Solution of the kinematic wave equation with an accelerating point source

In this paper, we present the solution of the kinematic wave equation ut+uux=Aδ(x−(t2/2)), with u=1 at t=0. This is a model for a point force accelerating through the sound speed at the sonic time t=1, when its velocity passes through the initial value of u. The linear approximation, when the nonlinearity parameter A is small, yields a singularity at the sonic point. Such sonic singularities are unphysical because the amplitude is limited by nonlinearity and/or damping. In this paper, we are interested in the effect of nonlinearity. We present a global solution, valid for arbitrary A and which is in most cases analytical. We construct the characteristic diagram and we resolve multivalued regions by shocks and empty regions by expansion fans. As A varies, the qualitative features of the characteristic diagram change. However, for all A the introduction of nonlinearity leads to a finite amplitude at the sonic point.