Optimal sinusoidal waveform for viscous snail locomotion ($A=1/\sqrt{2},V=4U/3,{\Phi }_{\min }=(8\sqrt{2}/3)\mu L{U}^2/h_0$)

For $h=h_0[1-A\sin (2\pi x/L)]$, the crawling speed is $U/V=3A^2/(1+2A^2)$. At prescribed $U$, eliminate $V$ to obtain $\Phi =2\mu L{U}^2(1+2A^2)/(3h_0{A}^2\sqrt{1-A^2})$. The derivative in $A^2$ vanishes only at $A^2=1/2$, and the dissipation diverges at both endpoints. The displayed optimum still dissipates $8\sqrt{2}/3$ times the simple Couette flow dissipation of an externally driven flat foot moving at speed $U$ across a uniform film of thickness $h_0$.