Stable Model Equations for Long Water Waves

§ 1. Introduct ion In two earl ier papers , ci ted as I [1] and I I [2], it was shown how the H a m i l t o n fo rma l i sm can be used to ob ta in sat isfactory app rox ima te equat ions o f the Boussinesq type for fa i r ly long, fa i r ly low water waves. The ma in purpose was to find equat ions which are stable in the shortwave ta i l o f the wave spec t rum. The p rocedure used to arr ive at these equat ions was based on the canonica l theorem for the exact equat ions for gravi ty waves on an incompress ible , non-viscous fluid. The t r ea tmen t as given in I and I[ falls shor t in two respects. In the p r o o f o f the canonica l theorem in I some i m p o r t a n t detai ls were omit ted. In § 2 we will give a more precise p r o o f of this theorem. In der iving the a p p r o x i m a t e equat ions a t ten t ion was focussed on shor t -wave instabi l i ty . The poss ib i l i ty tha t solut ions become unstable due to the height o f the waves was not l ooked into. This poss ib i l i ty is connec ted with the occurrence o f a t e rm Tf I d x 1 2 = ~q~b~ (1.1) in the H a m i l t o n i a n (2.6) in II . F o r solut ions where the mo t ion would tend to concent ra te in regions o f less than average depth , that is of nega-