Inextensible filament ($|{\mathbf{r}}_s|=1$)

An inextensible filament keeps the length of every material segment fixed. With material arc length $s$, its centerline obeys $|{\mathbf{r}}_s|=1$ and has unit tangent vector $\mathbf{t}={\mathbf{r}}_s$. A filament tension acts as a Lagrange multiplier for this constraint.

For a periodic inextensible filament with arclength period $\Lambda$ and projected axial period $\lambda$, material points slide tangentially through the wave frame at speed $Q$. An axial pattern speed $V$ relative to mean filament motion gives $Q\lambda /\Lambda =V$. Confusing the two speeds changes the swimming-to-wave-speed ratio by $\lambda /\Lambda$.

Write a small planar displacement in arclength coordinates as $X=s+X_2$ and $Y=y$. The inextensibility relation $X_s^2+Y_s^2=1$ gives $X_{2,s}=-y_s^2/2$ to quadratic order. Thus longitudinal material velocity enters one order later than the transverse velocity. For a periodic material stroke with no mean axial drift, its mean velocity vanishes, although its instantaneous contribution is needed for exact local force balance.

The filament bending modulus multiplies squared curvature of a space curve in the bending energy, $E_b=(B/2)\int {\kappa }^{2}ds$. It has dimensions force times length squared. In the small-slope Monge representation, the leading elastic force per unit length is $-B{y}_{xxxx}$.

Filament tension is the axial force enforcing local inextensibility. Its force density is ${\partial }_s(T{\mathbf{r}}_s)$. A pre-existing constant tension contributes $T{y}_{xx}$ to small-slope transverse dynamics; in an unloaded, initially straight filament, tension induced by transverse motion begins at second order in amplitude.