Multi-dimensional MHD simple waves

In this paper we consider a formalism for multi-dimensional simple MHD waves using ideas developed by Boillat. For simple wave solutions one assumes that all the physical variables (the density rho, gas pressure p, fluid velocity V, gas entropy S, and magnetic induction B in the MHD case) depend on a single phase function phi(r,t). The simple wave solution ansatz and the MHD equations then require that the phase function has the form phi = r x n(phi) - lambda(phi)t, where = n(phi) = Delta phi / (absolute value of Delta phi) is the wave normal and lambda(phi) = omega/k = -phi t / (absolute value of Delta phi) is the normal speed of the wave front. The formalism allows for more general simple waves than that usually dealt with in which n(phi) is a constant unit vector that does not vary along the wave front. The formalism has implications for shock formation for multi-dimensional waves.