A new analytical solution for tide-induced groundwater fluctuations in an unconfined aquifer with a sloping beach

Deriving analytical solutions for tide-induced groundwater fluctuations in unconfined aquifers confronts two problems: (1) As the Boussinesq equation itself contains nonlinear terms, the “secular term” would be generated in derivation, thus making perturbation solution unable to be deduced to higher order; (2) for aquifers with sloping beaches, the perturbation parameter in existing analytical solution integrating the beach slope and hydrogeological property would be sometimes larger than 1. So the application of perturbation solutions is relatively limited. Furthermore, as the beach slope decreases, the error of analytical solution would gradually increase. Given that water table over-height would increase the aquifer thickness and speed up wave propagation, this paper integrates over-height into the perturbation parameter and adjusts boundary conditions to settle the problem of “secular term” and to derive a new high-order analytical solution for nonlinear Boussinesq equation in terms of sloping beaches. Results show that the new analytical solution is more reasonable, and the analytical accuracy is obviously improved in comparison with the existing analytical solution for a gentle slope. The new analytical solution provides a theoretical basis for analyzing the propagation characteristics (e.g., wave length and over-height variation) of tide-induced groundwater wave in unconfined aquifers, particularly those with sloping beaches.