 Solution

ID: past-exam-of-the-year-2-of-the-earth-sciences-geology-course-of-the-university-of-oxford/2025/paper-3/8/e/solution

For each spatial normal mode, the temporal equation is

T_{n}^{''} + ω_{n}^{2} T_{n} = 0

ω_{n} = c k_{n} = c nπ / L

. Hence

T_{n} (t) = A_{n} cos (ω_{n} t) + B_{n} sin (ω_{n} t) .

The ordinary frequency is

ω_{n} / (2 π) = c n / (2 L)

; it should not be confused with the angular frequency used in the sine and cosine arguments.

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