= Impulse cancellation of a harmonic oscillator
For the <harmonic oscillator equation> $y''+\omega^2y=J\delta(t-t_*)$, with $\omega>0$, the displacement is continuous and the velocity increases by $J$ at the impulse. If the incoming displacement is zero and $J=-y'(t_*-)$, the outgoing displacement and velocity both vanish. The subsequent <homogeneous solution> is identically zero. Equivalently, the causal forced response $(J/\omega)H(t-t_*)\sin(\omega(t-t_*))$ cancels the pre-existing oscillation, where $H$ is the <Heaviside step function>.
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