Spring 2008 9 Misc

Velocity of surface (capillary) waves v is normally much smaller than the sound velocity c in water. For example, for the wavelength of λ = 1 cm we have v ~ 10−4c. On the other hand, both surface tension σ and bulk compressibility B of water have the same physical origin: shortrange intermolecular forces. Show that the smallness of the v/c ratio is related to small average distance a between water molecules. Estimate a by following these steps:

(a) Use scaling and dimensional analysis to derive how v depends on wavelength λ and physical parameters of water.

(b) Do the same for c.

(c) Get the desired estimate of a using the data given above.

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Answer

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(a)
As for any other waves, the capillary wave velocity is determined by the competition of the restoring force (provided here by the surface tension) and the inertia (characterized by mass density ρ).

Surface tension has units [σ] = (energy)/(length)2 and ρ has units [ρ] = (mass)/(length)3 . There is only one combination of these two quantities and the wavelength that has the units of velocity:

(1)
\begin{align} v(\lambda) \sim \sqrt{\sigma/(\lambda/\rho)} \end{align}

(b)
From undergraduate physics

(2)
\begin{align} c=\sqrt{B/\rho} \end{align}

This formula can also be derived from scaling analysis, taking into account that the compressibility has units [B] = (energy)/(length)3.

(c)
Based on the dimensions of σ and B, we expect

(3)
\begin{align} \sigma \sim B a \end{align}

where a is the distance between the molecules. Combined with the formulas above, this entails

(4)
\begin{align} a\sim\lambda\left(\frac{v}{c}\right)\sim 10^{-8}cm \end{align}
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