Sol. The speed of propagation of surface water waves depends on the surface tension γ and the density ρ of water, and on their wavelength, λ . The dimensions of these quantities are

[ γ ] = = , [ ρ ] = , [ λ ] = m.

An expression for velocity can only be derived from these quantities if γ and ρ , appear in the combination γ / ρ (otherwise the velocity, which involves only length and time, would have to depend upon the unit of mass). However, as

= ,

This expression has to be further divided by the wavelength, and then square-rooted in order to make the result have the dimension of speed. In summary, dimensional analysis dictates that the speed of propagation of capillary waves is proportional to the reciprocal of the square-root of the wavelength,

v ~ ~

From this functional dependence (and the given data), we can conclude that the speed of propagation of surface waves would reach that of sound in water when their wavelengths are of the order of 10 —8 cm.

Since the speed of propagation of surface waves cannot be greater than that of sound (the molecules cannot transmit a disturbance to each other faster at the surface than inside the matter), waves of wavelength less than approximately 10 —8 cm have no meaning. This is, in fact, the order of magnitude of the size of water molecules!