ζ_{t} = w

z = 0

. Part c therefore gives

\overset{ϵ}{˙} = - \frac{ρ g + B k ^{4}}{2 μ k} ϵ,

(1)

and hence

ϵ (t) = ϵ (0) e^{- t / τ (k)}, τ (k) = \frac{2 μ k}{ρ g + B k ^{4}} .

(2)

In terms of wavelength

λ = 2 π / k

τ (λ) = \frac{4 π μ / λ}{ρ g + B ( 2 π / λ ) ^{4}} .

(3)

Without elasticity,

τ \sim 2 μ k / (ρ g)

: shorter wavelengths penetrate less deeply but relax more slowly because their viscous gradients are larger. At short wavelength, elastic bending dominates and

τ \sim \frac{2 μ}{B k ^{3}},

(4)

so the strong

B k^{4}

restoring stress makes very short waves relax rapidly. The graph therefore tends to zero at both wavelength extremes and has one maximum. Differentiating with respect to