Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 314 / 2 / d / Solution

The tidal potential is proportional to the solid spherical harmonic $r^{\ell }{Y}_{\ell }^m$, whose gradient has exactly the spatial form of the mode in part (c). Projecting the forced linear equation onto that eigenfunction gives the oscillator factor ${\omega }_{\ell }^2-{\omega }^2$. Consequentlyand its radial component isTaking the real part gives the stated physical displacement. The amplitude displays tidal resonance of a stellar oscillation: it is enhanced near $|\omega |={\omega }_{\ell }$ and formally diverges in this undamped linear model. Physical damping makes the peak finite and supplies a phase shift through resonance.