ON THE WEAK SOLUTION OF THE WAVE EQUATION WITH VARIABLE COEFFICIENT

Abstract

We look at the weak solution of the 1D wave equation where the coefficient C(x) has a small BV and takes values close to 1. It begins by approximating BV C(x) as a sequence of step functions. Our goal is to now study the special case of the wave equation where C(x) is a step function in order to generalize our results to the BV case. In order to do so, we take the partial Laplace transform of the wave equation with respect to the time variable. We then compute the solution of our transformed equation using a reflection and transmission series. By observing that transmission waves (waves that do not get reflected) account solely for the dominant term in the solution, and that the contribution of the accumulation of the reflected waves is at most proportional to the contribution of the transmission wave, we obtain, with the inverse Laplace transform, the form of the weak solution to the wave equation with C(x) a step function in terms of the solution to the standard wave equation. Lastly, dominated convergence gives us that the solution to the approximated wave equations converges point-wise to the solution of the BV case.