Extending the analysis of two-dimensional water wave problems to aperiodic infinite surfaces that allow for overturning waves. This work is the culmination of the work found in Preston et al. (JIE 2008, IMA JNA 2010) and uses a novel partition of the boundary to allow a well-posed solution to be proven.

Our previous work on aperiodic, half plane water wave problems required that the infinite free surface be described by the graph of a function. In this work we relax this restriction to allow for overturning waves to be considered by using a novel partition of the surface into 5 sections (relative to each point on the boundary). We follow the same arguments and use the same integral equation formulation as in our early works (that required only 3 partitions: a near field, a transition field and a far field) but require more subtly in the manipulations to show that our integral equation formulation is well-posed.

Preston et al.: A Nyström method for a boundary value problem arising in unsteady water wave problems, IMA J Numerical Analysis, doi: 10.1093/imanum/drq009, 2010
Preston et al.: An integral equation method for a boundary value problem arising in unsteady water wave problems, J Integral Equations, Vol. 20, No. 1, 2008