Circular Velocity/Acceleration in Ocean Waves & Surf Physics

Below are some figures/diagrams of circular velocity and acceleration. Circular/elliptical motion is observed in the water movement of deep and shallow water ocean waves.

Related links:

http://highered.mheducation.com/olcweb/cgi/pluginpop.cgi?it=swf::640::480::/sites/dl/free/0072826967/30425/14_04.swf::Fig.%2020.4%20-%20Orbital%20Motion%20in%20Shallow%20Water

http://www.acs.psu.edu/drussell/Demos/waves/Water-v8.gif
http://www.acs.psu.edu/drussell/demos/waves/wavemotion.html

The green line is the relatively constant velocity of the ocean water moving along its circular or elliptical path.  The left and right walls of the square/rectangle describe the vertical motion at the front and back of the wave, respectively.  The top and bottom of the square/rectangle describe the horizontal motion at the top and bottom of the wave, respectively.


Circular Acceleration & Velocity

Elliptical Acceleration & Velocity

Red and Blue lines show how velocity changes in the vertical and horizontal dimensions.

Surfboard Trim Dynamics

My latest grapics.

In order to take off, the surfboard must achieve wave(form) speed (C) before the wave passes under the surfer.  Wave slope affects the time (t) required to achieve C.  The surfboard's shoreward speed (Vs) must be Vs = C to remain in trim.  The surfboard's transverse velocity (Vt) along the wave's face must be greater than or equal to the transverse velocity of the breaking crest (Vbc).  After the wave has been caught, if shoreward surfboard velocity (Vs) drops below wave speed (C), the wave will pass under the surfer/surfboard leaving the them both behind.

I think many mistake the angle of the surfboard relative to the X axis (transverse movement viewed from the shore) as the angle that affects the continued forward velocity.  The angle of the wave face slope relative to the Z axis (direction of forward/shoreward movement) provides the acceleration needed to maintain continuous critical velocity (gravity).  The slope of the face is the curved ramp (inclined plane).

The colored curved lines below represent the slopes of three waves, red being steepest and blue gentlest.  The colored vertical lines, tangential to the top of each curve, represent the wave height.  The long black arrow represents the point on the wave face where the surfer achieves a shoreward velocity equal to wave speed (C) on take-off.  For an individual/independent wave, it could also represent the point where wave slope provides the angle necessary to create the acceleration needed to maintain a shoreward speed (V_s) equal to wave speed C (necessary for trim).

The point where the velocity arrow intersects the curve on each wave is also representative of the horizontal distance traveled, and therefore, the time required to achieve waveform speed C, dropping down from the crest.  The length of the slope curve decreases as wave slope increases.  Net acceleration increases as slope increases (curved ramps/inclined planes).  Position on the wave-face slope determines where the shoreward velocity reaches C.

I didn't add labels to the axes because I would have had to reduce figure size to fit them in.  The figure would get a bit busy to look at with added labels.