A T matrix for scattering from a doubly infinite fluid–solid interface with doubly periodic surface roughness

Abstract

The T-matrix formalism is used to calculate scattering of a plane wave from a doubly infinite fluid–solid interface with doubly periodic surface roughness. The Helmholtz–Kirchhoff integral equations are used to represent the scattered pressure field in the fluid and the displacement field in the solid. The boundary conditions are applied and a system of four coupled integral equations is obtained. The incident and scattered pressure fields in the fluid, as well as the surface pressure field, are represented by infinite series of scalar Floquet plane waves, while the scattered displacement field in the solid and the surface displacement field are represented by infinite series of rectangular vector basis functions constructed from Floquet plane waves. This process discretizes the integral equations and transforms them into a system of four coupled doubly infinite linear equations. The extended boundary condition is applied and the T matrix that relates the spectral amplitudes of the incident field to the spectral amplitudes of the scattered fields is constructed. An exact analytic solution and numerical results are obtained for plane wave scattering from doubly periodic sinusoidal and triangular surface roughness.

Diffraction of homogeneous and inhomogeneous plane waves on a doubly corrugated liquid/solid interface

“…The third one is the paper by Bishop and Smith [5], in which research is reported on the diffraction on quite the same corrugated surfaces as is undertaken in the present paper. However, Bishop and Smith [5] use a different method which consists of a ÔT-matrixÕ formalism applying the Helmholtz-Kirchoff integral equations to represent the scattered pressure field in the liquid and the displacement field in the solid.…”

“…Another reason is because it can also be found in an earlier paper of Mampaert and co-workers [9], in which a similar approach is formulated as in the present paper, that for a singly corrugated surface, the shape of the corrugation is indeed relatively unimportant and has no effect at all on the position of anomalies in the reflection spectra. Furthermore, even though Bishop and Smith [5] observe some anomalies in their reflection spectra, they do not prove what exactly causes these anomalies, which could be done, for example, by means of depicting the displacement field of the diffracted sound. Finally, Bishop and Smith [5] apparently have not found any situation under which a reflection or transmission coefficient exceeds unity.…”

“…The reason for our focus on inhomogeneous waves is the well known fact [6][7][8] that leaky Rayleigh waves can only be practically stimulated by inhomogeneous plane waves and not by homogeneous plane waves. Bishop and Smith [5] particularly focus on the interesting fact that the roughness shape is qualitatively unimportant for the phenomena that appear during diffraction and is completely unimportant in the positioning of anomalies that appear in the diffraction spectra as a function of the frequency or as a function of the angle of incidence. The present paper does not investigate this phenomenon because it has been done by Bishop and Smith [5].…”

“…Bishop and Smith [5] particularly focus on the interesting fact that the roughness shape is qualitatively unimportant for the phenomena that appear during diffraction and is completely unimportant in the positioning of anomalies that appear in the diffraction spectra as a function of the frequency or as a function of the angle of incidence. The present paper does not investigate this phenomenon because it has been done by Bishop and Smith [5]. Another reason is because it can also be found in an earlier paper of Mampaert and co-workers [9], in which a similar approach is formulated as in the present paper, that for a singly corrugated surface, the shape of the corrugation is indeed relatively unimportant and has no effect at all on the position of anomalies in the reflection spectra.…”

“…However, Bishop and Smith [5] use a different method which consists of a ÔT-matrixÕ formalism applying the Helmholtz-Kirchoff integral equations to represent the scattered pressure field in the liquid and the displacement field in the solid. Moreover, Bishop and Smith [5] limit their discourse to pure homogeneous incident plane waves, whereas the current paper applies not only to homogeneous, but also to inhomogeneous plane waves. The reason for our focus on inhomogeneous waves is the well known fact [6][7][8] that leaky Rayleigh waves can only be practically stimulated by inhomogeneous plane waves and not by homogeneous plane waves.…”