 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2016-resit/11/e/solution

Release from rest gives

B = D = 0

. At

t = 0

A + 2 C = d

and

A - C = 0

, so

A = C = d /3

. Consequently

y_{1} (t) = \frac{d}{3} (cos ω_{-} t + 2 cos 2 ω_{-} t), y_{2} (t) = \frac{d}{3} (cos ω_{-} t - cos 2 ω_{-} t) .

The frequency ratio is exactly two, so both motions repeat after

2 π / ω_{-}

. These are commensurate normal modes, rather than a pair of nearly equal frequencies producing slow beats.

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