Study of the solution of the nonstationary problem of plane wave diffraction at an impedance wedge

Abstract

THE problem of diffraction at a wedge was solved in 1. [1] under restrictions regarding the wedge angle and the direction of wave incidence. These restrictions are removed below; in addition, the ray expansion of the solution close to the diffracted wave front, and all the terms of the high-frequency asymptotic solution of the analogous stationary problem are found (only the first term is found in 2. [2]; the entire expansion is obtained in 3. [3], but under the restrictions u = 0 and ∂u ∂n = 0 ). The convergence of the ray series is improved by separation of the singularities. The solution close to the shadow boundary, and the surface waves in the non-stationary problem, are examined (see [2] for the stationary case). The theory of generalized solutions is used to prove that the solution is unique when account is taken of its behaviour close to the rib of the wedge. Except in Section 4, the same problem is considered throughout as in Section 1 of [1]. We seek the solution of the equation U tt = C 2( U xx + U yy ) in the angular region ¦ϑ¦ < Φ under the boundary conditions U n = Y+ u t t ( ϑ = Φ), u n = Y − u t ( ϑ = − Φ) ( u n is the derivative with respect to the inward normal to the boundary and r, ϑ are polar coordinates), identical for t < 0 with the plane u 0 = ƒ(t + x c cosβ + y c sinβ) (incident wave) or with the sum of the incident wave and the waves reflected from the wedge sides ϑ = ± ϑ, if such waves exist for t < 0. It is assumed that Re y ± ⩾0, ƒ(t) = 0 for t < 0, and the solution is required to satisfy the same condition as in [1], Section 1, as r → 0.