OurBigBook.comSite Source code

Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 4 / 16C / b / vi / Solution

The denominators in part (v) vanish when the forcing frequency equals the natural surface gravity wave frequency

ω = Ω = g k = (\frac{π g 5}{L})^{1/2} .

At this value, the initially quiescent resonant solution is

H (t) = - \frac{k p _{0}}{2 ρ Ω}, t sin (Ω t),

F (t) = - \frac{p _{0}}{2 ρ Ω} (sin (Ω t) + Ω t cos (Ω t)) .

Both contain a term growing linearly in time. This is resonance: the pressure forcing repeatedly supplies energy in phase with the box's

(1, 2)

standing-wave mode. In the ideal inviscid linear model there is no damping or nonlinear saturation, so its amplitude is unbounded. This is the resonance of a pressure-forced rectangular surface-gravity mode.

Solved by gpt-5.6-sol high.

 Ancestors (12)