Past exam of the fluids and waves course of the University of Oxford / 2023 / 3 / c / ii

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- Solution ii

Solution

Define the dimensionless parameter

δ = (\frac{ρ g h ^{4}}{B})^{1/4} .

The stipulated hierarchy

kh ≪ δ ≪ 1

implies both

kh ≪ 1

and

\frac{B k ^{4}}{ρ g} = (\frac{kh}{δ})^{4} ≪ 1.

Thus the finite-depth correction uses

tanh (kh) \sim kh

, while the restoring effect of bending stiffness is small compared with gravity. The dispersion relation becomes

ω^{2} = g h k^{2} [1 + (\frac{kh}{δ})^{4}] \frac{t a n h ( kh )}{kh} \sim g h k^{2} .

Hence, to leading order, these surface gravity waves are nondispersive waves with

ω \sim k g h, c_{p} \sim c_{g} \sim c = g h .

For clarity, the first corrections to the phase speed are

\frac{c _{p}}{g h} = 1 + \frac{1}{2} (\frac{kh}{δ})^{4} - \frac{1}{6} (kh)^{2} + O ((kh / δ)^{8} + (kh)^{4} + (kh / δ)^{4} (kh)^{2}) .

They exhibit the weak residual wave dispersion at finite wavenumber. The claimed nondispersive behavior is the asymptotic leading-order result, not exact equality for the complete relation.

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