Fourier decomposition of a plane nonlinear sound wave and transition from Fubini´s to Fay´s solution of Burger´s equation

Burgers' equation describes plane sound wave propagation through a thermoviscous fluid. If the boundary condition at the sound source is given as a pure sine wave, the exact solution is given by the Cole-Hopf transformation as a quotient between two Fourier series. Two approximate Fourier series representations of this solution are known: Fubini's (1935) solution, neglecting dissipation and valid at short distance from the sound source, and Fay's (1931) solution, valid far from the source. In the present investigation a linear system of equations is found, from which the coefficients in a series expansion of each Fourier coefficient can be derived one by one. Curves which join smoothly to Fubini's solution (valid up to slightly before shock formation) and to Fay's solution (valid for approximately three shock formation distances). Maxima for the Fourier coefficients of the higher harmonics are given. These maxima are situated in a region where neither Fubini's nor Fay's solution is valid