Fluids: Liquids and Gases - Computer Animation: Algorithms and Techniques - page 255

it takes for one complete wave to pass a given point. The wavelength, period, and speed are related
by the equation C¼L / T .

2 pðs CtÞ
L

cos 1 f ðs; tÞ¼ (8.1)

The motion of the wave is different from the motion of the water. The wave travels linearly across
the surface of the water, while a particle of water moves in nearly a circular orbit ( Figure 8.7 ). While
riding the crest of the wave, the particle will move in the direction of the wave. As the wave passes and
the particle drops into the trough between waves, it will travel in the reverse direction. The steepness, S ,
of the wave is represented by the term H / L where H is defined as twice the amplitude.
Waves with a small steepness value have basically a sinusoidal shape. As the steepness value
increases, the shape of the wave gradually changes into a sharply crested peak with flatter troughs.
Mathematically, the shape approaches that of a cycloid.
In an idealized wave, there is no net transport of water. The particle of water completes one orbit in
the time it takes for one complete cycle of the wave to pass. The average orbital speed of a particle
of water is given by the circumference of the orbit, pH, divided by the time it takes to complete
the orbit, T (Eq. 8.2 ).

Q ave ¼ pH

T ¼ pHC

¼ pSC (8.2)

L

If the orbital speed, Q , of the water at the crest exceeds the speed of the wave, C , then the water will
spill over the wave, resulting in a breaking wave. Because the average speed, Q , increases as the steep-
ness, S , of the wave increases, this limits the steepness of a nonbreaking wave. The observed steepness
of ocean waves, as reported by Peachey [ 15 ] is between 0.5 and 1.0.
A common simplification of the full CFD simulation of ocean waves is called the Airy model, and it
relates the depth of the water, d , the propagation speed, C , and the wavelength of the wave, L (Eq. 8.3 ).

s

g 2

r

g
k

2 pd

L

C ¼ tanh ðkdÞ ¼

p

tanh

L ¼ CT (8.3)

In Equation 8.3 , g is the acceleration of a body due to gravity at sea level, 9.81 m/sec 2 ,and k¼ 2 p / L is

the spatial equivalent of wave frequency. As the depth of the water increases, the function tanh( kd ) tends
toward one, so C approaches gL /2 p . As the depth decreases and approaches zero, tanh( kd ) approaches kd ,
so C approaches g p . Peachey suggests using deep to mean d>L /4 and shallow to mean d<L /20.

L

H 2 A

FIGURE 8.7
Circular paths of particles of water subjected to waves.