Note on waves on the surface of running water

Extract

The method for the solution of problems of surface waves on water with which this note is concerned can be sufficiently illustrated by a particular example given by Lamb (1). In Lamb's example it is supposed that a steady stream in which the x, y components of the velocity are U, 0, where U is constant, is slightly disturbed by the application of a constant pressure ρPU² to a band of finite width 2a of the surface of the stream; ρ denotes the density of the water and P a non-dimensional constant. In the solution Lamb, using an artifice due originally to Rayleigh and widely used since, assumes that ‘the deviation of any particle of the fluid from the state of uniform flow is resisted by a force proportional to the relative velocity’, and states that this ‘law of friction does not profess to be altogether a natural one, but it serves to represent in a rough way the effect of small dissipative forces; and it has the great mathematical convenience that it does not interfere with the irrotational character of the motion’.