On the Source Terms in Lilley's Equation

A review of the early developments and later additions concerning Lilley's equation is presented, attention being given to the relationship between different existing expressions for the source function. The importance of expressing source terms in the form of proper equivalent sources, which is easily achieved if the source terms are directly identified at the fundamental equation level (i.e., in the continuity, momentum and energy equations), is emphasised. The fact that this identification procedure was not used in most of the earlier work is shown to have resulted in expressions for the source function in which one or more terms lack a clear physical meaning. Problems and misunderstandings associated with these expressions are identified and discussed. It is pointed out that, irrespective of the chosen dependent variable, the number of equivalent source terms in Lilley's equation is actually quite small. The formulation due to Dowling, Ffowcs Williams and Goldstein, presented in 1978, based on Lighthill's analogy and originally given for a plug flow, is extended for a shear flow, while the compact form developed by Goldstein in 2001 is generalised for non-perfect gases for which the density dependence on entropy and pressure is separable. It is shown that both approaches are equivalent for homoentropic flows. An expression equivalent to those given by Goldstein in 2002 and 2003, but advantageously simpler, is derived.