On the diffraction of acoustic waves by a quarter-plane

Abstract

This paper follows the work of A.V. Shanin on diffraction by an ideal quarter-plane. Shanin's theory, based on embedding formulae, the acoustic uniqueness theorem and spherical edge Green's functions, leads to three modified Smyshlyaev formulae, which partially solve the far-field problem of scattering of an incident plane wave by a quarter-plane in the Dirichlet case. In this paper, we present similar formulae in the Neumann case, and describe a numerical method allowing a fast computation of the diffraction coefficient using Shanin's third modified Smyshlyaev formula. The method requires knowledge of the eigenvalues of the Laplace-Beltrami operator on the unit sphere with a cut, and we also describe a way of computing these eigenvalues. Numerical results are given for different directions of incident plane wave in the Dirichlet and the Neumann cases, emphasising the superiority of the third modified Smyshlyaev formula over the other two.

Precise description of the different far fields encountered in the problem of diffraction of acoustic waves by a quarter-plane

“…The key to the derivation of these three embedding formulae is that the primary radiated waves can be annihilated by some simple differential operators with constant coefficients. In the conclusion of Assier & Peake (2012), we conjectured the existence of an ultimate modified Smyshlyaev formula giving the corner diffraction coefficient everywhere. In order to obtain such a formula, if one wishes to follow a similar approach as in Shanin (2005b), one needs to derive an ultimate embedding formula.…”

“…The main advantage of this method compared with the one mentioned previously is that in this case the formulae giving the diffraction coefficient are 'naturally convergent' in the sense that they do not require a special treatment to regularize or accelerate the convergence. This method has been extensively described, adapted to the Neumann case and implemented by Assier & Peake (2012). The main 606 R. C. ASSIER AND N. PEAKE achievement of this method is to allow one to have access to the diffraction coefficient describing the diffraction by the corner of the quarter-plane.…”

“…Indeed, for example, two very important papers on the subject by Shanin (2005b) and Skelton et al (2010) give a different representation of this coefficient. The second motivation is the conjecture formulated at the end of Assier & Peake (2012), where the possibility of an ultimate modified Smyshlyaev formula was expressed. The existence of such a formula is closely related to the other diffracted fields (not only the diffraction from the corner but also the scattering and re-scattering by the two edges) involved in the quarter-plane problem.…”

“…Throughout this paper, the cases of Dirichlet (soft surface) and Neumann (hard surface) boundary conditions shall, when possible, be treated simultaneously. Hence, following the notation of Assier & Peake (2012), we introduce the indexes d,n such that d refers to the Dirichlet case and n refers to the Neumann case. This approach is similar in spirit to the approach taken by Budaev & Bogy (2005), in the first part of their paper, when dealing with the plane sector.…”

“…Finally, the wave diffracted by the corner of the quarter-plane is described by introducing the diffraction coefficient f d,n . The evaluation of this diffraction coefficient was the main topic of Assier & Peake (2012).…”