5Water waves

IB Fluid Dynamics



5.3 Two-dimensional waves (straight crested waves)
We are going further simplify the situation by considering the case where the
wave does not depend on y. We consider a simple wave form
h = h
0
e
i(kxωt)
.
Using the boundary condition at
z
= 0, we know we must have a solution of the
form
φ =
ˆ
φ(z)e
i(kxωt)
.
Putting this into Laplace’s equation, we have
k
2
ˆ
φ +
ˆ
φ
00
= 0.
We notice that the solutions are then of the form
ˆ
φ = φ
0
cosh k(z + H),
where the constants are chosen so that
φ
z
= 0 at z = H.
We now have three unknowns, namely
h
0
, φ
0
and
ω
(we assume
k
is given,
and we want to find waves of this wave number). We use the boundary condition
φ
z
=
h
t
at z = 0.
We then get
kφ
0
sinh kH = h
0
.
We put in Bernoulli’s equation to get
ˆ
φ(z)e
i(kxωt)
+ gh
0
e
i(kxωt)
= f(t).
For this not to depend on x, we must have
φ
0
cosh kH + gh
0
= 0.
The trivial solution is of course h
0
= φ
0
= 0. Otherwise, we can solve to get
ω
2
= gk tanh kH.
This is the dispersion relation, and relates the frequency to the wavelengths of
the wave.
We can use the dispersion relation to find the speed of the wave. This is just
c =
ω
k
=
r
g
k
tanh kH.
We can now look at the limits we have previously obtained with large and small
H.
In deep water (or short waves), we have
kH
1. We know that as
kH
,
we get tanh kH 1. So we get
c =
r
g
k
.
In shallow water, we have
kH
1. In the limit
kH
0, we get
tanh kH kH
.
Then we get
c =
p
gH.
This are exactly as predicted using dimensional analysis, with all the dimension-
less constants being 1.
We can now plot how the wave speed varies with k.
kH
c
gH
1
We see that wave speed decreases monotonically with
k
, and long waves travel
faster than short waves.
This means if we start with, say, a square wave, the long components of the
wave travels faster than the short components. So the square wave disintegrates
as it travels.
Note also that this puts an upper bound on the maximum value of the speed
c
. There can be no wave travelling faster than
gH
. Thus if you travel faster
than
gH
, all the waves you produce are left behind you. In general, if you
have velocity U, we can define the Froude number
F r =
U
gH
.
This is like the Mach number.
For a tsunami, we have
λ 400 km, H 4 km.
We are thus in the regime of small kH, and
c =
p
10 × 4 × 10
3
= 200 m s
1
.
Note that the speed of the tsunami depends on the depth of the water. So
the topography of the bottom of the ocean will affect how tsunamis move. So
knowing the topography of the sea bed allows us to predict how tsunamis will
move, and can save lives.