An Efficient Method for Local Base Transform in Pekeris Waveguide with Radiation Condition

In this paper, for the fast computation of the coordinates under the basis of the eigenfunctions of Helmholtz operator, we derive the conjugate operator with the radiation boundary condition. Further, we prove the cross orthogonality between the linearly-independent eigenfunctions of the Helmholtz operator and the linearly-independent eigenfunctions of the conjugate operator. The numerical simulations demonstrate the effectiveness of our treatment.


Introduction
It is known that the computation of the wave propagation plays a particularly prominent role in ocean acoustics or optical waveguides [1] [2] [3]. Because of the inhomogeneity of the ocean medium, the sound velocity forms the underwater channel with the regular change of the ocean, and the remote acoustic propagation of hundreds of or even thousands of kilometers can be carried out by using the underwater channel. The study of acoustic wave propagation in the ocean is of great significance to such things as underwater positioning, communication and ranging [4].
The propagation of sound waves in the medium is mathematically formulated as a well-known Helmholtz equation [5]. When the propagation area or the solution region is bounded, the linearly-independent eigenfunctions of the Helmholtz operator are weighted orthogonal, so one can use the marching algorithm to solve the Helmholtz equation quickly [1], and a good numerical solution has been obtained. However, in reality, it is extremely difficult to deal with such a problem when considering the propagation region is unbounded in the transverse direction, such as the infinite depth of the ocean (including the submarine rock layer) or open light region in the upper or lower direction [6]. For the convenience of calculation, we need to truncate the unbounded waveguide area into a bounded waveguide area. It is common practice to add some artificial boundary conditions to the outside of the truncation area [7]. In order for the acoustic propagation of the truncated area to be consistent with that before truncation, special processing needs to be added to the artificial boundary so that the sound waves do not have obvious reflection when they leave the area, and the sound waves can be absorbed effectively.
For the addition of artificial boundary conditions, we use the radiation boundary condition (RBC) [8] to truncate the unbounded waveguide. However, the orthogonality of between the linearly-independent eigenfunctions of the Helmholtz operator is lost, so it is difficult to fast compute the coordinate coefficients under the local eigenfunctions' bases when the Helmholtz equation with the RBC is solved by some numerical marching method, such as the Operator Marching Method and the One-way Method [9].
For this reason, the conjugate operator of the Helmholtz operator with the RBC is constructed in this paper. Further, it is proved that there is the crossly-orthogonal property between the linearly-independent eigenfunctions of the Helmholtz operator and their linearly-independent conjugate eigenfunctions.
Furthermore, a simple and convenient formula is given to the coordinate transformation under two different eigenfunction bases. Finally, the numerical experiment results verify the correctness of this treatment.

Eigenfunctions of the Helmholtz Operator
For simplicity, we consider the wave propagation in the Pekeris waveguide. The mathematical model is as follows: Where z is the depth axis pointing downward with the ocean surface at 0 z = , and x is the range variable in horizontal direction. One layer with density 1 ρ is located in 0 z G < < , the other with density 2 ρ is located in z G > ; and their interface is flat at z G = . Here, 1 κ and 2 κ are represented as the wavenumbers in two different subregions, respectively.
For the coordinate transformation between two different bases, it is necessary to quickly calculate the coordinate coefficients of any given function under a basis. Here the basis is formed by the linearly-independent eigenfunctions of the Helmholtz operator P . If we assume the solution of Equation (2) , then the eigenvalue problem of the Equation (2) is the eigenfunction corresponding to the eigenvalue λ for the eigenvalue problem of Equation (2). (3) are crossly orthogonal to their conjugate func- corresponds to each eigenvalue j λ , respectively, ( ) ; and the overline represents the conjugate operation.
The following processes focus on constructing the conjugate eigenfunctions , and further prove their cross orthogonality.
For Equation (3), we have the general solutions as follows:

The Construction of the Conjugate Operator
Let the operator P be with the RBC at = z H and the interface conditions at = z G , we have where ( ) Let φ and ϕ be the eigenfunction and the corresponding conjugate eigenfunction of the operator P , respectively, where the boundary conditions are Consider where ( ) Then, by the integration by parts and making use of the interface and the boundary conditions, we have Thus, we have ( ) the above expression is simplified as follows: ( ) For simplicity, the interface and the boundary conditions are given to the conjugate eigenfunction ϕ as follows: it is equivalent to it is equivalent to and let it is equivalent to Remark 1: Although the two operators P and Q have the same representation, their RBCs are different.

Proof of the Cross Orthogonality between the Eigenfunctions and Their Conjugate Eigenfunctions
In this section, we prove the cross orthogonality of the linearly-independent ei-

Numerical Examples
In this section, we will verify the method presented in the previous sections. Considering a Pekeris waveguide, we let These parameters are taken from reference [1]. By the nonlinear Equation (6)  . The details of choosing initial values are listed in the reference [3]. As a result, we obtain the eigenvalues' distributions of the operators P and Q , which are shown in Figure 1. And these distributions really fit the physical meaning.
To numerically verify the cross orthogonality, we list some eigenvalues of the operator P in detail as follows:      λ λ λ and 8 λ ), respectively. and the linearly-independent eigenfunctions of the operator Q. Finally, Numerical simulation results demonstrate that the cross orthogonality has been almost founded when the RBC is used to truncate the unbounded domain. Namely, it is possible that the high-precision computation of the coordinate transformation under two different local bases is helpful for obtaining more actual propagation behavior. However, we also see that the AGI method has great influence on the