ANIMATIONS

1. Evolution of a surface gravity wave packet in water of finite depth (0.16 MB gif file). Water depth is 1 m. Initial wave profile is sech(x-x0). Note non-dispersive wave front moving with finite propagation speed (gH)^(1/2).
2. Evolution of a surface gravity wave packet in deep water (0.16 MB gif file). Initial wave profile is sech(x-x0). All waves have group velocity equal to half their phase speed so wave crests can be seen to propagate through the wave packet at disappear out the front.
3. Schematic of an internal plane wave (0.1 MB gif file). The parallel lines are lines of constant phase. They are blue when moving up to the left, red when moving down to the right. The tilted sinusoidal curve shows the propagating phase. It represents a line of particles which lay it a straight line prior to the waves arrival.
4. KdV Soliton Interaction (0.18MB gif file) This shows the nonlinear interaction of two KdV solitons. A soliton is a solitary wave with special properties. As the name suggests, solitary waves are waves which decay to zero at infinity. They are solutions of nonlinear wave equations, however not all nonlinear wave equations have solitary wave solutions. For the KdV equation the propagation speed of a solitary wave increases and the wave width decreases as the wave amplitude increases. Because large waves propagate faster than small waves a large wave trailing a small wave will eventually catch up to the small wave. A complicated nonlinear interaction between the two waves results in a transfer of energy, mass and momentum from the larger wave to the smaller wave. As a consequence the rear wave shrinks in amplitude and slows down while the small one grows in amplitude and speeds up, propagating ahead of the trailing wave. The final large and small waves have exactly the same amplitudes as the initial large and small waves and hence are called solitons. The preservation of wave identities after a nonlinear interaction between two waves is a special property of the KdV equation and of some other special nonlinear wave equations.
5. Breather (0.47MB gif file) An example of a nonlinear wave called a breather.
6. Solitary wave packet (0.40MB gif file) A wave packet is a group of waves with slowly varying amplitude. Many types of waves are dispersive, which means that waves of different wave lengths propagate with different speeds. Surface gravity waves on the surface of lakes are an example of such waves. Dispersive waves have the property that energy propagates with a different velocity than wave crests. For surface gravity waves energy propagates more slowly than wave crests (for capillary waves, driven by surface tension, the opposite is true). Thus, waves appear at the back of the wave packet, propagate through the packet growing and shrinking in size, until they disappear out the front of the wave packet. If you carefully watch the waves generated when you throw a stone into a still pond you can see this phenomenon. This animation shows the propagation of a solitary wave packet. This is a wave packet of special shape which, due to nonlinearities, propagates without changing shape. The wave envelope is a solution of the nonlinear Shrodinger equation. It is a soliton, so two wave packets of different amplitude propagate at different speeds and interact as KdV solitons do.
7. Internal wave packet (0.15MB gif file) Another example of dispersive waves are internal gravity waves which occur in density stratified fluids. They are ubiquitous features of the atmosphere, lakes, and oceans. For internal waves energy propagates along wave crests, hence energy actually propagates perpendicular to where you see the crests going!
8. Stokes Drift, case1 (0.5MB gif file) This shows the motion of particles at a number of depths due to the passage of a linear, sinusoidal surface gravity wave. Water depth is 1.0 m, wavenumber is 1.0 1/m, wave amplitude is 0.15 m. Arrows indicate horizontal current.
9. Stokes Drift, case2: larger waves (0.75MB gif file) . Water depth is 1.0 m, wavenumber is 1.0 1/m, wave amplitude is 0.3 m.
10. Stokes Drift, case3: shorter waves (0.5MB gif file) Water depth is 1.0 m, wavenumber is 4.0 1/m, wave amplitude is 0.15 m. Exponential decay with depth is quite apparent in this case.
11. Kink-antikink interaction for Sine Gordon equation
12. Breather solution of the Sine Gordon equation