- Trang Chủ
- Địa Lý
- Normal mode vs parabolic equation and their application in Tonkin Gulf
Xem mẫu
- P-ISSN 1859-3585 E-ISSN 2615-9615 SCIENCE - TECHNOLOGY
NORMAL MODE vs PARABOLIC EQUATION
AND THEIR APPLICATION IN TONKIN GULF
MODE CHUẨN SO VỚI PHƯƠNG TRÌNH PARABOLIC VÀ ÁP DỤNG VÀO VỊNH BẮC BỘ
Tran Cao Quyen
to an eigenfunction (mode shape) and an eigenvalue
ABSTRACT
(horizontal propagation constant).
Normal Mode (NM) and Parabolic Equation (PE) have been used widely by
Third, the PE method is introduced firstly by Tappert [5]
Underwater Acoustic Community due to their effectiveness. In this paper, author
investigates NM and PE in term of their mathematical approach as well as their and is considered the modern method since it applied for
computation. Further, Tonkin Gulf has been modeled and simulated using both the medium which has layers separated unclearly [5-8]. The
of NM and PE. The simulation results show that there are the agreement and the advantages of parabolic method consists of using a source
reliability between both methodologies. with one-way propagation, applying for range
dependence, as well as performing in the medium which is
Keywords: SONAR, Parabolic Equation, Normal Mode, Tonkin Gulf. not required exactly layered separation.
TÓM TẮT In this paper we investigate NM and PE in term of their
Phương pháp Mode chuẩn và phương trình Parabolic được dùng rộng rãi mathematical approach as well as their computation.
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
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
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,
chuẩn và phương trình Parabolic. Các kết quả mô phỏng cho thấy có sự đồng
z , the range, r (5 10)z , the parabolic algorithm
nhất và tin cậy giữa hai phương pháp trên. 4
Từ khóa: SONAR, Phương trình Parabolic, Mode chuẩn, Vịnh Bắc Bộ. converged fast. The achieved results of transmission loss
factors (TLs) shows that there is a consistent agreement of
TLs between NM and PE. The computation of PE is slightly
Faculty of Electronics and Telecommunications,
more than NM.
VNU University of Engineering and Technology
Email: quyentc@vnu.edu.vn The rest of the paper is organized as follows. Section 2
Received: 01 June 2019 presents the mathematical representations of NM and the
Revised: 21 June 2019 PE. We evaluate the NM and PE model in Tonkin gulf in
section 3. Section 4 is our discussions. We conclude the
Accepted: 15 August 2019
paper in section 5.
2. NORMAL MODE AND PARABOLIC EQUATION
1. INTRODUCTION 2.1. The Normal Mode
First, sound propagation in ocean waveguide is Staring from Helmholtz equation in two dimensions
investigated for a long time since its important role in with sound speed c and density ρ depending only on depth
SONAR (Sound navigation and ranging) techniques. As we z [1]:
known, there are numerous ways of the underwater sound
modeling which appeared in time order namely ray, normal 1 ψ 1 ψ ω2 δ(r)δ(z zs )
(r ) ρ(z) ( ) 2
ψ (1)
mode (NM) and parabolic equation (PE) [1]. r r r z ρ(z) z c(z) 2πr
Second, the NM is introduced the first time where zs is source depth, z is depth and r is distance.
independently by Pekeris [2] and Ide [3] and then is
Using separation of variables (r, z) (r). V(z) , we
classified by Williams [4]. After some decades of
development of the NM, it becomes one of the most obtain the modal equation
powerful approach of ocean acoustic computation. The d 1 dVm (z) 2
best idea of NM is that it considers an acoustic pressure as (z) [ ][ k rm2 ]Vm (z) 0 (2)
dz (z) dz c(z)2
an infinite number of modes which are similar to those
obtained from a vibrating string. Each mode corresponds with the boundary conditions such as
No. 53.2019 ● Journal of SCIENCE & TECHNOLOGY 3
- KHOA HỌC CÔNG NGHỆ P-ISSN 1859-3585 E-ISSN 2615-9615
dV 1
V ( 0) 0, z D 0 (3) Vrr Vr k 02 V 0 (13)
dz r
The former condition implies a pressure release surface The root of (13) is a Hankel function with its
and the latter condition is from a perfect rigid bottom. The approximation as
modal equation that is the center of the NM, has an infinite 2 i (k0r π4 )
number of modes. Each mode represents by a mode Vr0 H10 (k 0 r ) e (14)
amplitude Vm(z) and a horizontal propagation constant krm. πk 0 r
Vm(z) and krm are also called eigenfunction and eigenvalue After some manipulations, (12) becomes
respectively
2ik 0 r zz k 20 (n2 1) 0 (15),
Noting that the modes are orthonormal, i.e.,
D
i.e. a parabolic equation.
Vm (z)Vn (z)
0 ρ(z) dz 0, m n Taking the Fourier transform both side of (15) in z
(4) domain obtained
D
Vm (z)2 2ik 0 r k z2 k 20 (n2 1) 0 (16)
0 ρ(z) dz 1
Rewrite (16) in simpler form as
Since the modes forms a complete set, the pressure can
k 20 (n2 1) k 2z
represents as a sum of the normal modes r 0 (17)
2ik 0
ψ(r , z ) (r ) Vm ( z ) (5)
m 1
m Thus, from [9] we have
k 20 (n2 1)k z2
After some manipulations, we obtain 2ik 0
(rr0 )
(r,k z ) (r0 ,k z )e (18)
i
ψ(r , z) Vm (z s )H10 (k rmr ) (6) where (r0 , k z ) is the initial value of the source.
4ρ(zs )
Taking the Inverse Fourier transform both side of (18)
where H10 is the Hankel function of the first kind.
obtained
Substitute (6) back to (5) we have k0 2 irk 2z
i (n 1) r
2ik 0
i (r, z) e 2
(r ,k )e eik z z dk z (19)
ψ(r , z) Vm (z s ) Vm (z)H10 (krmr)
4ρ(zs ) m 1
(7)
0 z
where r r r0 .
Finally, using the asymptotic approximation of the
Hankel function, the pressure can be written as Finally, we arrived
2
i
eikrmr i
k0 2
(n 1) r irk z
iπ / 4
ψ(r , z) e V m (z s ) Vm (z) (8) (r,z) e 2
e 2ik0 (r0 ,z)
1
(20)
ρ(z s ) 8πr m 1 k rm
2.2. The Parabolic Equation This form is called Split-Step Fourier transform.
Starting from the Helmholtz equation in the most 3. SIMULATION RESULTS
general form [1]
3.1. The acoustic and noise source
2 ψ k 02 (n2 1)ψ 0 (9)
The point source with the center frequency of 250Hz and
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
wavenumber at the acoustic source. the receiver is placed at the same transmitter’s depth; the
In cylindrical coordinate, (1) becomes noise source is Gaussian and the SNR level of 3dB.
1 3.2. Medium parameters
ψrr ψr ψ zz k 20 (n2 1)ψ 0 (10) Table 1. The medium parameters
r
in which the subscripts denote the order of derivative. Paremeter Value
From the assumption of Tappert [5-6], ψ is defined as Ocean depth 100m
ψ(r , z) (r , z) V (r ) (11) Sound speed in winter c(z) = 1500 + 0.3z (m/s)
where z denotes depth and r denotes distance. Bottom Sand, ρ1 = 2000 kg/m3
Thus (10) becomes the system of equations as follows c1 = 1700 m/s
In this simulation, Tonkin gulf is used as Pekeris
1 2
rr Vr r zz k 20 (n2 1) 0 (4) and (12) waveguide model with its sound velocity which is
r V measured from [10]. Thuc was carried out many sound
4 Tạp chí KHOA HỌC & CÔNG NGHỆ ● Số 53.2019
- P-ISSN 1859-3585 E-ISSN 2615-9615 SCIENCE - TECHNOLOGY
speed measurements which were reported in his In the second case (when SNR of 3dB), from Figure 2,
monograph. On the basis of Thuc’s results, the medium the agreement of TLs of both methods is more consistent
parameters of Tolkin gulf are given in the Table 1. since the signal level in this case is higher than the noise
In Table1, c denotes sound velocity whereas ρ indicates level and it is compensated for a long range transmission.
medium density. The computation of PE is slightly more than NM (it is not
3.3. Simulation Results shown here).
The transmission loss factors (TLs) of NM and PE are 5. CONCLUSIONS
shown in Figure 1 and 2. In this paper, the rigorous mathematical analyses of NM
and PE are presented. The idea behind NM is vibrating of
modes along depth axis and behind PE are one-way
propagation and using Split-Step Fourier transform.
Further, in conditions of this simulation, there is a
consistent agreement of TLs between NM and PE in both
noise and noiseless cases.
ACKNOWLEDGEMENT
This work has been supported by Vietnam National
University, Hanoi (VNUH), under Project No. QG.17.40.
REFERENCES
[1]. F. B. Jensen at al, 2011. Computational Ocean Acoustics. Sringer.
[2]. C. L. Pekeris, 1948. Theory of propagation of explosive sound in shallow
Figure 1. Transmission loss factors of NM and PE with range up to 15km,
water. Geol. Soc. Am. Mem. 27
noiseless case
[3]. J. M. Ide, R. F. Post, W.J. Fry, 1947. The propagation of underwater sound
at low frequencies as a function of the acoustic properties of the bottom. J. Acoust.
Soc. Am. 19 (283).
[4]. A. O. Williams, 1970. Normal mode methods in propagation of
underwater sound. In Underwater Acoustics, ed. by R.W.B Stephens, Wiley-
Interscience, New York.
[5]. F. D. Tappert, 1977. The parabolic approximation method. Wave
propagation in underwater acoustics, pp.224-287, Springer, New York.
[6]. D. Lee, 1984. The state of the art parabolic equation approximation as
applied to underwater acoustic propagation with discussion on intensive
computations. J. Acoutic. Soc. Am, 76.
[7]. E. C. Young and D. Lee, 1988. A model of underwater acoustic
propagation. Math. Comput. Modelling, 1, pp.58-61.
[8]. J. Soneson and Y. Lin, 2017. Validation of a wide angle parabolic model
for shallow focus ultrasound transducer. J. Acoutic. Soc. Am, 142.
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
SNR = 3dB edition, Jones and Bartlett Learing, LCC, ISBN: 978-1-4496-9172-1.
4. DISCUSSIONS [10]. Pham Van Thuc, 2011. Ocean Sound and Sound Field in South East Asia
From Figure 1 and Figure 2 we can see clearly that the Sea. National and Science Technology Express.
TLs of both NM and PE with range up to 15km far from the
acoustic source. In the conditions of this simulation, this
TLs are stable after hundreds of simulations. Further, there
is the agreement of TLs between NM and PE. THÔNG TIN TÁC GIẢ
In the first case (noiseless case), from Figure 1, the TL of Trần Cao Quyền
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
NM. It is basically, could be thought of the nature of range
dependence of PE approach.
No. 53.2019 ● Journal of SCIENCE & TECHNOLOGY 5
nguon tai.lieu . vn
