טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
Ph.D Thesis
Ph.D StudentTsabary Guy
SubjectAn Iterative Solution for the Helmholtz Equation above a
Rough Surface
DepartmentDepartment of Applied Mathematics
Supervisor Professor Yehuda Agnon


Abstract

The essence of this thesis report is a development and inspection of two iterative solutions of the Helmholtz equation for a scalar field in R3 above a rough surface that admits the Dirichlet boundary condition. The bases for the two iterative methods are two different boundary integral equations that represent the (unique) solution. The first integral equation is classified as a Fredholm integral equation of the first kind. The second is classified as a Fredholm integral equation of the second kind. This classification suggests that it is easier to find stable solution methods to the second equation. In both methods, we separated the boundary integral into a major part which is easy to calculate and a local residual part. The major part is a convolution and thus can be calculated using FFT in complexity O(N log N), where N is the number of surface points in which the surface height and its first derivatives together with the incoming wave and its normal derivative are all known. The residual element of the equations can be approximated efficiently and cheaply only in surfaces where their amplitude is less than the wavelength of the incoming wave. The iterative schemes were tested numerically against a reference solution using Matlab computer programs that examined the applicability range, the error estimation and the stability of the schemes. All tests supported the supremacy of the second method. In particular the error estimation and stability tests indicated good performance for surfaces with slope up to 1.