The role of the frozen surface approximation in small wave-height perturbation theory for moving surfaces

In the standard development of the small wave-height approximation (SWHA) perturbation theory for scattering from moving rough surfaces [e.g., E. Y. Harper and F. M . Labianca, J. Acoust. Soc. Am. 58, 349-364 (1975) and F. M. Labianca and E. Y. Harper, J. Acoust. Soc. Am. 62, 1144-1157 (1977)] the necessity for any sort of frozen surface approximation is avoided by the replacement of the rough boundary by a flat (and static) boundary. In this paper, this seemingly fortuitous byproduct of the small wave-height approximation is examined and found to fail to fully agree with an analysis based on the kinematics of the problem. Specifically, the first order correction term from the standard perturbation approach predicts a scattered amplitude that depends of the source wave number, whereas the kinematics point to a scattered amplitude that depends on the scattered wave number. It is shown that a perturbation approach in which an explicit frozen surface approximation is made before the SWHA is invoked predicts (first order) scattered amplitudes that are in agreement with the analysis based on the kinematics.