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 Incoming links: Hydrostatic lower bound for a travelling-pressure response

Past exam of the fluids and waves course of the University of Oxford / 2021 / 3 / b / ii / Solution 2026-10-25

For

0 < c < c^{*}

, the nonnegative amplitude is

a (c) = \frac{p _{1} t a n h h}{ρ [( c ^{*} ) ^{2} - c ^{2} ]} = \frac{p _{1}}{ρ g} \frac{1}{1 - ( c / c ^{*} ) ^{2}} .

Its first two derivatives are positive in this interval:

\frac{d a}{d c} = \frac{2 p _{1} t a n h h c}{ρ [( c ^{*} ) ^{2} - c ^{2} ] ^{2}} > 0, \frac{d ^{2} a}{d c ^{2}} = \frac{2 p _{1} t a n h h [( c ^{*} ) ^{2} + 3 c ^{2} ]}{ρ [( c ^{*} ) ^{2} - c ^{2} ] ^{3}} > 0.

The graph is increasing and convex, starts with a horizontal tangent at the limiting value

a (0) = p_{1} / (ρ g)

, and has a vertical asymptote at

c = c^{*}

. Therefore the hydrostatic lower bound for a travelling-pressure response is

a_{crit} = \frac{p _{1}}{ρ g}, a (c) > a_{crit} (0 < c < c^{*}) .

Because the paper assumes

c > 0

, this is an infimum, approached as

c ↓ 0

, rather than an attained minimum. If the stationary limit

c = 0

is admitted, equality holds: the hydrostatic equilibrium condition is

ρ g η = - p_{1} cos x

. Near resonance,

a (c) \sim \frac{p _{1}}{ρ g} \frac{c ^{*}}{2 ( c ^{*} - c )} .

The divergence describes the linear model; sufficiently close to

c^{*}

the small-disturbance assumptions fail. The lower bound and sketch refer to the travelling forced component, since adding free oscillations can change the instantaneous surface elevation.

Figure 1.
Subcritical travelling-wave amplitude rises from its hydrostatic lower bound and diverges at resonance
. Details.

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