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Diffraction of nonstationary waves on wedges and half‐planes under mixed boundary conditions: Exact solutions and weakening of blast waves by wedge‐shaped barriers
Author(s) -
Israilov Mukhady Sh.
Publication year - 2021
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.201900279
Subject(s) - mathematical analysis , wedge (geometry) , boundary value problem , mathematics , diffraction , mixed boundary condition , free boundary problem , neumann boundary condition , robin boundary condition , geometry , physics , optics
The problem of diffraction of nonstationary plane waves on a wedge under mixed boundary conditions is considered when the Neumann condition is set on one side of the wedge and the Dirichlet condition is set on other side. For the case of an incident plane wave in the form of a step function, an exact solution of the problem (expressed through elementary functions) is obtained by the direct method based on self‐similarity of the problem, i.e., using the fact that its solution is a homogeneous function of space coordinates and time. This method reduces the dimension of the problem by unity and leads it to the initial‐boundary value problem for the equation of string oscillations in the hyperbolic region and to the mixed boundary problem for the two‐dimensional Laplace equation in the elliptic region. The exact solution is given by reducing the mixed problem for an elliptic domain to a problem with boundary conditions of one type on the whole boundary by means of analytic continuation of unknown function. The results for a half‐plane are significantly simplified in comparison with the solution for a wedge. The behaviour of the obtained solution at the tip of the wedge and near the diffraction front is investigated and qualitative differences from the results for diffraction problems with one‐type boundary conditions on the whole boundary are noted. The numerical comparison is given of the near‐front asymptotic solution of mixed diffraction problem and the similar asymptotic solution by Friedlander for a wedge with acoustically rigid walls. This comparison shows that in the case of a wedge with mixed boundary conditions there is a much greater weakening (attenuation) of the blast waves in the shadow region.