 Kelvin-Voigt rod Green function

For a rod with unit mass density and elastic modulus, the Kelvin-Voigt model gives ${\varphi }_{tt}-{\varphi }_{xx}-\gamma {\varphi }_{txx}=0$, $\gamma >0$. 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 ${\omega }_{\pm }=-i\gamma k^2/2\pm \sqrt{k^2-{\gamma }^2{k}^4/4}$. With ${\Omega }_k=\sqrt{k^2-{\gamma }^2{k}^4/4}$ real, the retarded fundamental solution in spatial Fourier variables is ${\tilde{G}}_R(k,t)=-\Theta (t)e^{-\gamma k^{2}t/2}\sin ({\Omega }_{k}t)/{\Omega }_k$. 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.