= Solution
The <Lighthill elongated-body theory> is a reactive, nearly <inviscid flow> description of a long slender swimmer, appropriate when the longitudinal <Reynolds number> is large but its lateral <displacement> and slope are small. To leading order, crossflow past each section is the two-dimensional <potential flow> generated by lateral motion of that section. Longitudinal variation is slow on the cross-sectional scale, and axial speed $U$ is prescribed at this order. With $x$ increasing from nose to tail, fluid passes sections in increasing $x$, and the linearized relative lateral <velocity> is
$$
w=h_t+Uh_x,\qquad \mathcal D=\partial_t+U\partial_x.
$$
The section's <added mass> per unit length $m(x)$ is defined by its crossflow <kinetic energy> $\tfrac12m(x)w^2$. Equivalently, if $\Phi$ is the <velocity potential> for unit lateral section motion, then $m=\rho\int|\nabla_\perp\Phi|^2dA$ over the exterior crossflow plane. It has units of <mass> per length and depends on cross-sectional shape and <density>. For a circular section of radius $a$, $\Phi=-a^2\cos\varphi/r$ gives $m=\rho\pi a^2$. It is not the body's <mass> per unit length.
The transverse fluid impulse attached to a section is $m(x)w$. A control segment has both changing impulse and an impulse flux $Umw$ through its ends. Taking their sum per unit length gives the reaction on the fluid:
$$
\boxed{F_z=\partial_t(mw)+U\partial_x(mw)
=\left(\partial_t+U\partial_x\right)\left[m(x)(h_t+Uh_x)\right].}
$$
The <force> on the fish is $-F_z$. In particular, $m$ is inside the convective <derivative>; dropping its spatial <derivative> or the endpoint fluxes is not justified merely because the fish is slender.
The <recoil correction in elongated-body theory> adds unknown whole-body translation and yaw to a prescribed active bending shape:
$$
h=h_0+Z_r(t)+(x-\bar x)\theta(t).
$$
They are determined by total lateral <linear momentum> and yaw <angular momentum> balance, rather than by imposing that the deforming swimmer remains on its original axis. Body inertia and hydrodynamic added inertia have the same sign. If the translation variable is required to be the literal <centre of mass>, the active deformation must first have its mass-weighted mean removed: the reference <displacement> $Z_r$ is otherwise not the <centre of mass> <displacement> during the stroke. The corresponding deformation contribution to <angular momentum> must also be retained when it is significant.
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