Low-Amplitude Nonlinear Theory of Plane Progressive Waves

Abstract

As the title suggests, this is a theory of plane waves of finite amplitude which is intermediate between traditional small-signal theory and exact nonlinear theory. It is based upon a certain assumption concerning the relation between wave amplitude and distance from the wave source. When this assumption holds, as it does in a large class of cases, one can simplify substantially Earnshaw's exact but highly unmanageable solution for plane waves. Application of the same assumption to the differential equations governing plane-wave propagation allows one to derive a useful approximate nonlinear wave equation. The exact solution of this approximate equation is the same as the simplified version of Earnshaw's solution. One interesting feature of this theory is that the form of the approximate nonlinear wave equation and its solution are the same in Eulerian and Lagrangian coordinates. A specific example will be discussed, namely the wave motion produced by sinusoidal motion of a piston in a lossless tube. It is found that the work of many past investigators can be unified by this theory. The limits of validity of the theory will also be discussed. (Part of this work was carried out at Harvard University with support from the Office of Naval Research.)