On the forming of sharp corners at a free surface

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 δmax 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.