Transient Solution for Sound Radiated into a Viscous Fluid

Sinusoidal vibration of a plane at the edge x = 0 of a viscous fluid causes sound to be radiated into the fluid. The motion begins at time t = 0. Of primary interest is the transient signal. The wave equation, which in this case is of third order because of the presence of viscosity, is solved by changing the form of the viscosity term slightly. The approximation made is justified as long as αλ≪1, where αλ is the dimensionless attenuation coefficient. The altered equation is reducible to the diffusion equation, whose solutions are well known. One of these solutions is adapted for the present problem. It turns out that the transient part of this solution can be represented in terms of the error function of complex argument, a tabulated function. The transient proves to be significant in the vicinity of the main wave front x=c0l. When the wave is first generated, the transient is very weak, but it grows in importance as the wave travels away. Near the point x = 1/α the transient has its maximum amplitude. The diffusive character of the transient is shown by the fact that if the transient signal is integrated from t = 0 to t = ∞, the integral never vanishes, not even at x = ∝. Improved solutions are also discussed.