  Paraxial ray tracing procedures have become widely accepted techniques for
acoustic models in seismology and underwater acoustics. To date a generic form
of these procedures including fluid motion and time dependence has not appeared
in the literature. A detailed investigation of the characteristic curves of the
equations of hydrodynamics allows for an immediate generalization of the
procedure to be extracted from the equation form geodesic deviation. The
general paraxial ray trace equations serve as an ideal supplement to ordinary
ray tracing in predicting the deformation of acoustic beams in random
environments. The general procedure is derived in terms of affine
parameterization and in a coordinate time parameterization ideal for
application to physical acoustic ray propagation. The formalism is applied to
layered media, where the deviation equation reduces to a second order
differential equation for a single field with a general solution in terms of a
depth integral along the ray path. Some features are illustrated through
special cases which lead to exact solutions in terms of either ordinary or
special functions.
