= Solitary-wave amplitude-speed relation for the conduit equation
{title2=$c/f_0=(2\ln\alpha-1+\alpha^{-2})/(1-2/\alpha+\alpha^{-2})$}
A positive <travelling wave> $f$ on a uniform background $f_0$ has first integral $cf'^2/(2f^2)+V(f)=V(f_0)$, with $V(f)=\ln f+c/f+(f_0^2-cf_0)/(2f^2)$. Equating the crest and background potentials gives the displayed speed relation for crest ratio $\alpha>1$. The elevation-wave speed exceeds $2f_0$ and approaches that long-wave speed as $\alpha\to1$. The formula requires the positive-area branch, rather than continuation through $f=0$.
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