The axially symmetrical spreading of waves into water at rest and of uniform depth at large distances from the centre of the disturbance is studied. The long-wave equations provide a plausible first approximation, but on grounds differing from those previously advanced for their justification. In particular, the wave length need not be large compared with the depth. The approximate solution of the nonlinear equations is derived for the initial phase of the motion in which the amplitude is small, assuming that any bores occurring are weak at the large radii considered.
Meyer, R. E., and M. C. Shen. 1962. "On the radial spreading of water waves." Journal of Marine Research 20, (3). https://elischolar.library.yale.edu/journal_of_marine_research/984