An exact integral of complete spectral equations for unsteady one-dimensional flow

Finite-amplitude, one-dimensional, isothermal flow of a perfect gas is considered. When the flow is organized initially as a “linear” sound wave, it is equivalent to the solution of an “advection equation”. The flow that evolves in this way from an initial state is analyzed by means of a Fourier series in space, with time-dependent coefficients. These coefficients can be determined exactly when the initial velocity is a pure sinusoid. The result is such that the Fourier series in space is a Kapteyn series in time. The series is valid up to the time of shock formation. Throughout this interval the lowest harmonics inferred from severely truncated spectral equations are in good agreement with the exact solution. However, the spectral representation becomes very inefficient as the discontinuous state is approached.