 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2017/iii/paper-331/3/b/iv/solution

When

R ≪ Q

, the ratio of cubic damping to linear amplification is

R^{2} / Q^{2} ≪ 1

\frac{R ^{'}}{R} = Q^{2} - R^{2} ≃ Q^{2} = μ - k^{2}, R (t) ≃ R (t_{0}) e^{(μ - k^{2}) (t - t_{0})}

while the small-amplitude regime lasts. With

c_{d} = 0

ω = U k + i (μ - k^{2})

and hence amplitude proportional to

e^{(μ - k^{2}) t}

. Thus

γ_{small amplitude} = γ_{linear Fourier mode} = μ - k^{2} .

This is the temporal growth rate of the same Fourier component. It is not the fixed-position absolute frequency rate

μ - U^{2} /4

of a localized impulse. Nonlinear damping becomes important when

R

is comparable with

Q

and then arrests exponential amplification.

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