Nonlinear water waves in channels of arbitrary shape

Abstract

The generalized channel Boussinesq (gcB) two-equation model and the forced channel Kortewegae 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.

On Boussinesq models of constant depth

“…For the case of aϭ0, bϭ1/3, system ͑d͒ becomes system ͑a͒ with depth-mean velocity. The relationship between wave speed and the wave amplitude ␣ is given by 27 ϭͱ 6͑1ϩ␣ ͒ 2 ␣ 2 ͑ 3ϩ2␣ ͒ ͓͑1ϩ␣ ͒ln͑ 1ϩ␣ ͒Ϫ␣͔, ͑61͒…”

Evolution of long water waves in variable channels

“…Model equations Generation and propagation of nonlinear long waves in a variable channel of arbitrary shape, whose width and water depth are supposed to be of the same order, can be described by the generalized channel Boussinesq (gcB) model (Teng & Wu 1990,1992: Evolution of long water waves in variable channels 305 with the tilde denoting the surface mean averaged across the channel surface width and the bar the section mean averaged over the cross-sectional area. All variables in (1)-(4) are written in non-dimensional form with the length variables scaled by the unperturbed mean water depth, h,, and the time scaled by (h,/g)i, g being the gravitational acceleration.…”

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This paper applies two theoretical wave models, namely the generalized channel Boussinesq (gcB) and the channel Korteweg-de Vries (cKdV) models (Teng & Wu 1992) to investigate the evolution, transmission and reflection of long water waves propagating in a convergent-divergent channel of arbitrary cross-section. A new simplified version of the gcB model is introduced based on neglecting the higher-order derivatives of channel variations.

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A Solitary Wave Propagates through a Gap into a Channel with an Upward Step

“…Chang et al (1979) [7], with the added experimental measurements, utilized Shuto's equation to investigate numerically on solitary waves moving along a rectangular channel with a linearly varying width but a uniform depth. Teng and Wu (1992) [8] applied two theoretical wave models, namely the generalized channel Boussinesq (gcB) and the channel Korteweg-de Vries (cKdV) models, to evaluate the general features of solitary and cnoidal waves propagating in a uniform channel of arbitrary shape. Their work was later extended by [9] to study the evolution, transmission and reflection of long water waves propagating in a convergent-divergent channel of arbitrary cross-section.…”