Stationary-phase flexural-beam pulse ($\eta (Vt,t)\sim a{e}^{-ϵ|V|/(2\lambda )}\cos (V^{2}t/(4\lambda )-\pi /4)/(2\sqrt{\pi \lambda t})$)

For ${\eta }_{tt}+{\lambda }^2{\eta }_{xxxx}=0$ with Lorentzian displacement $ϵa/[\pi (x^2+{ϵ}^2)]$ and zero initial velocity, the Fourier transform is $a{e}^{-ϵ|k|}\cos (\lambda k^{2}t)$. Along a fixed ray $x=Vt$, the two phases $Vk\pm \lambda k^2$ have saddles $k=\mp V/(2\lambda )$ and curvatures $\pm 2\lambda$. Applying stationary phase and adding the conjugate contributions gives the displayed leading term. The finite pulse width supplies the attenuation factor; it may be dropped only if $ϵ|V|/\lambda \ll 1$. The expansion is an additive $o(t^{-1/2})$ statement, which remains meaningful at zeros of the leading cosine.