Ray method for flow of a compressible viscous fluid

The author describes an asymptotic method to solve the linearized Navier-Stokes equations governing the flow of a compressible viscous fluid subject to free surface and rigid bottom boundary conditions. The solution of these equations is assumed to consist of a phase function and an amplitude function. It is found that the phase function satisfies the Hamilton-Jacobi equation, and the first order approximation to the amplitude function satisfies a transport equation. The Hamilton-Jacobi equation may be solved by means of the method of characteristics, which reduces the equation to a set of ordinary differential equations. Their solutions determine a family of time-space curves called rays. The transport equation can be easily integrated along each ray to yield the so-called conservation relation. At certain anomalies the amplitude function becomes infinite and a uniform expansion is then constructed to remove these difficulties.