Asymptotic solutions of the elastic wave equation and reflected waves near boundaries (Q1813572)

Firstly, the author considers the elastic wave equation \((\partial^ 2_ t-L(x,\partial_ x))u=0\), where \(x=(x_ 1,\ldots,x_ n)\in\Omega\subset\mathbb{R}^ n\), \(t\in\mathbb{R}\), \[ L(x,\partial_ x)u=\sum^ n_{i,j=1}a_{ij}(x)\partial_{x_ i}\partial_{x_ j}u +\sum^ n_{j=1}b_ j(x)\partial_{x_ j}u+c(x)u, \] and \(a_{ij}(x)\), \(b_ j(x)\) and \(c(x)\) are \(n\times n\) matrices of \(C^ \infty\)-functions, which satisfy some additional conditions. Asymptotic solutions \((\sigma\gg 1)\) (1) \(u^ l(t,x,\sigma)=\sum^ \infty_{j=0}e^{i\sigma(\varphi(x)-t)}u_ j(t,x) (i\sigma)^{-j}\) of the equation, connected with the eigenvalue \(\lambda_ l(x,\xi)\) \((\xi\in\mathbb{R}^ n)\) of the matrices \(\sum^ n_{i,j=1}a_{ij}(x)\xi_ i\xi_ j\) such that \(\lambda_ l(x,\partial_ x\varphi)=1\), are called the waves of \(\lambda_ l\)- mode. There is some boundary condition \(Bu=g\). Theorems on existence and uniqueness of the waves of \(\lambda_ l\)-mode in some domain \(U\) \((U\cap\partial\Omega\neq\emptyset)\) are proved in the case when \(g\) is represented in the form (1) on \(U\cap\partial\Omega\) and the first-order derivatives of \(\varphi\) along the boundary are near 0 on \(U\cap\partial\Omega\). In the second part, the author investigates waves reflected by boundaries for plane incident waves of \(\lambda_ l\)-mode. Especially, he examines whether or not the mode-conversion occurs near points where the incident waves hit the boundaries perpendicularly.