Wave splitting of the telegraph equation in R ³ and its application to inverse scattering

Abstract

The technique for splitting waves into up and down components is applied to solutions of the generalized wave equation in three dimensions. In particular, the splitting associated with the principal part of the equation is applied to the fundamental solution (Green function). The equations for the up and down wave components are obtained q d the singular parts removed. The use of these reduced wave components together with their equations. boundary data and initial data, in the inverse problem (where the reflected field 6 produced by a point-impulse source, exterior to the scattering medium) is discussed.Furthermore, we will assume that the scattering medium lies below the plane x3 = 0, and that the region x3 0 is free space, i.e.where co is a constant. For convenience we will set p = 1 for x3 > 0.is free space with c = CO. b = q = 0 and p = 1 here.Note for a compact scatterer lying in the slab -L e x3 < 0, then the region x3 < -LThe following notation will be employed x = (5, 1 3 ) 5 = (Xl, x2)

Reduced differential transform method to solve two and three dimensional second order hyperbolic telegraph equations

“…Similarly, the three dimensional (3D) telegraphic equation 69 is expressed as follows: uðx; y; z; tÞ ¼ n 1 ðx; y; z; tÞ; ðx; y; zÞ 2 C p ; t P 0; @u @g ðx; y; z; 0Þ ¼ n 2 ðx; y; z; tÞ; ðx; y; zÞ 2 C q ; t P 0; iðx; y; z; tÞ ¼ w 1 ðx; y; z; tÞ; ðx; y; zÞ 2 C p ; t P 0; @i @g ðx; y; z; 0Þ ¼ w 2 ðx; y; z; tÞ; ðx; y; zÞ 2 C q ; t P 0; wave propagation (Weston and He, 1993), random walk theory 93 (Banasiak and Mika, 1998) Das et al, 2011;Srivastava et al, 2013a,b;Ahmad and 103 Hassan, 2013; Keskin and Oturanc, 2009). 104 The present paper describes an analytical scheme, the 105 reduced differential transform method to provide approximate wðx; y; z; tÞ ¼ X 1 …”

Transient waves from internal sources in non-stationary media — Numerical implementation

“…A challenge of great interest today, is to solve transient wave propagation problems in three dimensions. Some interesting results have been reported [25,26].…”

“The source problem” — Transient waves propagating from internal sources in non-stationary media

“…The same technique applies for the solution of three-dimensional inverse scattering problems, and several interesting results have been reported, see e.g. Refs 13,[30][31][32][33]. In the present paper, the scattering analysis for non-stationary media is extended to include internal sources in the scattering media.…”