Abstract In a recent paper Lord Rayleigh has called attention to the fact of the instantaneous propagation of a limited disturbance over the surface of heavy incompressible fluid. This instantaneity occurs in spite of the fact that the velocity of any simple-harmonic gravity wave is finite, the depth being finite and constant. As the points thereby raised are of some delicacy, and are not completely settled in the paper quoted, some further remarks on the phenomenon may not be superfluous. 2. Solution of the Cauchy-Poisson Problem for Finite Depth. In his treatment of this case Lord Rayleigh used the integral form of solution. There are, however, great difficulties raised in this way on account of lack of convergence at the surface. I proceed to obtain a solution in the form of a series similar in type to the known serial solution of the problem for infinite depth.