= Solution
<Coupled oscillators> have more than one collective pattern of motion. For two equal <masses> $m$, each attached to a fixed support by a spring of stiffness $k$ and linked to each other by stiffness $\kappa$, small displacements obey
$$
m\ddot x_1+b\dot x_1+(k+\kappa)x_1-\kappa x_2=F_1\cos\Omega t,
$$
$$
m\ddot x_2+b\dot x_2+(k+\kappa)x_2-\kappa x_1=F_2\cos\Omega t.
$$
Here equal linear damping $b$ is included to make finite <steady-state responses> at resonance. The <normal mode coordinates> $q_+=(x_1+x_2)/\sqrt2$ and $q_-=(x_1-x_2)/\sqrt2$ diagonalize these equations. Their natural <angular frequencies> are
$$
\boxed{\omega_+^2=k/m,\qquad \omega_-^2=(k+2\kappa)/m.}
$$
The symmetric <normal mode> leaves the coupling spring unextended; the antisymmetric <normal mode> stretches it and therefore has the higher <frequency>. With $F_\pm=(F_1\pm F_2)/\sqrt2$, their complex <steady-state responses> are
$$
\boxed{\widetilde q_\pm=\frac{F_\pm}{m(\omega_\pm^2-\Omega^2)+ib\Omega}.}
$$
Thus the drive's spatial pattern matters as well as its <frequency>. Equal in-phase <forces> drive only the symmetric <normal mode>; opposite <forces> drive only the antisymmetric one; driving a single mass generally excites both. Finite damping broadens the <resonance> peaks and produces a phase lag. In the ideal undamped system, a resonant drive with nonzero projection onto a <normal mode> produces secular growth rather than a bounded resonant steady state.
\Image[/past-exam-of-the-physics-b-course-of-the-university-of-cambridge/2016/paper-2-coupled-response.png]
{title=Original driven two-mass response showing two resonances and an intervening antiresonance}
The displacement of a particular mass adds contributions from both <normal modes>. They can cancel at an <antiresonance>. For $F_2=0$ and no damping, $x_1$ vanishes at $\Omega^2=(k+\kappa)/m$, while the partner still moves. This differs from the two nearby resonances, where a mode responds strongly; a small measured response need not mean that no motion or stored <energy> exists elsewhere.
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