The application of a Fourier transform to the solution of a diffraction problem

Abstract

We shall consider the problem of diffraction of elastic waves in the exterior of a simply connected two-dimensional region. The problem is solved by using a Fourier transform of the type U(r,p)= 1 2π ∫ 0 ∞ U(r,t)e iptdt; where U( r, t) is the displacement vector with components u( r, t) and v( r, t) ; r—the radius vector with the coordinates ( x, y); p = α + iβ; α and β are real numbers, −∞ < α < ∞; β 0 is any small fixed number. The initial non-stationary problem is reduced in the usual way to a stationary diffraction problem which consists in finding the solution of a system of partial differential equations with a given boundary condition. It is shown that the solution of the stationary diffraction problem must be sought in a class of functions that can be continued analytically with respect to p to the half-plane Imp ⩾ β and become 0 when p → ∞. The solution of the stationary problem is unique in this class of functions and on applying to it a reciprocal Fourier transform of the kind U(r,p)= 1 2π ∫ −∞+iβ ∞+iβ U(r,p)e −iptdα the solution of the initial problem is obtained. It is further shown that the solution of the stationary diffraction problem in this class of functions is equivalent to a solution in the class of functions satisfying Sommerfeld's radiation conditions when r → ∞ for the corresponding potentials \\ ̃ gf( r, p) and ψ( r, p).