Fluids: Liquids and Gases - Computer Animation: Algorithms and Techniques - page 256

As a wave approaches the shoreline at an angle, the part of the wave that approaches first will
slow down as it encounters a shallower area. The wave will progressively slow down along its length
as more of it encounters the shallow area. This will tend to straighten out the wave and is called wave
refraction .
Interestingly, even though speed ( C ) and wavelength ( L ) of the wave are reduced as the wave enters
shallow water, the period, T , of the wave remains the same and the amplitude, A (and, equivalently, H ),
remains the same or increases. As a result, the orbital speed, Q (Eq. 8.2 ), of the water remains the same.
Because orbital speed remains the same as the speed of the wave decreases, waves tend to break as they
approach the shoreline because the speed of the water exceeds the speed of the wave. The breaking of a
wave means that water particles break off from the wave surface as they are “thrown forward” beyond
the front of the wave.

Modeling ocean waves
The description of modeling ocean waves presented here follows Peachey [ 15 ] . The ocean surface
is represented as a height field, y¼f ( x , z , t ), where ( x , z ) defines the two-dimensional ground

plane, t is time, and y is the height. The wave function f is a sum of various waveforms at different

amplitudes (Eq. 8.4 ) .

X n

f ðx; z; tÞ¼

1 A i W i ðx; z; tÞ

(8.4)

i¼

The wave function, W i , is formed as the composition of two functions: a wave profile, w i , and a
phase function, y i ( x , z , t ), according to Equation 8.5 . This allows the description of the wave profile
and phase function to be addressed separately.

W i ðx; z; tÞ¼o i ð fraction ½y i ðx; z; tÞÞ (8.5)

Each waveform is described by its period, amplitude, source point, and direction. It is convenient
to define each waveform, actually each phase function, as a linear rather than radial wave and to
orient it so the wave is perpendicular to the x -axis and originates at the source point. The phase function
is then a function only of the x -coordinate and can then be rotated and translated into position in world
space.
Equation 8.6 gives the time dependence of the phase function. Thus, if the phase function is known
for all points x (assuming the alignment of the waveform along the x -axis), then the phase function can
be easily computed at any time at any position. If the depth of water is constant, the Airy model states
that the wavelength and speed are also constant. In this case, the aligned phase function is given in
Equation 8.7 .

t t 0
T i

y i ðx; y; tÞ¼y i ðx; y; t 0 Þ (8.6)

x i
L i

y i ðx; z; tÞ¼ (8.7)

However, if the depth of the water is variable, then L i is a function of depth and u i is the integral of
the depth-dependent phase-change function from the origin to the point of interest (Eq. 8.8 ). Numerical
integration can be used to produce phase values at predetermined grid points. These grid points can be