= Solution
The <Lighthill elongated-body theory> approximation treats each cross-section of a slender swimmer as a locally two-dimensional <potential flow> problem. The transverse dimension is small compared with the length and the scale over which cross-sections vary. The displacement and slope are small enough to linearize the <impermeability condition> about a straight body. The <Reynolds number> is sufficiently large for lateral loading to be dominated by fluid inertia; <viscosity>, axial friction and higher-order end corrections are neglected at this order. This reactive approximation must be distinguished from the viscous <resistive-force theory> used for a flagellum.
In coordinates translating with the swimmer, the undisturbed fluid passes in the positive $x$ direction at speed $U$. Applying the <impermeability condition> to the moving centre plane gives the transverse fluid <velocity>
$$
v_y=h_t+Uh_x=\mathcal Dh,\qquad \mathcal D=\partial_t+U\partial_x.
$$
The local <added mass> $m(x)$ is defined so that the transverse fluid <hydrodynamic impulse> per unit axial length is $P_y=m(x)v_y$. A fixed body-coordinate interval receives an impulse flux $UP_y$ at its leading end and loses the corresponding flux at its trailing end. Its rate of change of impulse plus the net outward impulse flux is therefore
$$
\partial_tP_y+U\partial_xP_y=\mathcal D(m\mathcal Dh).
$$
This is the transverse <force> exerted by the body on the fluid. By <Newton's third law>, the transverse <force> exerted by the fluid on the body has the opposite sign:
$$
\boxed{F_{\rm fluid}=\mathcal D(m\mathcal Dh),\qquad F_{\rm body}=-\mathcal D(m\mathcal Dh).}
$$
The distinction is essential here: the printed positive expression is labelled as the force on the fish, so that label and expression are inconsistent. For example, take a uniform cross-section and a rigid lateral displacement $h=H(t)$. The swimmer must accelerate fluid of <added mass> $m$, which reacts on it with force $-mH''$, not $+mH''$. Thus the positive expression is recovered exactly with the \b[force-on-fluid convention]; the physically consistent force on the fish is the negative expression. In the following parts, $F_y$ denotes $F_{\rm body}$ so that the beam balance uses a genuine external force on the fish.
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