Past exam of the mathematical methods II course of the University of Oxford / 2017 / 10 / a

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- Solution a

Solution

At fixed frequency

k_{j} = ω / v_{j}

, so the displacement reflection and transmission coefficients are

r = \frac{v _{2} - v _{1}}{v _{2} + v _{1}}, t = \frac{2 v _{2}}{v _{2} + v _{1}} .

The heavy-string speed is

v_{h} = 20 m s^{- 1}

and the light-string speed is

v_{l} = 40 m s^{- 1}

. Therefore

r_{h \to l} = \frac{1}{3}, t_{h \to l} = \frac{4}{3}, r_{l \to h} = - \frac{1}{3}, t_{l \to h} = \frac{2}{3} .

A displacement amplitude greater than the incident amplitude does not mean energy is created: the transmitted pulse also has a different width and propagates in a different medium.

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