= Stationary-phase flexural-beam pulse
{title2=$\eta(Vt,t)\sim ae^{-\epsilon|V|/(2\lambda)}\cos(V^2t/(4\lambda)-\pi/4)/(2\sqrt{\pi\lambda t})$}
For $\eta_{tt}+\lambda^2\eta_{xxxx}=0$ with Lorentzian displacement $\epsilon a/[\pi(x^2+\epsilon^2)]$ and zero initial velocity, the <Fourier transform> is $ae^{-\epsilon|k|}\cos(\lambda k^2t)$. Along a fixed ray $x=Vt$, the two phases $Vk\pm\lambda k^2$ have saddles $k=\mp V/(2\lambda)$ and curvatures $\pm2\lambda$. 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 $\epsilon|V|/\lambda\ll1$. The expansion is an additive $o(t^{-1/2})$ statement, which remains meaningful at zeros of the leading cosine.
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