The frequent presence of ripples on the free surface of water. on both thin film flows and ponds or lakes motivates this theoretical investigation into the propagation of ripples on gravity waves. These ripples are treated as "slowly-varying" waves in a reference frame where the gravity wave flow is steady. The methods used are those of the averaged Lagrangian (Whitham 1965,1967,1974) and the averaged equations of motion (Phillips 1966) which are shown to be equivalent. The capillary wave modulation is taken to be steady in the reference frame which brings the gravity wave, or gravity driven flow, to rest. Firstly the motion over ponds or lakes is considered. Linear capillary-gravity waves are examined in order to set the scene. Crapper's (1957) exact finite-amplitude waves are examined next to show the actual behaviour of the flow field. The underlying gravity driven flow is that of pure gravity waves over an' "infinite" depth liquid. These gravity waves are modelled with "numerically exact" solutions for periodic plane-waves. The initial studies are inviscid and show that steep gravity waves either "absorb" or "sweep-up" a range of capillary waves or, alternatively, cause them to break in the vicinity of gravity wave crests. Improvements on the theory are made by including viscous dissipation of wave energy. This leads to a number of solutions approaching "stopping velocities" or the "stopped waves solution". In addition to these effects "higher-order dispersion" is introduced for weakly nonlinear waves near linear caustics. This clarifies aspects of the dissipation results and shows that wave reflection sometimes occurs. Secondly, waves on thin film flows are considered. Linear capillary-gravity waves are again examined in order to set the scene. Kinnersley's (1957) exact finite-amplitude waves are examined next to show the actual behaviour of the flow field. The underlying gravity driven flow is given by shallow water gravity waves. No modelling of these is necessary simply because they are included within Whitham's or Phillips' equations ab initio. This study is inviscid and shows the unexpected presence of critical velocities at which pairs of solution branches originate. iii
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:334009 |
| Date | January 1989 |
| Creators | Popat, Nilesh R. |
| Publisher | University of Bristol |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://hdl.handle.net/1983/78695ee9-b923-4374-b70c-6589b4215241 |
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