Analytical insights into semi-infinite plate scattering: the wiener–hopf technique and two-sided linear boundary conditions

Summary The Wiener–Hopf Technique is popular throughout applied mathematics, particularly for wave scattering problems. One such problem is the scattering of an incident wave impinging upon a semi-infinite surface. For scattering problems in which one applies separate boundary conditions on each side of the diffraction medium, we must adapt the standard Wiener–Hopf approach via a generalised technique to deal with the resulting matrix system. Such problems are present in the literature, where the Wiener–Hopf problem is reduced to solving Hilbert equations, leading to the so-called Wiener–Hopf–Hilbert method. However, both boundaries tend to have the same mathematical form, which is overly simplistic when describing realistic compliant boundaries or surface coatings. In some other literature, the matrix factorization problem remains the focus and is solved for arbitrary entries, which may prove challenging to use in complex scattering problems. We solve a more general problem while keeping an applied scattering framework, drawing attention to the additional analytical considerations needed to adapt previous methods. More specifically, we study the Wiener–Hopf matrix kernel, which shows that it holds the key to distinguishing these problems from those studied in previous literature. Finally, we present an example of a sound wave scattering off a plate that has a one-sided surface coating and is in flow. We treat this as a two-sided boundary and model the upper side with the Ingard–Myers impedance boundary condition and the lower side with the rigid Neumann boundary condition. Results are presented using impedance values from existing literature that reflect real materials.