Fourier orthogonality of sheet swimming modes

The second-order mean no-slip boundary condition correction contains products of first-order deformation modes and velocity gradients. Distinct integer wavenumbers have zero averaged mixed products. Their separate mean longitudinal and transverse contributions therefore add, even though mixed products create nonzero oscillatory flow modes.

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.