= Rectangular-pulse cancellation of an undamped oscillator
{title2=$T=2\pi n/\omega_0$}
For $x''+\omega_0^2x=F$ during $0<t<T$ and zero force afterwards, with zero initial displacement and velocity, the <causal Green function> gives $x(t)=F[1-\cos(\omega_0t)]/\omega_0^2$ during the pulse. For $t\geq T$ it gives $x(t)=F[\cos(\omega_0(t-T))-\cos(\omega_0t)]/\omega_0^2$. For $F\ne0$, both outgoing displacement and velocity vanish precisely when $T$ is an integer number of full natural periods. Cancellation requires both initial data for the subsequent homogeneous equation, not only zero displacement at switch-off.
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