= Membrane-guide phase and energy speeds
{title2=$v_{p,x}v_{g,x}=v^2$}
A fixed-edge membrane strip has dispersion $\omega_n^2=v^2[k_x^2+(n\pi/b)^2]$, n≥1. Differentiating gives $v_{g,x}=v^2k_x/\omega_n$, whereas $v_{p,x}=\omega_n/k_x$. Their product is $v^2$. In this lossless scalar mode the integrated energy flux divided by energy density agrees with group speed. At cutoff the longitudinal group speed vanishes while formal phase speed diverges; an infinite phase speed therefore indicates no superfast energy transport.
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