Different-wavenumber cancellation in sheet swimming ($U_2=\frac{1}{2}(k_{\perp }^2{b}^2-k_{\parallel }^2{a}^2)$)

When longitudinal and transverse sheet waves share unit phase speed but have different wavenumbers $k_{\parallel },k_{\perp }$, their second-order swimming coefficient is $(k_{\perp }^2{b}^2-k_{\parallel }^2{a}^2)/2$. Thus their leading propulsion cancels at $b/a=k_{\parallel }/k_{\perp }$. This statement concerns the small-amplitude leading order.