Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-19T15:11:28.219Z Has data issue: false hasContentIssue false
  • English
  • Français

On the initial-value problem for a wavemaker

Published online by Cambridge University Press:  26 April 2006

John Miles
John Miles
Affiliation:
Institute of Geophysics and Planetary Physics, University of California. San Diego, La Jolla, CA 92093–0225, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The linearized initial-value problem for the generation of straight-crested waves in a deep, inviscid liquid in response to the prescribed motion of a piston wavemaker of finite depth is solved through integral transforms. The indicial admittance (the surface-wave response to a step-function velocity of the wavemaker) is cast in similarity form and expressed in terms of confluent hypergeometric functions for pure (no surface tension) gravity waves. This gravity-wave result, due essentially to Roberts (1987), provides an outer approximation for x [Gt ] l and gt2 [Gt ] l (x = horizontal distance from wavemaker and l = capillary length) but implies an infinite wave slope at the contact line (x = 0) in consequence of the neglect of surface tension. The corresponding similarity solution for capillary waves (no gravity) with either fixed contact angle or fixed contact line is constructed and is found to be analytic in x for t > 0 if the contact angle is fixed or singular like x4 log x if the contact line is fixed. An inner approximation for gravity waves with either fixed contact angle or fixed contact line is constructed for x = O(l) and gt2 [Gt ] l. The Laplace transform of the general solution is expressed in terms of confluent hypergeometric functions, which permits a compact discussion of its analytical properties.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 229 , August 1991 , pp. 589 - 601
Copyright
© 1991 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

Akylas, T. R. 1984 On the excitation of long nonlinear water waves by a moving pressure distribution. J. Fluid Mech. 141, 455–466.Google Scholar
Baines, P. G. 1977 Upstream influence and Long's model in stratified flows. J. Fluid Mech. 82, 147–159.Google Scholar
Baines, P. G. 1979 Observations of stratified flow over two-dimensional obstacles in fluid of finite depth. Tellus 31, 351–371.Google Scholar
Baines, P. G. 1984 A unified description of two-layer flow over topography. J. Fluid Mech. 146, 127–167.Google Scholar
Castro, I. P. & Snyder, W. H. 1988 Upstream motions in stratified flow. J. Fluid Mech. 135, 261–282.Google Scholar
Castro, I. P., Snyder, W. H. & Baines, P. G. 1990 Obstacle drag in stratified flow.. Proc. R. Soc. A 429, 119–140.Google Scholar
Cole, S. L. 1985 Transient waves produced by flow past a bump. Wave Motion 7, 579–587.Google Scholar
Gill, A. E. 1977 The hydraulics of rotating-channel flow. J. Fluid Mech. 80, 641–671.Google Scholar
Grimshaw, R. 1987 Resonant forcing of barotropic coastally trapped waves. J. Phys. Oceanogr. 17, 53–65.Google Scholar
Grimshaw, R. 1990 Resonant flow of a rotating fluid past an obstacle: the general case. Stud. Appl. Maths 83, 249–269.Google Scholar
Grimshaw, R. & Smyth, N. 1986 Resonant flow of a stratified fluid over topography. J. Fluid Mech. 169, 429–464.Google Scholar
Grimshaw, R. & Yi Zengxin 1990 Finite-amplitude long waves on coastal currents. J. Phys. Oceanogr. 20, 3–18.Google Scholar
Hanazaki, H. 1989 Upstream advancing columnar disturbances in two-dimensional stratified flow of finite depth. Phys. Fluids A 1, 1976–1987.Google Scholar
Lee, S.-J., Yates, G. T. & Wu, T. Y. 1989 Experiments and analyses of upstream-advancing solitary waves generated by moving disturbances. J. Fluid Mech. 199, 569–593.Google Scholar
Long, R. R. 1953 Some aspects of the flow of stratified fluids. I. A theoretical investigation. Tellus 5, 42–57.Google Scholar
Mcintyre, M. E. 1972 On Long's hypothesis of no upstream influence in uniformly stratified or rotating flow. J. Fluid Mech. 52, 209–242.Google Scholar
Melville, W. K. & Helfrich, K. R. 1987 Transcritical two-layer flow over topography. J. Fluid Mech. 178, 31–52.Google Scholar
Mitsudera, H. & Grimshaw, R. 1990 Resonant forcing of coastally trapped waves in a continuously stratified ocean. Pageoph 133, 635–644.Google Scholar
Pratt, L. J. & Armi, L. 1987 Hydraulic control of flows with nonuniform potential vorticity. J. Phys. Oceanogr. 17, 2016–2029.Google Scholar
Warn, T. 1983 The evolution of finite amplitude solitary Rossby waves on a weak shear. Stud. Appl. Maths. 69, 127–133.Google Scholar
Wei, S. N., Kao, T. W. & Pao, H. P. 1975 Experimental study of upstream influence in the two-dimensional flow of a stratified fluid over an obstacle. Geophys. Fluid Dyn. 6, 315–336.Google Scholar
Wu, T. Y. 1987 Generation of upstream advancing solitons by moving disturbances. J. Fluid Mech. 184, 75–99.Google Scholar
Yi Zengxin & Warn, T. 1987 A numerical method for solving the evolution equation of solitary Rossby waves on a weak shear. Adv. Atmos. Sci. 4, 43–54.Google Scholar
Yih, C. S. 1960 Exact solutions for steady two-dimensional flows of a stratified fluid. J. Fluid Mech. 9, 161–174.Google Scholar
Zhu, J. 1986 Internal solitons generated by moving disturbances. PhD thesis, Cal. Inst. Tech., 209 pp.