The solution of a mixed boundary value problem in the theory of diffraction by a semi-infinite plane

Abstract

A solution is obtained for the problem of the diffraction of a plane wave sound source by a semi-infinite half plane. One surface of the half plane has a soft (pressure release) boundary condition, and the other surface a rigid boundary condition. Two unusual features arise in this boundary value problem. The first is the edge field singularity. It is found to be more singular than that associated with either a completely rigid or a completely soft semi-infinite half plane. The second is that the normal Wiener-Hopf method (which is the standard technique to solve half plane problems) has to be modified to give the solution to the present mixed boundary value problem. The mathematical problem which is solved is an approximate model for a rigid noise barrier, one face of which is treated with an absorbing fining. It is shown that the optimum attenuation in the shadow region is obtained when the absorbing lining is on the side of the screen which makes the smallest angle to the source or the receiver from the edge.

A brief historical perspective of the Wiener–Hopf technique

“…Matrix Wiener-Hopf kernels are fundamentally distinct from their scalar counterparts in that there is no algorithmic approach to determining the factorization (4) of the transformed kernel [16]. Exact factorization can be achieved for matrices with certain special features: those that are upper (or lower) triangular in form; those that are of Khrapkov-Daniele, i.e., commutative, form (see [17][18][19][20]); those whose elements comprise meromorphic functions [21,22]; kernels with special singularity structure that allows the Wiener-Hopf equation to be recast into uncoupled Riemann-Hilbert problems [23][24][25]; and N × N matrices with special algebraic or group structure [26][27][28]. For more details on exact matrix kernel factorization the interested reader is referred to the last mentioned article and to references cited in [29].…”

Diffraction of Plane Waves by a Slit in an Infinite Soft-Hard Plane

“…The problem of plane wave diffraction by a half plane which is soft at the top and hard at the bottom was first solved by Rawlins [1] who adopted an ad-hoc method for the solution of this problem. Later on Büyükaksoy [2] reconsidered the problem solved by [1] and proposed an appropriate method for the factorization of the kernel matrix appearing in it.…”

“…Later on Büyükaksoy [2] reconsidered the problem solved by [1] and proposed an appropriate method for the factorization of the kernel matrix appearing in it. The continued interest in the problem is due to the fact that it constitutes the simplest half plane problem which can be formulated as a system of coupled Wiener-Hopf (WH) equations that cannot be decoupled trivially [2].…”

The optimum orientation of an absorbing barrier

“…In an earlier publication [1], numerical calculations for a limiting situation, where an infinite barrier, which was soft on one side and rigid on the other side, was insonified by a plane wave , led the author to suggest an optimum situation. However this was not proved rigorously, even for this somewhat hypothetical noise barrier.…”