Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-08T05:38:46.288Z Has data issue: false hasContentIssue false
  • English
  • Français

On surface waves crossing a step with horizontal shear

Published online by Cambridge University Press:  21 April 2006

Jerome Smith
Jerome Smith
Affiliation:
Marine Physical Laboratory of the Scripps Institution of Oceanography, University of California, San Diego, La Jolla, CA 92093, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

When surface waves encounter a step in bottom topography and/or a change in velocity parallel to the step, refraction and partial reflection occur. Comparison of several approximate solutions indicates that no single approximation works well for all cases. The pattern of success among models suggests that the velocity profile at the boundary favours the free wave with smaller vertical scale. For current changes over a flat bottom, a two-term Galerkin expansion (cf. Evans 1975) is employed for comparison with the other more general models. For small currents (|ΔV| < ½c), an ‘action-based approximation’ (cf. Smith 1983) is favoured, although all models perform adequately. With a strong current, one one-term (one-sided) model performs best, another worst among models; the favoured model includes ephemeral modes on the side with larger-scale free waves. For changes in depth only, the one-sided model with ephemeral modes on the deep side was shown by Miles (1967) to perform well. The two-term expansion (cf. Evans 1975) is not easily extended to this case, and none of the other approximations perform adequately. In the unusual case of a step combined with a strong current, such that much shorter waves occur in the deeper region, it is inferred that none of the models are accurate. Reflection from a submerged wall provides a severe test of the models. Without the ephemeral modes, no net reflection occurs. The Miles-like model overpredicts reflection slightly.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 175 , February 1987 , pp. 395 - 412
Copyright
© 1987 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Dean, W. R. 1945 On the reflexion of surface waves by a submerged plane barrier. Proc. Camb. Phil. Soc. 41, 231–238.Google Scholar
Evans, D. V. 1975 The transmission of deep-water waves across a vortex sheet. J. Fluid Mech. 68, 389–401.Google Scholar
Hayes, W. D. 1970 Conservation of action and modal wave action. Proc. R. Soc. Lond. A 320, 187–208.Google Scholar
Kirby, J. T. 1986 Comments on ‘The effects of a jet-like current on gravity waves in shallow water’. J. Phys. Oceanogr. 16, 395–397.Google Scholar
Kirby, J. T. & Dalrymple, R. A. 1983 Propagation of obliquely incident water waves over a trench. J. Fluid Mech. 133, 47–63.Google Scholar
Kirby, J. T., Dalrymple, R. A. & Seo, S. N. 1987 Propagation of obliquely incident water waves over a trench. Part 2. Currents flowing along the trench. J. Fluid Mech. 176, 95–116.Google Scholar
Lamb, H. 1945 Hydrodynamics. Dover. 728 pp.
McKee, W. D. & Tesoriero, F. 1986 Reflection of water waves from a vertical vortex sheet in water of finite depth. J. Austral. Math. Soc. B (submitted).Google Scholar
Mei, C. C. & Lo, E. 1984 The effects of a jet-like current on gravity waves in shallow water. J. Phys. Oceanogr. 14, 471–477.Google Scholar
Mei, C. C. & Lo, E. 1986 Reply. J. Phys. Oceanogr. 16, 398–399.Google Scholar
Miles, J. W. 1967 Surface-wave scattering matrix for a shelf. J. Fluid Mech. 28, 755–767.Google Scholar
Miles, J. W. 1982 On surface-wave diffraction by a trench. J. Fluid Mech. 115, 315–325.Google Scholar
Smith, J. 1983 On surface gravity waves crossing weak current jets. J. Fluid Mech. 134, 277–299.Google Scholar