Foundations of rigorous ray tracing

Abstract

Rigorous ray tracing may be derived by inserting an ansatz into the Helmholtz equation to develop an alternate Hamilton‐Jacobi equation for rigorous ray tracing expressed by (∂W∂z2) + Cm−2 − C−2 = −12ω2 ∂3W/∂z3∂W/∂z + 34ω2 (∂2W/∂z2∂W/∂z)2, where W is Hamilton's characteristic function (the generator of motion), z is depth, C is the sound velocity profile, Cm is the constant of motion (vertex velocity), and ω is the radial frequency. The left side of the above equation manifests the Hamilton‐Jacobi equation for classical ray tracing. The terms on the right side of the above equation, which contain the common factor ω−2, compensate for finite wavelength. This alternate Hamilton‐Jacobi equation does not have any associated auxiliary equation and may be related to the original Hamilton‐Jacobi equation for rigorous ray tracing [E. R. Floyd, J. Acoust. Soc. Am. 75, 803–808 (1984)]. Equations of motion for both ray paths and wave normals follow from the generator of motion W.