Question

Exercise:

[ 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. ]

(i) For the wave on a string described in Exercise do all the points on the
string oscillate with the same (a) frequency (b) phase (c) amplitude ? Explain
your answers (ii) what is the amplitude of a point 0.375 m away from one end ?

Answer 1

(i) All the points on the string except the nodes have the same frequency and phase. However, amplitude at a point on the wave is a function of the position of the point.

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

The amplitude at $x=\left(0.375m\right)$ is

$A=0.06sin\frac{2\pi }{3}x\times 1$

$=0.06sin\frac{2\pi }{3}\times 0.375$

$=0.06sin\frac{\pi }{4}=\frac{0.06}{\sqrt{2}}=0.042m$