The effects of phase-front distributions on the radiation patterns of an echosonde (acoustic echo-sounding) antenna are studied. A radiation integral is used to compute an equivalent excitation at the transmitting antenna aperture at a height H above the illuminating aperture of smaller radius. A two-dimensional Fourier transform utilizing the Kirchhoff-Fresnel diffraction formula is used to obtain the far-field expressions. The results reported for linear, square-law, cubic, and periodic phase-front distributions, indicate that (while details of the antenna pattern depend upon the specific phase distribution used) all distributions tended to raise the sidelobes and broaden the mainlobe for stronger phase modulations. In some cases, the stronger phase modulations actually reduced the central portion of the mainlobe, producing a hollow conical beam. Some computed beam patterns at a carrier frequency of 2 kHz and with different phase modulation coefficients were compared with two recent sets of experimentally measured patterns. The agreement overall is good with any variations probably being due to the fact that the actual phase distributions on the antennas were not exactly known and also due to the lateral drifts that might be experienced by the tethered weather balloon supporting the microphone probe (used for some of the pattern measurements) as a result of wind breezes.