Impulse cancellation of a harmonic oscillator

For the harmonic oscillator equation $y^{\prime \prime }+{\omega }^{2}y=J\delta (t-t_{\ast })$, with $\omega >0$, the displacement is continuous and the velocity increases by $J$ at the impulse. If the incoming displacement is zero and $J=-y^{\prime }(t_{\ast }-)$, the outgoing displacement and velocity both vanish. The subsequent homogeneous solution is identically zero. Equivalently, the causal forced response $(J/\omega )H(t-t_{\ast })\sin (\omega (t-t_{\ast }))$ cancels the pre-existing oscillation, where $H$ is the Heaviside step function.