RSPA_addendum.zip from The generalized Wiener–Hopf equations for the elastic wave motion in angular regions

In this work, we introduce a general method to deduce spectral functional equations in elasticity and thus, the generalized Wiener–Hopf equations (GWHEs), for the wave motion in angular regions filled by arbitrary linear homogeneous media and illuminated by sources localized at infinity. The work extends the methodology used in electromagnetic applications and proposes for the first time a complete theory to get the GWHEs in elasticity. In particular, we introduce a vector differential equation of first-order characterized by a matrix that depends on the medium filling the angular region. The functional equations are easily obtained by a projection of the reciprocal vectors of this matrix on the elastic field present on the faces of the angular region. The application of the boundary conditions to the functional equations yields GWHEs for practical problems. This paper extends and applies the general theory to the challenging canonical problem of elastic scattering in angular regions.

本研究提出了一种通用方法，可用于推导弹性力学中的谱泛函方程，进而为任意线性均匀介质填充、受局域于无穷远的源照射的角域内的波动问题，推导出广义Wiener-Hopf方程（generalized Wiener–Hopf equations，GWHEs）。本研究拓展了电磁学应用中所采用的方法论，首次提出了一套完整的理论以获取弹性力学领域的广义Wiener-Hopf方程。具体而言，我们引入了一类特征矩阵取决于角域填充介质的一阶矢量微分方程。通过将该矩阵的倒易矢量投影至角域界面上的弹性场中，可便捷地得到上述泛函方程。将边界条件应用于该泛函方程，即可得到适用于实际工程问题的广义Wiener-Hopf方程。本文将该通用理论进行拓展，并将其应用于角域内弹性散射这一极具挑战性的典型问题。