CREST INSTABILITIES OF GRAVITY-WAVES .2. MATCHING AND ASYMPTOTIC ANALYSIS

In a previous study (Longuet-Higgins & Cleaver 1994) we calculated the stability of the flow near the crest of a steep, irrotational wave, t he 'almost-highest' wave, considered as an isolated wave crest. In the present paper we consider the modification of this inner flow when it is matched to the flow in the rest of the wave, and obtain the normal -mode perturbations of the modified inner flow. It is found that there is just one exponentially growing mode. Its rate of growth beta is a decreasing function of the matching parameter e and hence a decreasing function of the wave steepness ak. When compared numerically to the r ates of growth of the lowest superharmonic instability in a deep-water wave as calculated by Tanaka (1983) it is found that the present theo ry provides a satisfactory asymptote to the previously calculated valu es of the growth rate. This suggests that the instability of the lowes t superharmonic is essentially due to the flow near the crest of the w ave.