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 $\omega_\ell^2-\omega^2$. Consequently
$$
\boldsymbol\xi
=\frac{A}{\omega^2-\omega_\ell^2}
\nabla(r^\ell Y_\ell^m)e^{-i\omega t},
$$
and its radial component is
$$
\boxed{\xi_r=
\frac{A\ell r^{\ell-1}}{\omega^2-\omega_\ell^2}
Y_\ell^m e^{-i\omega t}}.
$$
Taking 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.

Solution

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

Periodic tidal forcing excites a stellar normal mode with response proportional to $(\omega^2-\omega_\ell^2)^{-1}$ in an undamped linear model. Dissipation regularizes the divergence and introduces a phase shift near resonance.


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 Incoming links: Tidal resonance of a stellar oscillation