Generalized Fourier transform and its application to the volume integral equation for elastic wave propagation in a half space

In the present study, a generalized Fourier transform for time harmonic elastic wave propagation in a half space is developed. The generalized Fourier transform is obtained from the spectral representation of the operator derived from the elastic wave equation. By means of the generalized Fourier transform, a volume integral equation method for the analysis of scattered elastic waves is presented. The proposed method is based on the Krylov subspace iteration technique. During the iterative process, the discrete generalized Fourier transform is used, where the derivation of a huge and dense matrix from the volume integral equation is not necessary.