The wave equation in a halfspace with mixed boundary conditions are used as a model for the crack and punch problem in linear elasticity. The Fourier-Laplace transform with respect to time and tangential spatial variables are used to reduce the problem to a pseudo-differential equation on the boundary. Sufficient conditions on the boundary data for unique solvability in a Sobolev space is investigated with the aid of an elliptic regularization of the symbol. In particular it is observed that unlike its elliptic counterpart, the solution loses one order of smoothness in time. Numerical methods of finite element type are formulated as a means of solving the above pseudo-differential equations. Analysis is then done on the method via energy arguments and shows that the rate of convergence is slightly less than a half.
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