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  • The transverse displacement of a string clamped at its both ends is given by Interpret the wave as a superposition of two waves travelling in opposite directions. What is the wavelength frequency and speed of each wave?

Q.15.11 (b) The transverse displacement of a string (clamped at its both ends) is given byy(x,t)=0.06\; \sin (\frac{2\pi }{3}x)\cos (120\; \pi t)

where x and y are in m and t in s. The length of the string is 1.5 m and its mass is 3.0\times 10^{-2}\; kg.

 Answer the following :

(b) Interpret the wave as a superposition of two waves travelling in opposite directions. What is the wavelength, frequency, and speed of each wave?
 

Answers (1)

best_answer

We know that when two waves of the same amplitude, frequency and wavelength travelling in opposite directions get superimposed we get a stationary wave.

\\y_{1}=asin(kx-\omega t)\\ \\y_{2}=asink(\omega t+kx)

\\y_{1}+y_{2}=asin(kx-\omega t)+asink(kx+\omega t)\\ =asin(kx)cos(\omega t)-asin(\omega t)cos(kx)+asin(\omega t)cos(kx)+asin(kx)cos(\omega t)\\ =2asink(kx)cos(\omega t)

Comparing the given function with the above equations we get

\\k=\frac{2\pi }{3}\\ \lambda =\frac{2\pi }{k}\\ \lambda =3\ m

\\\omega =120\pi \\ \nu =\frac{\omega }{2\pi }\\ \nu =60\ Hz

\\v=\nu \lambda \\ v=60\times 3=180\ m\ s^{-1}

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