= Kelvin-Voigt rod Green function
{c}
{title2=$\widehat G_R(k,\omega)=[\omega^2-k^2+i\gamma k^2\omega]^{-1}$}
For a rod with unit mass density and elastic modulus, the <Kelvin-Voigt model> gives $\phi_{tt}-\phi_{xx}-\gamma\phi_{txx}=0$, $\gamma>0$. For the operator with the opposite overall sign, $-\partial_t^2+\partial_x^2+\gamma\partial_t\partial_x^2$, the inverse under the $e^{-i\omega t}$ <Fourier transform> convention is the displayed expression. Its poles are $\omega_\pm=-i\gamma k^2/2\pm\sqrt{k^2-\gamma^2k^4/4}$. With $\Omega_k=\sqrt{k^2-\gamma^2k^4/4}$ real, the <retarded fundamental solution> in spatial Fourier variables is $\widetilde G_R(k,t)=-\Theta(t)e^{-\gamma k^2t/2}\sin(\Omega_kt)/\Omega_k$. Its initial derivative is $-1$, fixed by the leading coefficient $-\partial_t^2$. At a repeated root take the continuous limit; for imaginary $\Omega_k$ the sine becomes a hyperbolic sine.
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