Next consider waves on the surface of water, which are simple gravity waves
if the effect of surface tension can be neglected (this is valid for long waves, while for short ripples
the surface tension is dominant and gives rise to capillary waves; gravity and
surface tension are equally important at a wavelength of 5 cm). How
does the frequency 
depend upon the wavenumber
k
of the wave? The relationship 
is known as the dispersion relation for the wave. The relevant variables would appear to be
, which have dimensions 
, 
and 
; these quantities have independent dimensions, so n=3, k=3.
Now we can determine the exponents:
so that
with the solution a=0, b=c=1/2. Therefore,
with C another undetermined constant. We see that the frequency of water
waves is proportional to the square root of the wavenumber, in contrast to
sound or light waves, for which the frequency is proportional to the
wavenumber. This has the interesting consequence that the group velocity of
these waves is 
, while
the phase velocity is 
, so that 
. Recall that the group velocity describes the large scale
``lumps'' which would occur when we superimpose two waves, while the phase
velocity describes the short scale ``wavelets'' inside the lumps. For water
waves these wavelets travel twice as fast as the lumps.
You might worry about the effects of surface tension 
on the
dispersion relationship. We can include these in our dimensional analysis by
recalling that the surface tension is the energy per unit area of the
surface of the water, so it has dimensions 
. The
dimensions of the surface tension are not independent of the dimensions of ; in fact, it is easy to show that 
, so that 
is dimensionless. Then using the same
arguments as before, we have
with 
some undetermined function. A calculation of the dispersion
relation for gravity waves starting from the fundamental equations of fluid
mechanics [5] gives
so that our function 
is
Dimensional analysis enabled us to deduce the correct form of the solution,
i.e., the possible combinations of the variables. Of course, only a complete
theory could provide us with the function .
What have we gained? We originally started with being a function of
the four variables 
; what dimensional analysis tells us
is that it is really only a function of the combination ,
even though we don't know the function. Notice that this is an important
fact if you are trying to measure the dependence of on the physical
parameters . If you needed to make (say) 10 separate
measurements on each variable while holding the others fixed, then without
dimensional analysis you would naively need to make 
separate
measurements. Dimensional analysis tells you that you only really need to
measure the combinations gk and , so only need to make
102 measurements to characterize . Dimensional analysis can be a
labor-saving device!
