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.

中文翻译：

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混合边界条件下楔形和半平面上非平稳波的衍射：楔形障碍物的精确解和冲击波减弱

当在楔形的一侧设置Neumann条件而在另一侧设置Dirichlet条件时，考虑了混合边界条件下楔上非平稳平面波的衍射问题。对于阶跃函数形式的入射平面波，通过直接方法基于问题的自相似性，即通过使用以下事实，可以得到问题的精确解（通过基本函数表示）解决方案是空间坐标和时间的齐次函数。此方法统一减小了问题的维数，并导致了双曲区域中弦振动方程的初边值问题和椭圆区域中二维Laplace方程的混合边界问题。通过未知函数的解析连续性，将椭圆域的混合问题简化为在整个边界上具有一种边界条件的问题，从而给出了精确的解决方案。与楔形解决方案相比，半平面的结果大大简化了。研究了所得溶液在楔形尖端和衍射前沿附近的行为，并指出了与在整个边界上具有一类边界条件的衍射问题的结果在质量上的差异。数值比较给出了混合衍射问题的近前渐近解和Friedlander对于具有声学刚性壁的楔的相似渐近解。