 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2015/11/e/ii/solution

ρ_{1} ≪ ρ_{2}

, then

k_{1} ≪ k_{2}

and the wave meets a much larger string impedance. The displacement and power coefficients approach

r \to - 1, t \to 0^{+}, R \to 1, T \to 0 .

The junction behaves approximately as a fixed end. The reflected displacement reverses sign, corresponding to a phase shift of

π

; the small transmitted displacement is not inverted. For a large but finite density ratio,

t ≃ 2 k_{1} / k_{2}

and the transmitted power fraction is about

4 k_{1} / k_{2}

, tending to zero rather than vanishing identically at every finite ratio.

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