Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2015 / Paper 2 / 17 / d

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- Solution d

Solution

Expanding the determinant along the first row gives

det (D - λ I) = (2 - λ) [(2 - λ)^{2} - 2] .

The eigenvalues and normalized spatial eigenvectors, in increasing frequency order, are

λ_{j} 2 - 2 2 2 + 2 a^{(j)} \frac{1}{2} (1, 2, 1) \frac{1}{2} (1, 0, - 1) \frac{1}{2} (1, - 2, 1) ω_{j}^{2} (k / m) (2 - 2) 2 k / m (k / m) (2 + 2) .

For each mode both signs of

ω_{j}

give the same cosine displacement. An arbitrary overall mode amplitude remains free. These vectors are the

N = 3

instances of the sine eigenvectors of the fixed-end spring chain.

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