Asymmetric Doppler amplitudes in the surface scatter channel

The generation of asymmetric Doppler sidebands in scatter channels in which the transmitter and receiver are at different depths has been demonstrated in several recent papers [E. Y. Harper and F. M. Labianca, J. Acoust. Soc. Am. 57, 1044–1051 (1975); 58, 349–364 (1975)]. In these papers the wave equation was solved by using a perturbation procedure. In the present paper a similar result is obtained by using a modified Fresnel expansion of the Kirchoff integral. The resulting expansion, which is correct to third order but contains only terms up to second order is then easily manipulated by standard methods. If the source and receiver are located in the xz plane at points (0, z0) and (x1, z1), respectively, and if the surface motion is along the x axis, the ratio of side‐band amplitudes is approximately exp [(4p/k) (z02 − z12)/(z03 + z13)x1] where 2π/p is the surface wavelength and k is the acoustic wave number. Other geometries and surface models are also easily computed. [Supported by ONR, Code 480.]