On the Validity of Modal Expansion in Pekeris Waveguide with PML

In underwater acoustics [1], the Pekeris waveguide is a simple model, which has been studied extensively [2]. For actual calculation, the region in consideration is always very large, such as the entire ocean, and the problem is usually reduced to a mathematically well-known Helmholtz equation over the infinite domain. The modal expansion of the waveguide consists of a few propagation modes and an integral of continuous radiation modes [3,4], which is difficult to evaluate in practical computations. When the transverse direction is bounded, many methods can be used [5-8]. A perfectly matched layer (PML), introduced by Berenger in 1994 [9,10], is a common tool to truncate the unbounded domain. It is an addition layer around the interested domain, in which the solutions decay. Mathematically, introducing a PML is equivalent to applying a complex coordinate transformation inside the additional layers. Modal expansion of the bounded problem with PML is quite different from the unbounded one. All the modes are discrete, and the integration is approximated by the infinite sum of leaky modes and PML modes. Although the modal expansion method has been used for calculation widely, there are few literatures on the theoretical analysis and numerical verifications, when it combines with PML.


Introduction
In underwater acoustics [1], the Pekeris waveguide is a simple model, which has been studied extensively [2]. For actual calculation, the region in consideration is always very large, such as the entire ocean, and the problem is usually reduced to a mathematically well-known Helmholtz equation over the infinite domain. The modal expansion of the waveguide consists of a few propagation modes and an integral of continuous radiation modes [3,4], which is difficult to evaluate in practical computations. When the transverse direction is bounded, many methods can be used [5][6][7][8]. A perfectly matched layer (PML), introduced by Berenger in 1994 [9,10], is a common tool to truncate the unbounded domain. It is an addition layer around the interested domain, in which the solutions decay. Mathematically, introducing a PML is equivalent to applying a complex coordinate transformation inside the additional layers. Modal expansion of the bounded problem with PML is quite different from the unbounded one. All the modes are discrete, and the integration is approximated by the infinite sum of leaky modes and PML modes. Although the modal expansion method has been used for calculation widely, there are few literatures on the theoretical analysis and numerical verifications, when it combines with PML.
In this paper, we give a discussion on the validity of modal expansion in Pekeris waveguide which is truncated by a PML [11]. Due to the introduction of PML, the coordinate-transformed operator is not self-adjoint, and the eigenfunctions lose the property of orthogonality. In this situation, the coefficients of modal expansion are difficult to evaluate. We use the conjugate eigenfunctions [12], which are analytical expressed, to compute the expansion coefficients. And the eigenvalues are given by the asymptotic approximation [13]. The theoretical results on the convergency and stability of the coefficients are derived in this paper. This paper is organized in the following manner. The mathematical formulation of bounded waveguide and the modal expansion method is presented in the section Model and Methods, and the numerical methods and theoretical results on convergency and stability are also given here. Numerical examples are given in the next section. Finally, we leave the conclusions.

Mathematical model of waveguide
When wave propagates in the Pekeris waveguide [13], the following two-dimensional Helmholtz equation is considered: and z is the depth, x is the range, k 1 and k 2 are wave numbers in different fluid layers, ρ 1 and ρ 1 are densities. For a range-independent waveguide where κ only depends on the depth z, the general solution of (1) has the form u = φ(z)e iβx , in which φ and λ = β 2 satisfy the following eigenvalue problem: 2 =0 = , > 0, The solution of problem (1) satisfies u ∈ L 2 [0,+∞], and it has the modal expansion form: Under the assumption that bottom is homogeneous for z>G, the PML technique is introduced in the depth direction. It is equivalent to a complex coordinate transformation, that is If the interested interval is 0<z<H for some H>G, the PML is added on the boundary x=H, and is terminated at D. (2) is therefore approximated by the following form: The technique described above result in an eigenvalue problem (4) on a bounded interval, whose solutions satisfy u ∈ L 2 [0, D]. F. Olyslager has proved that the discrete spectrum of the problem with PML (4) converges to the continuous spectrum of (2) in [14]. As a result, we have the modal expansion form exactly when the thickness of PML tends to infinity. In practical applications, the series must be truncated to a finite sum. In the following sections, we are going to show that the truncation is reasonable by showing the convergency and stability of the coefficients. As the eigenfunctions lose the property of orthogonality, and the determination of coefficients is technical, the method for which is given afterwards.

Eigenfunctions and conjugate eigenfunctions
The eigenfunctions φ j (z) (j = 1,2,…) of (4) do not have the property of weighted orthogonality, that brings difficulty in determining the expansion coefficients. Fortunately, the analytical expression of conjugate eigenfunctions ( ) ( = 1, 2,...) j z j ϕ can be derived [12], which are defined by The conjugate operator of can be obtained easily by its definition, which is 0 0 The resulting operator M is defined by In other words, should satisfy the following ODE: As before, the solutions of equation (9) turn out:

Eigenvalues and expansion coefficients
From the formulas of (6), (7), (10), once the eigenvalues are obtained, the corresponding expansion coefficients can be determined directly. Next, numerical calculation of eigenvalues is discussed.
Although eigenvalue problems are usually difficult to solve, under certain conditions, there exist some good methods. In this paper, asymptotic solution method is used. In order to get accuracy modes, a nonlinear equation for the eigenvalues of this problem is derived, which is where ˆ= ( ) .
The above equation is called dispersion relation, and its deduction process has used the boundary information φ(D)=0 in (4). The roots of (11) corresponding to the eigenvalues of (4).
Using the dispersion relation, Jianxin Zhu et al. [13,15] have given the asymptotic formulas of both PML modes and leaky modes of Pekeris waveguide. Here, we just quote the results as follows: Leaky modes satisfy which have the following asymptotic formulas: All the parameters mentioned above can be found in [13].
As mentioned earlier, there is a close relationship between eigenvalues and the corresponding expansion coefficients.
, the following theorems can be derived.
Next, we give error estimations for modal expansion method applying to these incident waves. The number of expansion is denoted by n. From tables 2 and 3, we find that the errors will be reduced if we increase the number of terms, but when a certain number of terms reached, the accuracy will not increase evidently. This is due to the fact that the latter coefficients in the expansion are very small and