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On the deformation of periodic long waves over a gently sloping bottom

Published online by Cambridge University Press:  12 April 2006

Ib. A. Svendsen  and
J. Buhr Hansen
Ib. A. Svendsen
Affiliation:
Institute of Hydrodynamics and Hydraulic Engineering, Technical University of Denmark, DK-2800 Lyngby
J. Buhr Hansen
Affiliation:
Institute of Hydrodynamics and Hydraulic Engineering, Technical University of Denmark, DK-2800 Lyngby
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Abstract

Two-dimensional time-periodic water waves on a gently sloping bottom are investigated under the classical long-wave assumptions that ε = h′0/λ′ and δ = H′/h′0 are small parameters (H′ being the wave height, h′0 a characteristic water depth and δ′ the horizontal scale for the oscillatory motion) and the assumption that δ/ε2 = O(1) as δ and ε tend to zero.

It is shown that for a bottom slope hx for which hx = o(ε3) the governing KdV equation with slowly varying coefficients (derived by Johnson 1973a) has a time-periodic solution which in the first approximation is a slowly varying cnoidal wave. The second-order approximation in an asymptotic expansion with respect to the bottom slope represents the deformation of this wave due to the sloping bottom. For hx [Gt ] ε5 this deformation is larger than the second-order contribution from the basic expansion with respect to wave amplitude which underlies the KdV equation itself; the calculation is then a consistent approximation to physical reality.

Numerical results for the deformation are given. Also, the wave profiles are compared with experiments on a plane of slope $h_x = \frac{1}{35}$ and show good agreement even for the large values of H’/h’ appropriate to waves rather close to breaking.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 87 , Issue 3 , 15 August 1978 , pp. 433 - 448
Copyright
© 1978 Cambridge University Press

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