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A string is composed of two sections joined at x = 0. The left section (x < 0) has linear mass density muₗ and the right section (x > 0) has linear mass density mu_r. The string tension T is uniform throughout. A transverse wave of unit amplitude travels along the left section toward the junction. What is the amplitude of the wave transmitted into the right section?

1. 1
2. 2 / (1 + sqrt(muₗ / mu_r))
3. 2 * sqrt(muₗ / mu_r) / (1 + sqrt(muₗ / mu_r))
4. (sqrt(muₗ / mu_r) - 1) / (sqrt(muₗ / mu_r) + 1)

Correct answer: 2 * sqrt(muₗ / mu_r) / (1 + sqrt(muₗ / mu_r))

Solution

From boundary conditions, the transmitted amplitude is Aₜ = 2*v_r/(vₗ + v_r). Dividing numerator and denominator by v_r and substituting vₗ/v_r = sqrt(mu_r/muₗ) gives Aₜ = 2/(1 + sqrt(muₗ/mu_r)).

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