Modelling of gravity waves in water of finite depth

An extension of shallow water theory proposed by Wilde (Wilde, Chybicki 2000), for finite water depth and based on the Lagrangian type formalism is presented. As in Bussinesq-type models the vertical dimension is being eliminated and the horizontal displacement is expanded in the even power series of vertical variable Y, but only two terms - with power null and two are taken into account. Based on continuity equation, vertical displacement is expressed in terms of horizontal displacement and its derivatives. The equations of motion are derived from a Hamilton principle applied to Lagrangian function being a difference of kinetic and potential energy. In order to solve the set of governing equations a direct method of variational calculus has been applied. The solutions preserve total energy. The numerical simulations have been verified experimentally, in terms of wave measurements in the flume, for various wave heights and ratios of wavelength to water depth, showing good conformity between measured and calculated values. The theory presented here can also be applied for the case of varying depth.