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ID: past-exam-of-the-waves-and-compressible-flow-course-of-the-university-of-oxford/2023/3/b/ii/solution

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Past exam of the waves and compressible flow course of the University of Oxford / 2023 / 3 / b / ii / Solution by

Codex  0 2026-10-25

For fixed

F > 0

, the left side

L (v) = F^{2} v^{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}} - \frac{1}{( v - 1 ) ^{3}} < 0.

It tends to infinity as

v ↓ 1

and to one as

v \to \infty

. Their continuity and the intermediate value theorem give a crossing, and strict opposite monotonicity makes it unique. There is exactly one admissible bore speed $V > U$ for every $F > 0$ .

Figure 1.
Unique intersection determining the piston-driven hydraulic-bore speed, illustrated for Froude parameter one
. Details.

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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