Fourier analysis of Rlemman-Earnshaw simple waves

In the ideal gas, Riemman-Earnshaw (R-E) solution for the simple waves has been expanded as a Fourier series. By means of a numerical computation, a comparison between the R-E and Fubini solutions was carried out, from which a conclusion can be drawn that the Fubini solution approaches closely the exact solution when the Mach number not exceeds 0.2. However, along with the increase of Mach number and the propagating distance, the difference between them increases.