Acoustic wave propagation in a fluid-filled borehole is affected by the type of rock which surrounds the hole. More specifically, the slowness dispersion of the various body-wave and borehole modes depends to some extent on the properties of the rock. We have developed a technique for estimating the dispersion relations from data acquired by full-waveform digital sonic array well-logging tools. The technique is an extension of earlier work and is based on a variation of the well-known Prony method of exponential modeling to estimate the spatial wavenumbers at each temporal frequency. This variation, known as the forward-backward method of linear prediction, models the spatial propagation by purely real-valued wavenumbers. The Prony exponential model is derived from the physics of borehole acoustics under the assumption that the formation does not vary in the axial or azimuthal dimensions across the aperture of the receiver array, but can vary arbitrarily in the radial dimension. The exponential model fits the arrivals of body waves (i.e., head waves) well, because the body waves are dominated by a pole rather than a branch point. Examples of this processing applied to synthetic waveforms, laboratory scale-model data, and field data illustrate the power of the technique and verify its ability to recover dispersion relations from sonic array data. The interpretation of the estimated dispersion in terms of rock properties is not discussed.