Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 3 / 39C / a / i

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- Solution i

Solution

Substitute the monochromatic wave

ϕ = e^{i (k x - ω t)}

into the differential equation. Since

(ik)^{9} = i k^{9}

, cancellation of the common exponential gives

- iω + i U k + \frac{i}{9} k^{9} = 0.

Thus the dispersion relation is

ω (k) = U k + \frac{k ^{9}}{9} .

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