Past exam of the mathematical methods II course of the University of Oxford / 2015 / 11 / d / i

Solution

The transmitted and incident waves have the same angular frequency and tension, but the transmitted wavenumber is twice as large. The time-average string-power formula therefore gives

\frac{P _{t}}{P _{i}} = \frac{k _{2} ∣ A _{t} ∣ ^{2}}{k _{1} ∣ A _{i} ∣ ^{2}} = 2 (\frac{2}{3})^{2} = \frac{8}{9} .

The extra factor

k_{2} / k_{1}

is essential. Squaring the displacement transmission coefficient alone would give

4/9

, which is not the transmitted power fraction in strings with different linear mass densities.

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