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calm water we can completely neglect the advection terms, leaving us with
∂u
∂t + g ∂h
∂x =0 ,
∂w
∂t
+ g ∂h
∂z
=0 ,
d ∂u
.
∂h
∂t
∂x + ∂w
=
∂z
Divide the height equation through by the depth d and differentiate in
time:
1
d
=
∂t
∂h
∂t
∂x
∂u
∂t
∂z
∂w
∂t .
Then substitute in the simplified velocity equations to get
1
d
=
g ∂h
∂x
+
g ∂h
∂z
.
∂t
∂h
∂t
∂x
∂z
Expanding the left-hand side, but further neglecting the quadratic term as
being much smaller, gives
2 h
∂t
= gd
∇·∇
h,
where the Laplacian ∇·∇ here is just in two dimensions ( x and z ). Finally,
with the assumption that the depth d in the right-hand side remains near
enough constant, this is known as the wave equation .
The wave equation also pops up naturally in many other phenomena—
elastic waves in solid materials, electromagnetic waves, acoustics (sound
waves), and more—and has been well studied. Fourier analysis can provide
a full solution, but to keep things simple let's just try a single sinusoid
wave. 3 Take a unit-length vector k (in two dimensions) which will represent
the direction of wave motion; the peaks and troughs of the waves will lie
on lines perpendicular to k .Let λ be the wavelength, A the amplitude,
and c the speed of the wave. Putting this all together gives
A sin 2 π ( k
.
·
( x, z )
ct )
λ
3 In fact, any wave shape will do—we pick sinusoids simply out of convention and to
match up with the ocean wave modeling in the next chapter where sinusoids are critical.
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