Wave Equation in Halfspace and in Two Adjacent Wedges

Abstract

AbstractConsider the two adjacent rectangular wedges K1, K2 with common edge in the upper halfspace of R3 and the operator A = ‐aj Δ in Kj, acting on a subspace of Π2j=1L2(Kj) satisfying prescribed transmission conditions and the Dirichlet boundary condition on the bottom of R3+. We interprete the corresponding wave equation with A defining its spatial part as a simple model for wave propagation in two adjacent media with different material constants. Showing that A is selfadjoint and using the Fourier (‐sine) transformations we reduce our problem with singularities along the z‐axis to the Klein‐Gordon equation in one space dimension with potential step. The spectral theorem for unbounded selfadjoint operators yields the solution of the time dependent wave equation via expansion in generalized eigenfunctions of A. In the case a1 = a2 we have a halfspace problem with reflection due to the Dirichlet boundary condition and no transmission effects. We recognize this special case as a part of our original problem and conclude the discussion by a derivation of L∞‐time decay results for this case.