= Solution
For fixed $F>0$, the left side $L(v)=F^2v^2$ is a <strictly increasing function> on $v>1$, with limit $F^2$ at $1$ and unbounded growth. The right side $R(v)=1+3/[2(v-1)]+1/[2(v-1)^2]$ is a <strictly decreasing function>, since
$$
R'(v)=-\frac{3}{2(v-1)^2}-\frac1{(v-1)^3}<0.
$$
It tends to infinity as $v\downarrow1$ and to one as $v\to\infty$. Their <continuity> and the <intermediate value theorem> give a crossing, and strict opposite <monotonicity> makes it unique. \b[There is exactly one admissible bore speed $V>U$ for every $F>0$.]
\Image[/past-exam-of-the-waves-and-compressible-flow-course-of-the-university-of-oxford/2023-piston-bore-uniqueness.png]
{title=Unique intersection determining the piston-driven hydraulic-bore speed, illustrated for Froude parameter one}
The illustrated case is $F=1$; changing $F>0$ moves the increasing curve without changing the existence or uniqueness argument. The vertical asymptote at $v=1$ expresses the growing depth ratio $H/h_0=v/(v-1)$.
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