Mathematical model for non-linear standing waves in a tube

A mathematical model is developed to describe the behavior of one-dimensional, non-linear standing waves in a fluid-filled, rigid-wall tube. The tube is bounded on one end by a piston vibrating with periodic motion and terminated on the other end by an impedance boundary. A numerical procedure for calculations based on this model also is described. Calculations are carried out on a digital computer. The model and computer program are presently restricted to pre-shock conditions. Numerical results obtained for the special case of a rigid reflector and a sinusoidally moving piston are in excellent agreement with the results of Coppens and Sanders (private communication). The general approach is applicable to any one-dimensional system bounded by a velocity condition on one end and an impedance condition on the other end.