Groundwater waves in aquifers of intermediate depths

Abstract

In order to model recent observations of groundwater dynamics in beaches, a system of equations is derived for the propagation of periodic watertable waves in uncofined aquifers of intermediate depths, i.e. for finite values of the dimensionless aquifer depth nwd K which is assumed small under the Dupuit-Forchheimer approach that leads to the Boussinesq equation. Detailed consideration is given to equations of second- and infinite-order in this parameter. In each case, small amplitude ( η d ⪡ 1 ) as well as finite amplitude versions are discussed. The small amplitude equations have solutions of the form η( x, t) = η 0 e − kx e iwt in analogy with the linearized Boussinesq equation but the complex wave numbers k are different. These new wave numbers compare well with observations from a Hele-Shaw cell which were previously unexplained. The “exact” velocity potential for small amplitude watertable waves, the equivalent of Airy waves, is presented. These waves show a number of remarkable features. They become non-dispersive in the short-wave limit with a finite and quite slow decay rate affording an explanation for observed behaviour of wave-induced porewater pressure fluctuations in beaches. They also show an increasing amplitude of pressure fluctuations towards the base, in analogy with the evanescent modes of linear surface gravity waves.