 Solution

ID: past-exam-of-the-short-options-paper-shared-by-prelims-and-parts-a-and-b-of-the-physics-course-of-the-university-of-oxford/2017/s7/2/c/solution

Substitution of the exponential ansatz gives the homogeneous amplitude system

(ω_{1}^{2} - ω^{2} - 2 i Ω ω 2 i Ω ω ω_{2}^{2} - ω^{2}) (\overset{x}{ˉ}_{1} \overset{x}{ˉ}_{2}) = 0.

A nonzero mode requires zero determinant. Thus

(ω_{1}^{2} - ω^{2}) (ω_{2}^{2} - ω^{2}) = 4 Ω^{2} ω^{2}

with the stated two effective stiffnesses. The off-diagonal product is positive, which determines the sign on the right.

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