Nonlinear water waves in channels of arbitrary shape

Abstract

The generalized channel Boussinesq (gcB) two-equation model and the forced channel Korteweg–de Vries (cKdV) one-equation model previously derived by the authors are further analysed and discussed in the present study. The gcB model describes the propagation and generation of weakly nonlinear, weakly dispersive and weakly forced long water waves in channels of arbitrary shape that may vary both in space and time, and the cKdV model is applicable to unidirectional motions of such waves, which may be sustained under forcing at resonance of the system. These two models are long-wave approximations of a hierarchy set of section-mean conservation equations of mass, momentum and energy, which are exact for inviscid fluids. Results of these models are demonstrated with four specific channel shapes, namely variable rectangular, triangular, parabolic and semicircular sections, in which case solutions are obtained in closed form. In particular, for uniform channels of equal mean water depth, different cross-sectional shapes have a leading-order effect only on the variations of a k-factor of the coefficient of the term bearing the dispersive effects in the model equations. For this case, the uniform-channel analogy theorem enunciated here shows that long waves of equal (mean) height in different uniform channels of equal mean depth but distinct k-shape factors will propagate with equal velocity and with their effective wavelengths appearing k times of that in the rectangular channel, for which k = 1. It also shows that the further channel shape departs from the rectangular, the greater the value of k. Based on this observation, the solitary and cnoidal waves in a k-shaped channel are compared with experiments on wave profiles and wave velocities. Finally, some three-dimensional features of these solitary waves are presented for a triangular channel.