5Water waves
IB Fluid Dynamics
5.4 Group velocity
Now suppose we have two waves travelling closely together with similar wave
numbers, e.g.
sin k
1
x + sin k
2
x = 2 sin
k
1
+ k
2
2
x
cos
k
1
− k
2
2
x
,
since
k
1
and
k
2
are very similar, we know
k
1
+k
2
2
≈ k
1
, and
k
1
−k
2
2
is small, hence
the cosine term has long period. We would then expect waves to look like this
So the amplitudes of the waves would fluctuate. We say the wave travels in
groups. For more details, see IID Fluid Dynamics or IID Asymptotic Methods.
It turns out the “packets” don’t travel at the same velocity as the waves
themselves. The group velocity is given by
c
g
=
∂ω
∂k
.
In particular, for deep water waves, where ω ∼
√
gk, we get
c
g
=
1
2
r
g
k
=
1
2
c.
This is also the velocity at which energy propagates.