Past exam of the physics A course of the University of Cambridge / 2021 / Paper 2 / Section B / B6 / d

Let the middle segment occupy

0 < x < l

. Normalize the incident displacement complex amplitude to one, suppress the common time factor

e^{- iω t}

, and choose

y_{L} (x) y_{M} (x) y_{R} (x) = e^{i k_{L} x} + r e^{- i k_{L} x}, = a e^{i k_{M} x} + b e^{- i k_{M} x}, = t e^{i k_{R} (x - l)}, x < 0, 0 < x < l, x > l .

Here

r

is the reflected complex amplitude at

0

a

and

b

are the rightgoing and leftgoing middle-segment complex amplitudes referenced to

0

, and

t

is the outgoing complex amplitude referenced to

l

. The middle wavenumber is

k_{M} = ω / c_{M}

, and

γ = e^{i k_{M} l}, ∣ γ ∣ = 1

is the propagation phase factor across the lossless middle segment.

At each junction, continuity of displacement and transverse force means continuity of

y

and

T y_{x}

. Using

T k_{j} = ω Z_{j}

gives

1 + r Z_{L} (1 - r) γa + γ^{- 1} b Z_{M} (γa - γ^{- 1} b) = a + b, = Z_{M} (a - b), = t, = Z_{R} t .

These boundary conditions also include all multiple reflections inside the middle segment:

a

and

b

are the total travelling components, not just the first waves to cross a junction.

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