NEAR FIELD REPRESENTATIONS OF THE ACOUSTIC GREEN S FUNCTION IN A SHALLOW OCEAN WITH FLUID-LIKE SEABED

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1 Georgian Mathematical Journal Volume , Number, NEAR FIELD REPRESENTATIONS OF THE ACOUSTIC GREEN S FUNCTION IN A SHALLOW OCEAN WITH FLUID-LIKE SEAED ROERT GILERT AND MIAO-JUNG OU Abstract. In this paper, the near-field approximation of the acoustic Green s function in a two-layer waveguide is constructed by using a variation of the method of Ahluwalia and Keller []. The relation between the constructed multiple-scattering representation suitable for near-field and the Hankel transform representation suitable for mid-range is also discussed in this paper. The construction scheme presented in this paper can be generalized for an N-layer waveguide Mathematics Subject Classification: 34E05, 35A35, 35G5, 35J05, 4A60, 74H0. Key words and phrases: Layered waveguide, acoustic Green s function, Helmholtz equation, near-field approximation, multiple-scattering representation.. Introduction In this paper, we show how to construct acoustic Green s functions for a water column with fluid-like basement atop a rigid rockbed. The main interest is to obtain a representation of the acoustic Green s function which is valid in the near field. It is hoped with a near field Green s function, we will be able to solve the unknown object, inverse problem for an object submerged in an ocean with a layer of fluid-like sediment atop rigid rock, much in the same way as in [5], [3] for the completely reflecting seabed. In these papers, the method of integral equations is used, hence the need for an accurate near-field approximation. An alternate method which could also make use of an accurate near-field Green s function is the method of complete families of solutions. The idea of this method is discussed in [2] by ergman and Schiffer, but goes back much farther. This method has been used effectively by Angell, Kleinman, Lesselier, and Rozier [6, 7] who used the fact that the Green s function evaluated on a dense set of points in the interior of the bounding curve of the unknown object provides a complete family [2]. Our approach makes use of an idea put forth in a paper by Ahluwalia and Keller []. Their method generalizes the method of images representation of the Green s function for a homogeneous shallow ocean with reflecting seabed with ISSN X / $8.00 / c Heldermann Verlag

2 0 R. GILERT AND MIAO-JUNG OU finite depth h, namely Gx, x 0 = n e ik x x 0 2 +y y 0 2 +z z 0 +2nh 2 n= x x0 2 + y y z z 0 + 2nh 2 e ik x x 0 2 +y y 0 2 +z+z 0 +2nh 2, x x0 2 + y y z + z 0 + 2nh 2 where x 0, y 0, z 0 is the source location and k is the wave number, which is defined as the ratio of acoustic frequency and the acoustic wave speed in the ocean. This representation is called the ray representation of the Green s function. This method can be further generalized to a stratified ocean and the resulting representation is called the multiple scattering expansion of the Green s function []. Although it is well known that the ray/multiple scattering representation is mathematically equivalent to the modal expansion [], actual computations with each of these representations can return different results due to truncation errors. This is especially true for the two extreme cases, i.e. when the point of measurement is very close to or far away from the acoustic source point. We are interested in the model of an ocean with a fluid-like seabed. It is modeled as a two-layered waveguide. Within each layer, the acoustic property can be smoothly varied with respect to depth. On the interface between the two layers, there is a jump in the refractive index nz := cz/c 0 and continuity of acoustic displacement and acoustic pressure are imposed there as transmission conditions. The modal expansion of the Green s function for a waveguide of two homogeneous layers is already given in [4]. In this paper, we construct the multiple scattering expansion of the Green s function. The relation between the modal expansion and the multiple scattering expansion of the Green s function is also discussed. z ocean x 0,0,z 0 z=0 source y z=d fluid like seabed rigid seabed z=h Figure. Schematic description of the Pekeris waveguide

3 NEAR FIELD REPRESENTATIONS OF THE ACOUSTIC GREEN S FUNCTION.. Construction of the near-field representation of the Green s function. We consider a waveguide, which consists of a finite, uniform depth water column lying over a fluid-like seabed as shown in Figure. The waveguide can be stratified, i.e. the acoustic property is not necessarily homogeneous in each layer. We shall use the subscript for the material parameters in the water column and the subscript 2 for those in the basement; whereas for the ease of notation, we will use p and p 2 to denote the acoustic pressure in the ocean and the seabed, respectively. Moreover, we assume that the acoustic source is situated in the water column at r = 0 and z = z 0 in cylindrical coordinates. Hence, in the water column, the time harmonic acoustic pressure p must satisfy the nonhomogeneous Helmholtz equation p + k 2 n z p = δz z 0 δr 2πr. In the fluid-like sediment, the acoustic pressure p 2 satisfies the homogeneous Helmholtz equation p 2 + k 2 n 2 z p 2 = 0, 2 where k is a constant defined as k := ω c 0 with ω being the frequency of acoustic waves and c 0 the reference sound speed. The refractive index function n i z in the layer i, i =, 2, is the ratio of the reference sound speed c 0 and the sound speed function c i z, which is assumed to be continuous within each layer. At the ocean surface there is a pressure release boundary condition, p = 0 at z = 0. 3 Across the two-fluid interface z = d< 0, two transmission conditions are required such that the normal component of particle velocity and acoustic pressure be preserved, lim z d + p x, y, z = lim z d p 2 x, y, z, 4 lim p x, y, z = lim p2 x, y, z. 5 z d + z d At the bottom z = h< 0, the complete reflecting condition reads as p 2 = 0 at z = h. 6 It is convenient to use the Hankel transform pka, z := 2π 0 J 0 kar pr, z r dr to reduce the dimensionality of our problem, where J 0 is the essel function of order zero. Imposing the out-going radiation conditions on p and applying the

4 2 R. GILERT AND MIAO-JUNG OU Hankel transformation to equations 6 lead to the following system: 2 2 p ka, z + k 2 n z 2 a 2 p ka, z = δ z z 0, d < z < 0, p2 ka, z + k 2 n 2 z 2 a 2 p 2 ka, z = 0, h < z < d, 8 p ka, 0 = 0, 9 p lim z d + = lim p 2 z d, 0 lim = lim, z d + z d p 2 h = 0. 2 For the ease of notation, we define fc + and fc as fc + := lim fz z c + fc := lim fz z c for any given function fz and fixed number c. This notation will be extensively used in the rest of the paper. 2. The Method of Ahluwalia and Keller To demonstrate the singular behavior of the Green s function, we apply a variation on the method of images suggested in [] for a completely reflecting seabed. To this end, we seek for the Hankel transform of the Green s function Gr, z; 0, z 0 as a multiple scattering representation in the form Gr, z; 0, z 0 = ψ n r, z. n=0 Here ψ 0 r, z represents a direct wave from the source at 0, z 0 ; ψ r, z represents a wave which has been either reflected or refracted once, and ψ n r, z is a wave which has gone through n such actions. A particular nicety of this method is that the refractive index function n j, j =, 2, can vary with respect to z. For convenience, we work with the Hankel-transformed system 7 2. In order to construct the scattered waves we need two linearly independent solutions of the homogeneous form of 7. Let us designate these two solutions in the water column as U ka, z and D ka, z as up-going and down-going solutions respectively, relative to z 0. We normalize these solutions such that their Wronskian at z = z 0 is. Similarly, U 2 ka, z and D 2 ka, z are two linearly independent solutions of 8. In terms of the convention z > := max{z, z 0 }, z < := min{z, z 0 },

5 NEAR FIELD REPRESENTATIONS OF THE ACOUSTIC GREEN S FUNCTION 3 the first term in the multiple scattering representation is seen to be ψ 0 ka, z = Obviously ψ 0 satisfies 7. { U ka, z > D ka, z <, d < z < 0, 0, h < z < d. To simplify the notation, we will hereafter omit the variable ka in U ka, z, D ka, z, U 2 ka, z and D 2 ka, z. To calculate the next term in this representation we proceed as follows. When ψ 0 is incident upon the water column surface at z = 0, it produces a down-going wave. Since U and D are linearly independent solutions to a second-order linear ordinary differential equation, this downward wave must be proportional to D z. For convenience, we write it in the form D z 0 D z/, where is referred to as the reflection coefficient of the water surface. compute, we use the boundary condition at the surface z = 0, To ψ 0 ka, 0 + D ka, z 0 D 0 = 0, which implies = U 0 D 0. 3 Similarly, as ψ 0 is incident upon the interface z = d, it produces an up-going wave in the water column and a down-going wave in the fluid-like seabed. We denote the first one by R 2 U z 0 U z/ and the second one by T U z 0 D 2 z/, where R 2 is referred to as the reflection coefficient and T as the transmission coefficient on the interface. To calculate these coefficients, we apply and 0 to obtain ψ 0 ka, d + + R 2U z 0 U d + = T U z 0 D 2 d ψ 0 = T U z 0 ka, d + + R 2U z 0 D 2 d. U d +

6 4 R. GILERT AND MIAO-JUNG OU These equations yield D d + D 2 d D ρ d + D 2 R 2 = U d + D 2 d U ρ d + D 2 U d + D d + U ρ d + D T = U d + D 2 d d + U We conclude that ψ is given by ψ ka, z = D 2 d d d + d,. D z 0 D z R 2 2ik U z 0 U z, d < z < 0, T U z 0 D 2 z, h < z < d. 4 For ψ 2, as is shown in Figures 2a and 2b, there are only four possi- Sea Surface Sea Surface 2 Z 0 Interface Z 0 Interface 4 Seabed 3 Seabed a b Figure 2. Schematic construction of ψ 2 bilities. For cases and 4, by a similar argument as before, we write this up-going wave as R 2 D z 0 U z/ and the down-going wave as T 24 D z 0 D 2 z/. To compute R 2 and T 24, we use the transmission conditions to obtain the system U d + R 2 The above system implies U d + R 2 D 2 d T 24 = D d +, D 2 d D T 24 = ρ d +. R 2 = R 2, 5

7 NEAR FIELD REPRESENTATIONS OF THE ACOUSTIC GREEN S FUNCTION 5 T 24 = T. 6 For case 2 in Figure 2b, denoting this down-going wave by R 22 U z 0 D z/ and considering the pressure-release condition 9 lead to that is, R 2 U z 0 U 0 + R 22U z 0 D 0 = 0, R 22 = R 2 U 0 D 0 = R 2. 7 We denote the wave for case 3 in Figure 2b by R 23 U z 0 U 2. Since in the calculation of ψ, the incident wave for 3 has already been found to be T U z 0 D 2 z/, we use the boundary condition 2 at z = h to get Define as T U z 0 D2 h+ + R 23 U 2 U z 0 h+ = 0. 8 then R 23 from 8 can be written as D 2 h + := U 2 h+, 9 R 23 = T. 20 Combining the results, ψ 2 is given by R R 2 ψ 2 ka, z = D z 0 U z + R 2 U z 0 D z, d < z < 0, R T D z 0 D 2 z + T U z 0 U 2 z, h < z < d. To construct ψ 3, we consider all the six possible cases as shown in Figure 3a, 3b and 3c. The up-going wave 6 represented by T 36 U z 0 U z/ and the down-going wave 5 represented by R 35 U z 0 D 2 z/ can be determined by applying the transmission conditions to get T 36 = T T 2, R 35 = T R 5. Here T 2 and R 5 are given by U 2 d D 2 d U 2 ρ 2 d D 2 T 2 := U d + D 2 d d + U D 2 d d,

8 6 R. GILERT AND MIAO-JUNG OU Sea Surface Z 0 Interface Sea Surface Z 0 3 Interface 2 Seabed 4 Seabed a b Sea Surface Z 0 6 Interface 5 Seabed c Figure 3. Schematic construction of ψ 3 R 5 := U d + U 2 d U ρ d + U 2 U d + D 2 d d + U D 2 d d. Applying similar arguments to case to 4 in Figures 3a and 3b, we obtain the following representation of each individual wave: Wave : R2 R 2 D z 0 D z, Wave 2 : T D z 0 U 2 z, Wave 3 : R 2 2 U z 0 U z, Wave 4 : R 2 T Using these six waves, ψ 3 is represented as ψ 3 ka, z U z 0 D 2 z.

9 NEAR FIELD REPRESENTATIONS OF THE ACOUSTIC GREEN S FUNCTION 7 = R 2 R 2 D z 0 D z + R 2 2 U z 0 U z + T T 2 U z 0 U z, d < z < 0 T D z 0 U 2 z + R 2 T U z 0 D 2 z + T R 5 U z 0 D 2 z, h < z < d. Sea surface R 2 T 2 Interface R 5 T Seabed Figure 4. Schematic description of the six coefficients in the two-layer waveguide Sea surface Interface Seabed Figure 5. Schematic description of one of the rays in ψ 7. It can be represented by using the six parameters as T T 3 R 2 D z 0 D 2 z/ y induction, it can be shown that any possible wave can be uniquely represented in terms of the six coefficients, R 2, R 5, T, T 2 and defined as before. A schematic description of these 6 numbers is given in Figure 4. For example, the down-going wave in Figure 5 is T T 3 R 2 D z 0 D 2 z/. Therefore we may construct ψ n, n =, 2, 3,... by the following scheme based on a tree structure. The tree structure is built by the four rules: always branches out to T and R 2 in the next level of the tree. 2 T can only be followed by. R 2 can only be followed by. 3 always branches into T 2 and R 5 in the next level. 4 T 2 can only be followed by. R 5 can only be followed by. We divide the waves in ψ n, n > into two groups: one group includes all the waves which have D z 0 as part of the coefficients and the other group includes all the waves involving U z 0. For the first group, the root of the tree is and the tree is shown in Figure 6a. The other group contains two trees one starts with R 2 and the other starts with T. These two trees are shown in Figure 6b. Using these tree structures, ψ n, n, can be constructed by tracing every path from the root level to the nodes at level n. For

10 8 R. GILERT AND MIAO-JUNG OU example, the length-four path -T --T 2 of Tree a corresponds to the wave T T 2 D z 0 U z. After all paths of length n in all the three trees are traced, ψ n z in d < z < 0 is then given by the sum of all waves which contain U z or D z, and ψ n z in h < z < d is given by the sum of all waves which contain U 2 z or D 2 z. R 2 T T R 2 T R 2 T2 R 5 T2 R 5 T R 2 T2 R 5 T R 2 T R 2 T2 R5 T R 2 T2 R5 T2 R 5 T R a b Figure 6. Tree Structure The ray representation is then obtained by inverse Hankel transforming ψ l back. 3. Relation etween the Multiple Scattering Representation and the Hankel Transformed Representation of the Green s Function The multiple scattering representation constructed in the previous section can actually be derived in a simpler way by using binomial expansions as presented in this section. It is known that the Green s function for the system of 7 2 can be constructed by using two functions f and f 2 such that both of them satisfy the differential equations 7, 8 and the transmission conditions 0. Furthermore, f satisfies the boundary condition 9, whereas f 2 satisfies the boundary condition 2. ecause {U, D } and {U 2, D 2 } are linearly independent solutions of 7 and 8, respectively, we can write f and f 2 as f = { U + D, d < z < 0, αu 2 + βd 2, h < z < d, f 2 = { γu + δd, d < z < 0, D 2 + U 2, h < z < d.

11 NEAR FIELD REPRESENTATIONS OF THE ACOUSTIC GREEN S FUNCTION 9 Here and are defined as in 3 and 9. To determine the constants, we apply the transmission conditions 0 and to obtain α= β = U d + D 2 d d + U U D 2 d +R D d + D 2 d D ρ d + D 2 U 2 d D 2 d d d U 2 U d + D 2 d d + D 2 D 2 d d +R D d + D 2 d D ρ d + D 2 U 2 d D 2 d d d U 2 D 2 d γ = D d + D 2 d d + D d + D d + D 2 d D ρ d + D 2 U d + D d + d + d + D 2 U D d δ = U d + D 2 d d + U D 2 d U d + D 2 d U ρ d + D 2 U d + D d + d + d + U D d In terms of f and f 2, the Green s function Gz, z 0 is Gz, z 0 = f z > f z < W f, f 2 z 0, where W f, f 2 z 0 is the Wronskian of f and f 2 evaluated at z = z 0. Therefore, for z 0 in the ocean layer, i.e., for d < z 0 < 0, we have Gz, z 0 [U z > + D z > ] [ γ U z δ < + D z < ] R = γ, d < z < 0, δ δ [U z 0 + D z 0 ][D 2 z + U 2 z] γ, h < z < d. δ 2 The key step in connecting the above expression of the Green s function with the multiple scattering representation in d < z < 0 is to identify γ with the δ

12 20 R. GILERT AND MIAO-JUNG OU following combination of the parameters listed in Figure 4: γ δ = R 2 + T T 2 R Applying the binomial expansion to the first function in 2, the coefficient of the term U z > D z < becomes + C k l := k= k Cl kr 2 l k l T R 5 j T 2, 23 j=0 k! k l! l!. 24 Similarly, the coefficient of U z > U z < in 2 is + + R 2 + T R 5 j T 2 j=0 k Cl kr 2 l k l T R 5 m T T 2 T R 5 j T 2 m=0 j=0 k Cl kr 2 l k l T R 5 m T T 2 R 2, 25 k= k= m=0 and the coefficient of D z > D z < is + k= k k l Cl k R 2 l T R 5 j T j=0 Finally, the coefficient of D z > U z < is k= k Cl kr 2 l k l T R 5 j T For the Green s function in h < z < d, applying 22 and the identity j=0 δ = T R 5, followed by binomial expansions, we obtain the series expansion of the coefficient of the term U z 0 U 2 z of the Green s function Gz, z 0 in h < z < d: k=0 [ k Cl k R 2 l T T 2 T ] k l R 5 n R 5 m. 28 n=0 m=0

13 NEAR FIELD REPRESENTATIONS OF THE ACOUSTIC GREEN S FUNCTION 2 In a similar fashion, the coefficient of the term U z 0 D 2 z can be written as [ k ] k l Cl k R 2 l T T 2 T R 5 n R 5 m, 29 k=0 and for D z 0 U 2 z we have [ k Cl k R 2 l T T 2 T k=0 n=0 n=0 R 5 n ] k l m=0 m=0 R 5 m. 30 Finally, the coefficient of the term D z 0 D 2 z can be expressed as [ k ] k l Cl k R 2 l T T 2 T R 5 n R 5 m. 3 k=0 Clearly, there is a one-to-one correspondence between each of the terms in these expansions and the waves constructed by using the tree structures in Figures 6a and 6b. n=0 References. D. S. Ahluwalia and J.. Keller, Exact and asymptotic representations of the sound field in a stratified ocean. Wave propagation and underwater acoustics Workshop, Mystic, Conn., 974, Lecture Notes in Phys., Vol. 70, Springer, erlin, S. ergman and M. Schiffer, Kernel functions and elliptic differential equations in mathematical physics. Academic Press Inc., New York, N. Y., R. P. Gilbert, C. Mawata, and Y. Xu, Determination of a distributed inhomogeneity in a two-layered waveguide from scattered sound. Direct and inverse problems of mathematical physics Newark, DE, 997, 07 24, Int. Soc. Anal. Appl. Comput., 5, Kluwer Acad. Publ., Dordrecht, R. P. Gilbert and M.-J. Ou, A uniqueness theorem of the 3-dimensional acoustic scattering problem in a shallow ocean with a fluid-like seabed. J. Comput. Acoust. 2003, No. 4, R. P. Gilbert and Y. Xu, An inverse problem for harmonic acoustics in stratified oceans. J. Math. Anal. Appl , No., C. Rozier, D. Lesselier, T. S. Angell, and R. E. Kleinman, Reconstruction of an impenetrable obstacle immersed in a shallow water acoustic waveguide. Inverse problems of wave propagation and diffraction Aix-les-ains, 996, 30 42, Lecture Notes in Phys., 486, Springer, erlin, C. Rozier, D. Lesselier, T. S. Angell, and R. E. Kleinman, Shape retrieval of an obstacle immersed in shallow water from single-frequency farfields using a complete family method. Inverse Problems 3997, No. 2, m=0 Authors addresses: R. Gilbert Received

14 22 R. GILERT AND MIAO-JUNG OU Department of Mathematical Sciences University of Delaware Newark, Delaware 976 USA

15 NEAR FIELD REPRESENTATIONS OF THE ACOUSTIC GREEN S FUNCTION 23 Miao-Jung Ou Department of Mathematics University of Central Florida Orlando, FL 3286 USA mou@mail.ucf.edu