Wave Theories and Transformation Questions and Answers

Answer: A

These four describe a regular wave completely once the water depth is also known. Real sea states require statistical descriptions instead, since no single height and period suffices. A real sea contains waves of many heights and periods at once.

Answer: B

Correct answer: wave frequency to wavenumber and the water depth present

\[ \omega^2 = gk\tanh(kd). \] This single equation governs almost everything about linear wave propagation: it gives the celerity, distinguishes deep from shallow water through the behaviour of the hyperbolic tangent, and shows why waves of different period travel at different speeds.

Answer: C

Correct answer: Both A and R are true, and R is the correct explanation of A

Both are true and the reason gives the mechanism. In deep water \( \tanh(kd) \to 1 \), so \( C = \sqrt{g/k} = gT/2\pi \) - celerity grows with period. Long swell therefore outruns short waves, which is why a distant storm announces itself first as a long clean swell and only later as a confused sea.

Answer: D

The hyperbolic tangent approaches one, so celerity reduces to g times the period over two pi. Deep water waves are therefore dispersive while shallow water waves are not. Dispersion is the defining property of the deep water regime.

Answer: A

The result is independent of period, so all shallow water waves travel at the same speed. The sea state therefore stops dispersing once it reaches shallow water. Depth alone fixes the speed once the wave is long relative to it.

Answer: B

Correct answer: half the wavelength

At \( d = L/2 \), \( \tanh(kd) = \tanh(\pi) = 0.996 \), close enough to unity that the deep water forms may be used. Below about \( L/20 \) the shallow water approximations apply, and the range between is intermediate depth, where the full dispersion relation must be solved.

Answer: C

The first two are correct and are the two limiting cases of the dispersion relation. The third is wrong in every regime, since the celerity always depends on one or the other and usually on both. The dispersion relation covers the whole range of depths.

Answer: D

Both are true and the reason gives the dispersive sorting, and this allows the distance to the generating storm to be estimated. The rate at which the period falls gives the travel time directly. Swell period is a record of how far the waves have travelled.

Answer: A

Correct answer: the wave height is small compared with both depth and wavelength

Linear theory keeps only first-order terms, which requires the wave to be of small amplitude. It also assumes an inviscid, incompressible, irrotational flow over a horizontal bed. The theory is remarkably serviceable well outside its formal limits, but it cannot represent the sharp crests and flat troughs of a real steep wave, nor the mass transport that goes with them.

Answer: B

The first two are correct and are the standard assumptions. The third fails badly, since near breaking the wave is strongly non-linear and higher order theories are required instead. Stokes and cnoidal theories extend the range considerably.