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The Malyuzhinets theory for scattering from wedge boundaries: a review. (English) Zbl 1074.76611

Summary: The Malyuzhinets technique is reviewed based on his fundamental papers of the 1950s. Subsequent developments are surveyed and recent advances are presented. The review is focused around the basic problem of determining the wave field scattered from the edge of a wedge of exterior angle \(2\Phi\) with arbitrary impedance conditions on either face. We begin by establishing a direct relationship between the Sommerfeld integral representation and the Laplace transform. This provides fresh insight into Malyuzhinets’ inferences about functions representable via the Sommerfeld integral and, simultaneously, allows us to prove both the inversion formula for the Sommerfeld integral and the crucial nullification theorem. The special functions \(\eta_\Phi(z)\) and \(\psi_\Phi(z)\) occurring in Malyuzhinets’ theory of diffraction from a wedge-shaped region are described. Based on this theoretical background, we present a detailed derivation of the well-known Malyuzhinets expressions for the wave field diffracted by an impedance wedge. An alternative representation of the Malyuzhinets solution as a series of Bessel functions is also presented that is completely equivalent to the integral form of the Malyuzhinets solution. This permits a description of the wave field in the vicinity of the edge of an impedance wedge, when \(kr\leq1\), and simple expressions are given for the tip values of the field and its first derivatives. The edge value \(u_0\) can be expressed in terms of Malyuzhinets functions, and its magnitude is easily evaluated if the impedances of the wedge faces are purely imaginary. Thus, \(|u_0|\leq\pi/\Phi\) with equality only for a wedge with Neumann boundary conditions.

MSC:

76Q05 Hydro- and aero-acoustics
74J20 Wave scattering in solid mechanics
78A45 Diffraction, scattering
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