Wave scattering by a two-dimensional band-limited fractal surface based on a perturbation of the Green’s function

Abstract

A fast approximate method is described to calculate the acoustic scattering from a one-dimensional Dirichlet band-limited fractal surface. The formulation is based on a perturbation of the Green’s function allowing an approximation of the propagator in the kernel of the Helmholtz integral equation, which reduces the integral equation to a convolution equation. This allows us to find a solution using Fourier transforms rather than the usual matrix inversion that is required. We have shown that in the limit of small kσ, where k is the incident wave number and σ is the rms height, it is possible to find accurate closed form expressions for the reflection coefficients Rn, the spectral components of the normal gradient of the field ψn, the scattered field psca, and the angular scattering coefficient Isca representing the scattering from a band-limited fractal surface. For small values of kσ≪1, we have used the generalized Rayleigh method [D. L. Jaggard and X. Sun, J. Appl. Phys. 68, 5456 (1990)] to determine the theoretical linear relationship which exists between the slope of the absolute value of the reflection coefficients in dB versus the reflection mode and the fractal dimension D. This theoretical relationship has been verified by using the Green’s function perturbation method. This relationship and an analogous relationship between the scattering coefficient and the scattering angle allows the determination of the fractal dimension D and the rms height σ from the scattering pattern when kσ≤0.2.