A note on the breaking of waves

A model equation for water waves has been suggested by Whitham (1967) to study, qualitatively at least, the different kinds of breaking. This is an integro-differential equation which combines a typical nonlinear convection term with an integral for the dispersive effects and it is of independent mathematical interest. For an approximate kernel of the form e-b|x|, it is shown first that solitary or periodic waves have a maximum height with sharp crests and secondly that waves which are sufficiently asymmetric break into ‘bores’. The second part applies to a wide class of bounded kernels, but the kernel giving the correct dispersion effects of water waves has a square root infinity and the present argument does not go through. Nevertheless, the possibility of the two kinds of breaking in such integro-differential equations is demonstrated.