On the Breaking of Progressive Waves in Shallow Water Part 3.

The basic equations for wave motions are formed with the surface displacement-η and the velocity potential-φ. The previous paper proposed a model of the breaking of waves from these (η,φ)-systems. In this paper the fourth-order wave equation, expessed in the surface displacement only, that is, η-system, is obtained from the (η,φ)-system in the same way of the second report. The breaking model is recast from Hamiltonian function, which is the integral surface of the η-system, by appling catastrophe thoery to this function based on the view of perturbations for the surface. Analytical characteristics of wave profile, wave velocity and the breaking criterion for the catastrophe model are discussed in detail and compared with that of Stokes waves, Laitone's cnoidal waves and others. The main results of this analysis are summarized as follows: (1) Breaking hight and maximum wave velocity of the solitary wave are 0.8375 and 1.309, respectively, and nearly equal to that gained by Byatt-Smith and Fenton. (2) The breaking criterion of periodical waves agrees well with Miche formula and Shuto's criterion for Stokes waves in the range where the shallowness is less than 0.05. (3) Solitary and periodical wave profiles have steeper crests than that of Stokes and cnoidal waves. (4) Other periodical wave characteristics, that is, wave length, hight from trough to still water level, wave crest hight and wave velocity etc., show the values obtained from longer period waves about 5-8% than the other wave theories