= Endpoint momentum flux in elongated-body recoil
{title2=$\int F_zdx=\dot Q+U[mw]_0^L$}
With $r=x-\bar x$, lateral relative <velocity> $w=(\partial_t+U\partial_x)h$, $Q=\int mw\,dx$ and $P=\int rmw\,dx$, the <Lighthill elongated-body theory> <force> on the fluid is $F_z=(\partial_t+U\partial_x)(mw)$. <Integration by parts> gives $\int F_zdx=\dot Q+U[mw]_0^L$ and $\int rF_zdx=\dot P+U[rmw]_0^L-UQ$. The endpoint terms are <momentum> fluxes and must not be discarded when a tail has nonzero <added mass>. If both endpoint added masses vanish, total lateral body-plus-fluid <momentum> is conserved; a finite-mass body cannot then finish with zero relative crossflow and a new nonzero transverse <velocity>.
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