GENERAL-SOLUTIONS AND CAUSALITY FOR A VOIGT MEDIUM

A 1-D wave equation solution for a propagating seismic pulse in a Voigt medium can be obtained by using a separation of variables to find time harmonic particular solutions and then superimposing the particular solutions. This superposition is a time convolution of the boundary condition (or incident pulse) and the medium's impulse response. Even though causality is not introduced during the solution of the wave equation, the general solution is causal since the boundary condition is causal and the medium's impulse response can be shown to be causal. The relationship between attenuation and phase velocity as well as their dependence on frequency arise from the form chosen for the particular solutions. The arbitrary constants associated with the particular solutions are determined by the boundary condition, and the initial condition is also dependent on the boundary condition; however, the initial condition is properly determined and does not depend on times after the initial time (thereby satisfying causality). The convolutional nature of the general solution allows it to also be expressed as a time convolution of the boundary condition's time derivative and the medium's step function response. This expression can be viewed as a superposition of step function responses where the step function response is a particular solution to the wave equation obtained using an approach that is similar to one recently developed for propagating electric pulses. This new solution is obtained with the initial and boundary conditions being independently introduced during the solution of the wave equation. There is no frequency dependence in this solution, and the general solution is causal since it is a superposition of causal step function responses.