Residual amplitude of an oscillator after an integrable force ($R={\omega }^{-1}|{\int }_0^{\infty }F(s)e^{-i\omega s}ds|$)

Let $y^{\prime \prime }+{\omega }^{2}y=F(t)$ with $\omega >0$, zero initial displacement and velocity, and real-valued forcing with ${\int }_0^{\infty }|F(t)|dt<\infty$. The causal Green function gives $y(t)={\omega }^{-1}{\int }_0^{t}F(s)\sin [\omega (t-s)]ds$. Expanding the sine difference, the two coefficients tend to finite limits by absolute convergence. Thus $y(t)$ approaches a sinusoid whose amplitude is the displayed modulus of the one-sided Fourier transform, divided by $\omega$. The displacement error is at most ${\omega }^{-1}{\int }_t^{\infty }|F(s)|ds$, which tends to zero. This describes the surviving free oscillation after a transient force; an arbitrary nonzero initial oscillation must be added before taking the final amplitude.