Chapter 18
Shallow Water Equations
The shallow water equations model tsunamis and waves in bathtubs.
This chapter is more advanced mathematically than earlier chapters, but you
might still find it interesting even if you do not master the mathematical details.
The shallow water equations model the propagation of disturbances in water
and other incompressible fluids. The underlying assumption is that the depth of
the fluid is small compared to the wave length of the disturbance. For example, we
do not ordinary think of the Indian Ocean as being shallow. The depth is two or
three kilometers. But the devastating tsunami in the Indian Ocean on December
26, 2004 involved waves that were dozens or hundred of kilometers long. So the
shallow water approximation provides a reasonable model in this situation.
The equations are derived from the principles of conservation of mass and
conservation of momentum. The independent variables are time, t, and two space
coordinates, x and y. The dep endent variables are the fluid height or depth, h, and
the two-dimensional fluid velocity field, u and v. With the proper choice of units, the
conserved quantities are mass, which is proportional to h, and momentum, which
is proportional to uh and vh. The force acting on the fluid is gravity, represented
by the gravitational constant, g. The partial differential equations are:
∂h
∂t
+
∂(uh)
∂x
+
∂(vh)
∂y
= 0
∂(uh)
∂t
+
∂(u
2
h +
1
2
gh
2
)
∂x
+
∂(uvh)
∂y
= 0
∂(vh)
∂t
+
∂(uvh)
∂x
+
∂(v
2
h +
1
2
gh
2
)
∂x
= 0
Copyright
c
2011 Cleve Moler
Matlab
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October 2, 2011
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