 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ii/paper-3/35a/b/i/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

(2 π)^{2} / A

in two-dimensional

k

-space. Including the two phonon polarizations, the number of modes in the disk

∣ k ∣ \leq K

N (K) = 2 \frac{A}{( 2 π ) ^{2}} π K^{2} = \frac{A K ^{2}}{2 π} .

ω = C ∣ k ∣^{α}

gives

K = (ω / C)^{1/ α}

, so

N (ω) = \frac{A}{2 π} (\frac{ω}{C})^{2/ α} .

g (ω) = B ω^{b}, B = \frac{A}{π α C ^{2/ α}}, b = \frac{2}{α} - 1.

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