Normal mode vs parabolic equation and their application in Tonkin Gulf

P-ISSN 1859-3585 E-ISSN 2615-9615 SCIENCE - TECHNOLOGY<br /> <br /> <br /> <br /> <br /> NORMAL MODE vs PARABOLIC EQUATION<br /> AND THEIR APPLICATION IN TONKIN GULF<br /> MODE CHUẨN SO VỚI PHƯƠNG TRÌNH PARABOLIC VÀ ÁP DỤNG VÀO VỊNH BẮC BỘ<br /> <br /> Tran Cao Quyen<br /> <br /> to an eigenfunction (mode shape) and an eigenvalue<br /> ABSTRACT<br /> (horizontal propagation constant).<br /> Normal Mode (NM) and Parabolic Equation (PE) have been used widely by<br /> Third, the PE method is introduced firstly by Tappert [5]<br /> Underwater Acoustic Community due to their effectiveness. In this paper, author<br /> investigates NM and PE in term of their mathematical approach as well as their and is considered the modern method since it applied for<br /> computation. Further, Tonkin Gulf has been modeled and simulated using both the medium which has layers separated unclearly [5-8]. The<br /> of NM and PE. The simulation results show that there are the agreement and the advantages of parabolic method consists of using a source<br /> reliability between both methodologies. with one-way propagation, applying for range<br /> dependence, as well as performing in the medium which is<br /> Keywords: SONAR, Parabolic Equation, Normal Mode, Tonkin Gulf. not required exactly layered separation.<br /> TÓM TẮT In this paper we investigate NM and PE in term of their<br /> Phương pháp Mode chuẩn và phương trình Parabolic được dùng rộng rãi mathematical approach as well as their computation.<br /> trong cộng đồng thủy âm vì sự hiệu quả của chúng. Trong bài báo này, tác giả Besides, Tonkin Gulf has been modeled and simulated<br /> nghiên cứu mode chuẩn và phương trình Parabolic ở khía cạnh toán học và tốc độ using not only NM but also PE. The obtained results show<br /> tính toán. Hơn nữa, Vịnh Bắc Bộ được mô hình hóa và mô phỏng dùng cả mode that when we divided the grid small enough (the depth,<br /> chuẩn và phương trình Parabolic. Các kết quả mô phỏng cho thấy có sự đồng <br /> z  , the range, r  (5  10)z , the parabolic algorithm<br /> nhất và tin cậy giữa hai phương pháp trên. 4<br /> Từ khóa: SONAR, Phương trình Parabolic, Mode chuẩn, Vịnh Bắc Bộ. converged fast. The achieved results of transmission loss<br /> factors (TLs) shows that there is a consistent agreement of<br /> TLs between NM and PE. The computation of PE is slightly<br /> Faculty of Electronics and Telecommunications,<br /> more than NM.<br /> VNU University of Engineering and Technology<br /> Email: quyentc@vnu.edu.vn The rest of the paper is organized as follows. Section 2<br /> Received: 01 June 2019 presents the mathematical representations of NM and the<br /> Revised: 21 June 2019 PE. We evaluate the NM and PE model in Tonkin gulf in<br /> section 3. Section 4 is our discussions. We conclude the<br /> Accepted: 15 August 2019<br /> paper in section 5.<br /> 2. NORMAL MODE AND PARABOLIC EQUATION<br /> 1. INTRODUCTION 2.1. The Normal Mode<br /> First, sound propagation in ocean waveguide is Staring from Helmholtz equation in two dimensions<br /> investigated for a long time since its important role in with sound speed c and density ρ depending only on depth<br /> SONAR (Sound navigation and ranging) techniques. As we z [1]:<br /> known, there are numerous ways of the underwater sound<br /> modeling which appeared in time order namely ray, normal 1  ψ  1 ψ ω2 δ(r)δ(z  zs )<br /> (r )  ρ(z) ( ) 2<br /> ψ (1)<br /> mode (NM) and parabolic equation (PE) [1]. r r r z ρ(z) z c(z) 2πr<br /> Second, the NM is introduced the first time where zs is source depth, z is depth and r is distance.<br /> independently by Pekeris [2] and Ide [3] and then is<br /> Using separation of variables  (r, z)  (r). V(z) , we<br /> classified by Williams [4]. After some decades of<br /> development of the NM, it becomes one of the most obtain the modal equation<br /> powerful approach of ocean acoustic computation. The d 1 dVm (z) 2<br /> best idea of NM is that it considers an acoustic pressure as (z) [ ][  k rm2 ]Vm (z)  0 (2)<br /> dz (z) dz c(z)2<br /> an infinite number of modes which are similar to those<br /> obtained from a vibrating string. Each mode corresponds with the boundary conditions such as<br /> <br /> <br /> <br /> No. 53.2019 ● Journal of SCIENCE & TECHNOLOGY 3<br />  KHOA HỌC CÔNG NGHỆ P-ISSN 1859-3585 E-ISSN 2615-9615<br /> <br /> dV 1<br /> V ( 0)  0, z D 0 (3) Vrr  Vr  k 02 V  0 (13)<br /> dz r<br /> The former condition implies a pressure release surface The root of (13) is a Hankel function with its<br /> and the latter condition is from a perfect rigid bottom. The approximation as<br /> modal equation that is the center of the NM, has an infinite 2 i (k0r  π4 )<br /> number of modes. Each mode represents by a mode Vr0  H10 (k 0 r )  e (14)<br /> amplitude Vm(z) and a horizontal propagation constant krm. πk 0 r<br /> Vm(z) and krm are also called eigenfunction and eigenvalue After some manipulations, (12) becomes<br /> respectively<br /> 2ik 0  r   zz  k 20 (n2  1)  0 (15),<br /> Noting that the modes are orthonormal, i.e.,<br /> D<br /> i.e. a parabolic equation.<br /> Vm (z)Vn (z)<br /> 0 ρ(z) dz  0, m  n Taking the Fourier transform both side of (15) in z<br /> (4) domain obtained<br /> D<br /> Vm (z)2 2ik 0  r  k z2   k 20 (n2  1)  0 (16)<br /> 0 ρ(z) dz  1<br /> Rewrite (16) in simpler form as<br /> Since the modes forms a complete set, the pressure can<br /> k 20 (n2  1)  k 2z<br /> represents as a sum of the normal modes r  0 (17)<br /> <br /> 2ik 0<br /> ψ(r , z )   (r ) Vm ( z ) (5)<br /> m 1<br /> m Thus, from [9] we have<br />  k 20 (n2 1)k z2<br /> After some manipulations, we obtain 2ik 0<br /> (rr0 )<br /> (r,k z )   (r0 ,k z )e (18)<br /> i<br /> ψ(r , z)  Vm (z s )H10 (k rmr ) (6) where (r0 , k z ) is the initial value of the source.<br /> 4ρ(zs )<br /> Taking the Inverse Fourier transform both side of (18)<br /> where H10 is the Hankel function of the first kind.<br /> obtained<br /> Substitute (6) back to (5) we have k0 2   irk 2z<br /> i (n 1) r<br /> 2ik 0<br /> i  (r, z)  e 2<br />  (r ,k )e eik z z dk z (19)<br /> ψ(r , z)   Vm (z s ) Vm (z)H10 (krmr)<br /> 4ρ(zs ) m 1<br /> (7) <br /> 0 z<br /> <br /> <br /> where r  r  r0 .<br /> Finally, using the asymptotic approximation of the<br /> Hankel function, the pressure can be written as Finally, we arrived<br /> 2<br /> <br /> i <br /> eikrmr i<br /> k0 2<br /> (n 1) r  irk z <br />  iπ / 4<br /> ψ(r , z)  e V m (z s ) Vm (z) (8) (r,z)  e 2<br />  e 2ik0 (r0 ,z)<br /> 1<br /> (20)<br /> ρ(z s ) 8πr m 1 k rm  <br /> 2.2. The Parabolic Equation This form is called Split-Step Fourier transform.<br /> Starting from the Helmholtz equation in the most 3. SIMULATION RESULTS<br /> general form [1]<br /> 3.1. The acoustic and noise source<br />  2 ψ  k 02 (n2  1)ψ  0 (9)<br /> The point source with the center frequency of 250Hz and<br /> where n is the refraction index of the medium and k0 is the the depth of 99m is used in this simulation. We assume that<br /> wavenumber at the acoustic source. the receiver is placed at the same transmitter’s depth; the<br /> In cylindrical coordinate, (1) becomes noise source is Gaussian and the SNR level of 3dB.<br /> 1 3.2. Medium parameters<br /> ψrr  ψr  ψ zz  k 20 (n2  1)ψ  0 (10) Table 1. The medium parameters<br /> r<br /> in which the subscripts denote the order of derivative. Paremeter Value<br /> From the assumption of Tappert [5-6], ψ is defined as Ocean depth 100m<br /> ψ(r , z)  (r , z) V (r ) (11) Sound speed in winter c(z) = 1500 + 0.3z (m/s)<br /> where z denotes depth and r denotes distance. Bottom Sand, ρ1 = 2000 kg/m3<br /> Thus (10) becomes the system of equations as follows c1 = 1700 m/s<br /> In this simulation, Tonkin gulf is used as Pekeris<br /> 1 2 <br />  rr    Vr   r   zz  k 20 (n2  1)   0 (4) and (12) waveguide model with its sound velocity which is<br /> r V  measured from [10]. Thuc was carried out many sound<br /> <br /> <br /> <br /> 4 Tạp chí KHOA HỌC & CÔNG NGHỆ ● Số 53.2019<br /> P-ISSN 1859-3585 E-ISSN 2615-9615 SCIENCE - TECHNOLOGY<br /> <br /> speed measurements which were reported in his In the second case (when SNR of 3dB), from Figure 2,<br /> monograph. On the basis of Thuc’s results, the medium the agreement of TLs of both methods is more consistent<br /> parameters of Tolkin gulf are given in the Table 1. since the signal level in this case is higher than the noise<br /> In Table1, c denotes sound velocity whereas ρ indicates level and it is compensated for a long range transmission.<br /> medium density. The computation of PE is slightly more than NM (it is not<br /> 3.3. Simulation Results shown here).<br /> The transmission loss factors (TLs) of NM and PE are 5. CONCLUSIONS<br /> shown in Figure 1 and 2. In this paper, the rigorous mathematical analyses of NM<br /> and PE are presented. The idea behind NM is vibrating of<br /> modes along depth axis and behind PE are one-way<br /> propagation and using Split-Step Fourier transform.<br /> Further, in conditions of this simulation, there is a<br /> consistent agreement of TLs between NM and PE in both<br /> noise and noiseless cases.<br /> <br /> <br /> ACKNOWLEDGEMENT<br /> This work has been supported by Vietnam National<br /> University, Hanoi (VNUH), under Project No. QG.17.40.<br /> <br /> <br /> <br /> REFERENCES<br /> [1]. F. B. Jensen at al, 2011. Computational Ocean Acoustics. Sringer.<br /> [2]. C. L. Pekeris, 1948. Theory of propagation of explosive sound in shallow<br /> Figure 1. Transmission loss factors of NM and PE with range up to 15km,<br /> water. Geol. Soc. Am. Mem. 27<br /> noiseless case<br /> [3]. J. M. Ide, R. F. Post, W.J. Fry, 1947. The propagation of underwater sound<br /> at low frequencies as a function of the acoustic properties of the bottom. J. Acoust.<br /> Soc. Am. 19 (283).<br /> [4]. A. O. Williams, 1970. Normal mode methods in propagation of<br /> underwater sound. In Underwater Acoustics, ed. by R.W.B Stephens, Wiley-<br /> Interscience, New York.<br /> [5]. F. D. Tappert, 1977. The parabolic approximation method. Wave<br /> propagation in underwater acoustics, pp.224-287, Springer, New York.<br /> [6]. D. Lee, 1984. The state of the art parabolic equation approximation as<br /> applied to underwater acoustic propagation with discussion on intensive<br /> computations. J. Acoutic. Soc. Am, 76.<br /> [7]. E. C. Young and D. Lee, 1988. A model of underwater acoustic<br /> propagation. Math. Comput. Modelling, 1, pp.58-61.<br /> [8]. J. Soneson and Y. Lin, 2017. Validation of a wide angle parabolic model<br /> for shallow focus ultrasound transducer. J. Acoutic. Soc. Am, 142.<br /> Figure 2. Transmission loss factors of NM and PE with range up to 15km, [9]. D. G. Zill and W. S. Wright. Advanced Engineering Mathematics. Fifth<br /> SNR = 3dB edition, Jones and Bartlett Learing, LCC, ISBN: 978-1-4496-9172-1.<br /> 4. DISCUSSIONS [10]. Pham Van Thuc, 2011. Ocean Sound and Sound Field in South East Asia<br /> From Figure 1 and Figure 2 we can see clearly that the Sea. National and Science Technology Express.<br /> TLs of both NM and PE with range up to 15km far from the<br /> acoustic source. In the conditions of this simulation, this<br /> TLs are stable after hundreds of simulations. Further, there<br /> is the agreement of TLs between NM and PE. THÔNG TIN TÁC GIẢ<br /> In the first case (noiseless case), from Figure 1, the TL of Trần Cao Quyền<br /> PE seems reducing to distance more slightly than the TL of Khoa Điện tử - Viễn thông, Trường Đại học Công nghệ, Đại học Quốc gia Hà Nội<br /> NM. It is basically, could be thought of the nature of range<br /> dependence of PE approach.<br /> <br /> <br /> <br /> No. 53.2019 ● Journal of SCIENCE & TECHNOLOGY 5<br />