Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 329 / 2 / a / Solution

Interpret the stated ansatz asMultiplying the pressure relation by $a$ gives $2\mu a_t=Pa-\gamma$. Equating its constant and sinusoidal parts yieldsThe volume per wavelength is proportional to the mean of $a^2$, namely $f^2+g^2$. HenceChoosingthen gives exactly

For $g\ll f\simeq a_0$, the disturbance amplitude obeysso its long-wave linear growth rate isThe minimum radius is $a_{\min }=f-\sqrt{2}g$, andSince the trajectory follows the circle $f^2+g^2={\alpha }_0$ toward increasing $g$, it reaches $f=\sqrt{2}g$ and hence $a_{\min }=0$. Thus the nonlinear disturbance narrows monotonically to pinch-off within this approximation.