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The solution of the Laplace–Young equation determines the equilibrium height of the free surface of a liquid contained in a vessel under the action of gravity and surface tension. There are only two non-trivial exact solutions known; one corresponds to a liquid occupying a semi-infinite domain bounded by a vertical plane wall while the other relates to the case when the liquid is constrained between parallel walls. A technique called boundary tracing is introduced; this procedure allows one to modify the geometry of the domain so that both the Laplace–Young equation continues to be satisfied while the necessary contact condition on the boundary remains fulfilled. In this way, new solutions of the equation are derived and such solutions can be found for certain boundaries with one or more sharp corners and for others that possess small-scale irregularities that can be thought of as a model for roughness. The method can be extended to construct new solutions for a variety of other physically significant partial differential equations.

The groundwater response to tidal forcing is described by Laplace's equation for the hydraulic head subject to appropriate boundary conditions. A previous solution to this problem based on long-wave and shallowaquifer approximations is extended to higher-order and a general solution scheme, automated in Mathematica, is described. The solution can, in principle, be extended to arbitrary order, but is restricted to a detailed study of the properties of the solution when truncated to sixth-order. These results show that the mean water table height increases throughout the aquifer to a steady inland height elevated above mean sea level. The phase shift of the fluctuations and the asymmetry of the pore drainage process are investigated using Fourier analysis.

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