Question

The transverse displacement of a string (clamped at its both ends) is given by
$y(x,t)=0.06\sin \left(\frac{2\pi x}{3}\right)\cos \left(120\pi t\right)$
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 following :
(a) Does function represent a travelling wave or a stationary wave?
(b) Interpret the wave as a superposition of two waves travelling in opposite
directions. What is the wavelength, frequency, and speed of each wave ?
(c) Determine the tension in the string.

Answer 1

(a)

Travelling wave is given by $y(x,t)=A\sin (\omega t\pm kx+\varphi )$

Standing wave is given by $y(x,t)=A\sin \left(kx\right)\cos \left(\omega t\right)$

Hence, this is an example of standing wave.

(b)

$y(x,t)=0.06\sin \left(\frac{2\pi x}{3}\right)\cos \left(120\pi t\right)$

$y(x,t)=0.03\sin (\frac{2\pi x}{3}+120\pi t)+0.03\sin (\frac{2\pi x}{3}-120\pi t)$

Wavelength, $\lambda =\frac{2\pi }{k}=\frac{2\pi }{2\pi /3}=3m$

Frequency, $f=\frac{\omega }{2\pi }=\frac{120\pi }{2\pi }=60Hz$

Speed, $v=f\lambda =180m/s$

(c)

Speed of a wave in a string is given by:

$v=\sqrt{T/\mu }$

$⟹T=v^{2}m/l$

$T=\frac{{180}^2\times 3\times {10}^{-2}}{1.5}=648N$