Numerical Solution Of The Helmholtz Equation In The Multi-layer Unbounded Region And Its Application

In this paper ,we provide a method to solve Helmholtz equation in unbounded region.The first, we transform unbounded region to bounded region. Many kinds of boundary conditions such as the first condition, the second condition, and the third condition have been used to solve this problem. These conditions can only approximate the equation roughly. This result is not very good, because these boundary conditions are not fit for the real cases. According to the lack of this approximation, J.P. Berenger put forward the Perfectly Matched Layer (PML) in 1994.Francis Collino used a complex transform x = x + ∫₀^x σ(Ï„)dÏ„ to change Helmholtzequation to another equation in 1997, and this equation is called as an improved Helmholtz equation.The second, there are many methods to solve Helmholtz equation in the bounded region with a very large length, as compared with the typical wavelength. Standard mumerical methods, such as finite element and finite difference method, give rise to very large linear systems. These linear systems are difficult to solve, since they are non-symmetric and indefinite. Furthermore, when these boundary value problems of the Helmholtz equation are discretized, the large number of unknowns are solved together giving rise to a very large demand for computer memory. For waveguide problems, the length scale along the main propagation is much larger than the transverse length scale. So we use One-way method to calculate acoustic wave propagation.The last, when One-way Method is used to calculate acoustic wave propagation, the eigenvalues are needed. The research on Slab optical waveguide has been done. In this paper , we investigate the acoustic wave propagation on unbounded region with a curved interface. We use a local orthogonal transform to flat the curved interface ofthe waveguide. Then, we discrete operator characteristic equation, and get a complex matrix. And we make a Mulit-Generalized Rayleigh Quotient Iteration method to calculate the operator's eigenvalues. It provides the base of calculating acoustic wave propagation.The numerical examples show that our treatment can not only calculate Helmholtz equation's eigenvalues (Propagation Mode, Leaky Mode, Berenger Mode), but also calculate acoustic wave propagation in unbounded region. This method has some advantage such as triangle-diagonal, saving computer memory and so on. It can be used in calculating wave propagation in unbounded region with a curved interface.