Interaction of Progressive Rarefaction Waves

where n is a positive integer, may be integrated readily over a limited region of the X0, t plane. Simple waves of this type may be generated in a tube of uniform cross section by moving a piston at one end of the tube containing the gas. The absence of shocks implies that the piston motion must be such as to increase the volume occupied by the gas and the velocity of the piston must be a monotonic increasing function of the time, however, it may have discontinuities. In a wave produced in this manner, the pressure in the disturbed gas will decrease as one goes from the front of the disturbed region to the rear. Such waves are called rarefaction waves. The method of integration of the hydrodynamics equations is due to Riemann' and involves interchanging the role of the dependent and independent variables. When this is done one is lead to a second order linear partial differential equation. In case (1.1) holds this equation reduces to the Euler-Poisson2 equation whose general solution may be given explicitly in terms of two arbitrary functions. The phenomena occurring, when two waves generated by piston motions at two ends of a tube interact, or, when one such wave meets either a rigid wall or a density discontinuity, may be described by determining these two arbitrary functions. It will be shown in the following that these functions can be determined by solving rather simple ordinary differential equations. Platrier3 has used a similar method for obtaining the interaction of two progressive waves each wave being generated by the motion of a piston subjected