Computing Wave Propagation Along Stratified Range Dependent Waveguides In Ocean Acoustics

In acoustics, electro-magnetism, seismic migration and other applications, there are many large scale wave propagation problems. A direct numerical computing like finite-element, finite-difference could be very expensive since these methods often result in a very large linear system. The forward fundamental operator marching method (FFOMM) which is developed by Lu Y.Y. etc based on Dirichlet-to-Neumann (DtN) reformulation is a highly efficient numerical marching scheme for solving Helmoholtz equation in a large scale domain with curved interfaces or boundaries. However, this FFOMM previously suggested can only march in a large range step size in some simple waveguides, such as horizontal stratified waveguide, the waveguide with one curved interface or one curved boundary. If there are more curved interfaces or curved boundaries in the domain of waveguide, then the FFOMM has to march in a very short step unless some treatments are done.In the real world, some waveguides, like waveguides in ocean acoustics, often have multilayered stratified structure with some curved interfaces or boundaries. To efficiently compute wave propagation along the range with some marching methods, flattening the interfaces or boundaries and transforming equation are needed. In this paper, an analytical local orthogonal.coordinate transform and an analytical equation transform are constructed to flatten the interfaces and change the Helmholtz equation as a solvable form. For a waveguide with a flat top, a flat bottom and n curved interfaces, the coefficients of transformed Helmholtz equation is given in a closed formulation. However, when distance of two adjoint interfaces is so closely that there is not a horizontal straight-line to divide the corresponding layer into two parts, the analytical local orthogonal transform method will not be feasible. For this reason, a numerical coordinate transform and equation transform based on the classical Rugger-Kutta method are developed to solve Helmholtz equation by some marching methods in the situation. In the transformed horizontal stratified waveguide, the FFOMM uses a large range step method to discretize the range variable and compute the forward problem of Helmholtz equation efficiently. The analytical transform method and the numerical transform method are particularly useful for long range wave propagation problems in slowly varying waveguides with multilayered medium structure. Furthermore, the methods can also be applied for the wave propagation problems in acoustical waveguides associated with varied density.It is noticed that the inverse problem is also very important for large scale wave propagation problems in acoustics or electro magnetics. In our work, a called "inverse fundamental operator marching method" (IFOMM) based on the Dirichlet-to-Neumann map is suggested for solving the large scale inverse boundary value problem associated with the two dimensional Helmholtz equation in a range dependent waveguide. This method solves the badly ill-conditioned matrix equations arising in the marching progress by truncated singular value decomposition with the number of propagation modes in the waveguide chosen as the regularization parameter. Numerical examples show that the proposed method is computationally efficient, highly accurate with respect to the propagating part of a starting field.