Complex analysis of elastic symbols and construction of plane wave solutions in the half-space (Q1396421)

In this paper, plane wave solutions for the general reduced wave equation \[ \Biggl(\sigma^2 I+\sum^n_{i,j=1} a_{ij}\partial_i \partial_j\Biggr) u(x)= 0\qquad\text{in }\mathbb{R}^n_+\tag{1} \] are considered. In (1), \(\sigma\) is an arbitrary positive fixed parameter, \(x= (x_1,x_2,\dots, x_n)\), \(\partial_i= \partial/\partial x_i\) \((i= 1,2,\dots, n)\), \(\mathbb{R}^n_+= \{x\mid x_n> 0\}\), \(a_{ij}\) \((i,j= 1,2,\dots, n)\) is a positive definite symmetric matrix. Plane wave solutions of (1) which satisfy the Dirichlet boundary condition on \(x_n= 0\) are constructed by the application of complex analysis to the inverse matrix of elastic symbols.