Steep capillary waves on gravity waves.
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
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