Gravity waves on water

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 wavenumber¹ 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!

Footnotes:

¹ Recall that k = 2 p/l, where l is the wavelength