In some cases, it may be preferred to characterize the ocean surface or bottom by a complex reflection coefficient as a function of grazing angle. Such a reflection coefficient may be specified by tabulating the magnitude and phase at a discrete set of -values. To completely define the normal mode problem, must be defined continuously for some range of -values. A routine (REFCO) accomplishes this by linearly interpolating the magnitude and phase of the reflection coefficient. In addition, the reflection coefficient must be converted to an equivalent Robin style boundary condition with coefficients which are a function of horizontal propagation number, k.
Note that a surface reflection coefficient asserts a relationship between an upgoing and downgoing wave of the following form.
is the vertical wavenumber. Then,
which implies that,
Thus, the reflection coefficient implies an impedance or Robin style boundary condition of the form,
For bottom reflection we use the form
which leads to,
In general, the reflection coefficient is a function of the grazing angle which is related to by