Normal-mode response to a step seismic moment ($a_{\alpha }=C_{\alpha }[1-\cos ({\omega }_{\alpha }\tau )]H(\tau )/{\omega }_{\alpha }^2$)

For mass-normalized normal modes, an instantaneous step in a point seismic moment tensor excites mode $\alpha$ with $C_{\alpha }=\mathbf{M}:{\mathbf{e}}_{\alpha }^{\ast }({\mathbf{x}}_s)$ and $\tau =t-t_s$. Convolving the causal oscillator Green's function with the step gives $a_{\alpha }=C_{\alpha }[1-\cos ({\omega }_{\alpha }\tau )]H(\tau )/{\omega }_{\alpha }^2$. The expression contains a static response and undamped oscillations; attenuation requires an extension of the conservative model.