between

adjoining rooms-Part1:Frame document" (pp. 1- 30). Geneva,

Switzerland: International Organization for Standardization.

ISO. (2006b). "ISO 10848-2:2006(E)-Acoustics-Laboratory measurement of

the flanking transmission of airborne and impact sound between

adjoining rooms- Part2: Application to light elements when the

junction has a small influence" (pp. 1 -16). Geneva, Switzerland:

International Organization for Standardization.

Mahn, Jeffrey P. (2008). Prediction of Flanking Noise Transmission in

Lightweight Building Constructions: A Theoretical and Experimental

Evaluation of the Application of EN12354- 1. (PhD Research Thesis),

Universit y of Canterbury, Christchurch, New Zealand.

Nightingale, T.R.T. (1995). Application of the CEN draft building acoustics

prediction model to a lightweight double leaf construction. Applied

Acoustics, 46, 265 -284.

142

Nightingale, T.R.T, & Bosmans, I. (2003). Expressions for first-order

flanking paths in homogeneous isotropic and lightl y damped buildings.

Acta Acoustica, 89, 110-122.

Pavasovic, Vladimir. (2006). The radiation of sound from surfaces at grazing

angles of incidence (Master of Applied Science Research), RMIT

Universit y, Melbourne. Retrieved from

http://adt.lib.rmit.edu.au/adt/public/adt-VIT20060911.115939

Villot, Michel. (2002). Modeling flanking transmissions in lightweight

constructions. Paper presented at the Forum Acustica 2002, Seville

Spain.

Villot, Michel, & Guigou-Carter, Catherine. (2000). Contribution of forced

and resonant vibration in sound transmission through partitions and in

vibration transmission through plate junctions. Paper presented at the

NOVEM 2000. Noise and Vibration, Emerging Methods, Lyon, France.

143