We consider the boundary-value problem for the Helmholtz equation in a circular cone with an impedance boundary condition on its face. A new approach for its solution is proposed. The scheme of solution includes applying the Kontorovich Lebedev transform, derivation of a second-order difference equation in a strip of a complex variable, and reduction of the latter to an integral equation of the convolution type with variable coefficients. It is also shown that the equation is equivalent to a 2 x 2 matrix Riemann Hilbert problem with a discontinuous coefficient. We analyze the behavior of the solution of the integral equation at the ends of the contour and construct an approximate solution using a collocation method. The diffraction coefficient is found in terms of the solution of the integral equation. Numerical results for the diffraction coefficient and comparative analysis of the results for the impedance cone with the limiting cases of the acoustically soft and hard cones are reported. A full high frequency asymptotic expansion for the scattering field in the region where no reflected waves are observed is derived for the impedance, soft, and hard cones. [References: 27]
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机译：我们考虑圆形锥面上具有阻抗边界条件的Helmholtz方程的边值问题。提出了一种新的解决方案。解决方案包括应用Kontorovich Lebedev变换，在复变量带中推导二阶差分方程，以及将后者简化为具有可变系数的卷积型积分方程。还表明，该方程等效于具有不连续系数的2 x 2矩阵Riemann Hilbert问题。我们分析轮廓末端的积分方程解的行为，并使用搭配方法构造一个近似解。根据积分方程的解求出衍射系数。报道了衍射系数的数值结果以及阻抗锥在声学软锥和硬锥的极限情况下的对比分析。对于阻抗锥，软锥和硬锥，导出了在未观察到反射波的区域中散射场的完整高频渐近扩展。 [参考：27]
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