Reflection of plane waves from a linear transition layer in liquid media

It is shown that care should be taken in using the term "velocity" in connection with wave propagation in inhomogeneous media. An expression is derived for phase velocity which depends on frequency and depth. Exact solutions are found for normal and oblique incidence, for plane-wave propagation in a liquid medium. It is shown that the geometrical optics approximation solution, valid for high frequencies, is the first term in an asymptotic expansion of the exact solution.The reflection coefficients are obtained for a linear transition layer between two homogeneous half-spaces. Both first-order and second-order discontinuities in density and bulk modulus are considered at the boundaries of the transition layer.