5Water waves
IB Fluid Dynamics
5.1 Dimensional analysis
Consider waves with wave number
k
=
2π
λ
, where
λ
is the wavelength, on a layer
of water of depth
H
. We suppose the fluid is inviscid. Then the wave speed
c
depends on k, g and H. Dimensionally, we can write the answer as
c =
p
gHf(kH).
for some dimensionless function f.
Now suppose we have deep water. Then
H λ
. Therefore
kH
1. In this
limit, we would expect the speed not to depend on
H
, since
H
is just too big.
The only way this can be true is if
f ∝
1
√
kH
.
Then we know
c = α
r
g
k
,
where α is some dimensionless constant.
What happens near the shore? Here the water is shallow. So we have
kH
1.
Since the wavelength is now so long, the speed should be independent of
k
. So
f
is a constant, say β. So
c = β
p
gH.
We don’t know
α
and
β
. To know that, we would need proper theory. And we
also need that to connect up the regimes of deep and shallow water, as we will
soon do.
Yet these can already explain several phenomena we see in daily life. For
example, we see that wave fronts are always parallel to the shore, regardless of
how the shore is shaped and positioned. This is since if wave is coming in from
an angle, the parts further away from the shore move faster (since it is deeper),
causing the wave front to rotate until it is parallel.
shore
wave fronts
We can also use this to explain why waves break. Near the shore, the water is
shallow, and the difference in height between the peaks and troughs of the wave
is significant. Hence the peaks travels faster than the trough, causing the waves
to break.
H + h
H −h