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Wave Equation in Halfspace and in Two Adjacent Wedges
Author(s) -
Mihalinčić K.
Publication year - 1998
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.19980781577
Subject(s) - eigenfunction , mathematical analysis , wave equation , mathematics , boundary value problem , gravitational singularity , dirichlet problem , operator (biology) , fourier transform , reflection (computer programming) , dirichlet boundary condition , physics , eigenvalues and eigenvectors , quantum mechanics , biochemistry , chemistry , repressor , computer science , transcription factor , programming language , gene
Consider the two adjacent rectangular wedges K 1 , K 2 with common edge in the upper halfspace of R 3 and the operator A = ‐aj Δ in Kj, acting on a subspace of Π 2 j=1 L 2 (K j ) satisfying prescribed transmission conditions and the Dirichlet boundary condition on the bottom of R 3 + . 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 a 1 = a 2 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.