On the forming of sharp corners at a free surface

Abstract

By applying the technique for time-dependent irrotational flows proposed in the preceding paper, a new class of exact free-surface flows is derived. In these, the free surface has the form of a variable hyperbola, whose axes rotate in space. The angle γ between the asymptotes, and the angle δ of orientation of the axes, are found explicitly as functions of the time. The solutions fall into three groups. First there are those in which γ diminishes smoothly from 90° (a rectangular hyperbola) to zero (a slender hyperbola) while the angle of orientation 8 increases towards a finite limit. Secondly there are solutions in which γ diminishes to a positive minimum, and then returns again to 90°. Thirdly γ may begin from small values, increase to less than 45° and return again towards zero. In each case the total angle δ max remains finite. An exceptional but very interesting solution, in which the vertex angle γ remains constant at 45° and the free surface rotates with uniform angular velocity δ, is described in terms of elementary functions of the time t . It is suggested that these flows, which are generalizations of the symmetrical Dirichlet hyperbolae, are relevant to the flow near the tip of a breaking gravity wave. Since for large values of t the angle δ tends to its limit like t ~ 1 , the flows may be matched asymptotically to the parabolic arch of a plunging breaker. In other cases, the tip of the wave can curl over and appear to form a vortex. By the inclusion of terms cubic in the space coordinates it is also possible to represent a sharp crest pointing upwards and tending towards a cusp.