Two strings with different masses per unit length are connected together and hel

Question

Two strings with different masses per unit length are connected together and held at a tension of 100N. An wave with an amplitude of 2 cm and a frequency of 100.0 Hz is incident from the left. The waves on the left-hand string are observed to have a wavelength of 20 cm, while those on the right-hand string have a wavelength of 30cm. What is the mass per unit length of the left string? What is the mass per unit length of the right string? What is the amplitude of the transmitted waves on the right string? What is the transmitted power? What are the coefficients of reflection and transmission?

Explanation / Answer

speed in first string = lambda f = 0.20*100 = 20 m/s

speed in second string =30 m/s

mass density in left = T/v^2 = 100/20^2 = 0.25 kg/m

mass density in right= T/v^2 = 100/30^2 = 0.1111 kg/m

Amplitude of transmitted wave = 2Au1/(u1+u2)

= 2*2*0.25/(0.25+0.11111)

= 2.769 cm

transmitted power = 0.5u2*A^2*v*(2pi f)^2

= 0.5*0.11111*0.02769^2*30*(2 pi*100)^2

= 504.5 W

reflection coefficient =[(u1-u2)/(u1+u2)]^2 = [(0.25-0.111111)/(0.25+0.11111)]^2 = 0.148

transmission coefficient = 4u1u2/(u1+u2)^2 = 4*0.11111*0.25/(0.25+0.11111)^2 = 0.852

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