Question

A soap bubble (radius R, thickness d, soap density ρ) can perform radial oscilla- tions because of the restoring forces caused by the surface tension ơ. Calculate the oscillation frequency as a function of R, ρ, and σ (neglect the change of gas pressure inside the soap bubble due to the change of the bubbles volume, i.e. consider the case of small oscillation amplitudes). (3P)

Answer 1

Suppose that at the equilibrium radius is R. Excess pressure

$\frac{4\sigma}{R}$

A disturbance changes the radius to  $R+r$

The change in force due to surface tension is

$-\frac{4\sigma}{R^2}r$

The equation of motion becomes

$\ddot{r}=-\frac{1}{\rho dR^2\delta\Omega}\frac{4\sigma r}{R^2}(R^2\delta\Omega)=-\frac{4\sigma r}{\rho dR^2}$

for a small solid angle $\delta\Omega$

Therefore, the frequency of oscillation is given by

$\omega=\sqrt{\frac{4\sigma}{\rho dR^2}}$