A.M.Gomilko, V.T.Grinchenko, E.V.Lobova A reflection principle in plane boundary problems for the Helmholtz equation |
Acoustic bulletin, Vol. 1 ¹ 2, (1998) |
| The paper deals with possibilities of using the reflection principle when developing solutions of the internal and external boundary problems for the Hemholtz equation in plane domains with the boundaries containing the straight segments. The approach essence is to extend the desired solution in the circular canonical domain using the solution reflection formula for the Helmholtz equation with respect to straight boundary segments (under the uniform boundary conditions). In this case, the solution of the boundary problem is expressed through the series with respect to particular solutions of the Helmholtz equation in polar coordinates, for determining of which unknown coefficients the infinite system of algebraic equations may be obtained. The closure equation on the circle segments that are not the physical boundaries of the initial domain should be formulated originating from the chosen reflection manner for desired solution. Different variants are considered for the boundary conditions for the Helmholtz equation in the circular-straight meniscus (the internal and external problems). It is shown how it might be possible to take into account the local wave field singularities, that occur due to existence of angular points and mixed type of boundary conditions. The numerical calculations demonstrating the efficiency of the offered approach are presented for one of the mentioned problems. |
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KEY WORDS: a reflection principle, the Helmholtz equation |
| TEXT LANGUAGE: Russian |