On the scattering of sound by two semi-infinite parallel staggered plates - I. Explicit matrix Wiener-Hopf factorization

Abstract

This paper discusses the two-dimensional scattering of sound waves by two semi-infinite rigid parallel plates. The plates are staggered, so that a line in the plane of the motion passing through both edges is not in general perpendicular to the plane of either plate. The problem is formulated as a matrix Wiener-Hopf functional equation, which exhibits the difficulty of a kernel containing exponentially growing elements. We show how this difficulty may be overcome by constructing an explicit product decomposition of the matrix kernel with both factors having algebraic behaviour at infinity. This factorization is written in terms of a single entire auxiliary function that has a simple infinite series representation. The Wiener-Hopf equation is solved for arbitrary incident wave fields and we derive an asymptotic expression for the field scattered to infinity; the latter includes the possibility of propagating modes in the region between the plates. In part II of this work we will evaluate our solution numerically and obtain some analytical estimates in a number of physically interesting limits.

On the Solution of Wiener--Hopf Problems Involving Noncommutative Matrix Kernel Decompositions

“…Various techniques have been proposed in the literature to overcome this difficulty, including a general scheme by the author (Abrahams and Wickham [7]) and one specific to matrices of Khrapkov form (Daniele [15]). A full bibliography illustrating the relevant physical applications can be found in [6].…”

Intensification and suppression of jet noise sources in the vicinity of lifting surfaces

“…[6][7][8] To demonstrate this let us divide the whole space into 3 domains: I -, II -0 , and IIIy h > y h < < 0 y < . Denote the normal displacement of fluid particles at the interface between domains I and II as 1 η , and between domains II and III as 2 η , and denote pressure difference between both sides of the interfaces as and respectively.…”

On the matrix factorization of Wiener-Hopf Kernel

“…Thus a considerable effort has been made in recent years both to extend this class and to find new methods for cMculating matrix factorization, especially with reference to particular physical problems. Examples of these specific problems solved by achieving the matrix factorization can be found in the papers by Rawlins [3], Przezdziecki and Hurd [4], Hurd and Przezdziecki [5], Hurd and Lª [6], Abraham and Wickham [7] and Jones [8]. Some important techniques for factorization of certain class of matrices have been developed by Hurd [9], Daniele [10], Khrapkov [11], Rawlins and Williams [12] and Jones [13].…”