The transverse displacement of a string (clamped at both ends) is given by $\mathrm{y}\left(\mathrm{x},\mathrm{t}\right)=0.06\mathrm{sin}\left(\frac{2\mathrm{\pi x}}{3}\right)\mathrm{cos}\left(120\mathrm{\pi t}\right)$. All the points on the string between two consecutive nodes vibrate with:

(a) same frequency.

(b) same phase.

(c) same energy.

(d) different amplitude.

Choose the correct alternatives:

1. (a, b, d)

2. (a, c)

3. (b, d)

4. (c, d)

(1) Hint: Use the standard equation of standing wave.
Step 1: Compare the given equation with the standard equation of standing wave.
Given equation, $\mathrm{y}\left(\mathrm{x},\mathrm{t}\right)=0.06\mathrm{sin}\left(\frac{2\mathrm{\pi x}}{3}\right)\mathrm{cos}\left(120\mathrm{\pi t}\right)$
Comparing it with the standard equation of stationary wave,

$\mathrm{y}\left(\mathrm{x},\mathrm{t}\right)=\mathrm{asin}\left(\mathrm{kx}\right)\mathrm{cos}\left(\mathrm{\omega t}\right)$
It is represented by a diagram where N denotes nodes and A denotes antinodes.

Step 2: Find the frequency and amplitude of the particles.
Clearly, the frequency is common for all the points.
Consider all the particles between two nodes. They are having the same phase of (120$\mathrm{\pi t}$) at a given time but are having different amplitudes of 0.06$\mathrm{sin}\left(\frac{2\mathrm{\pi }}{3}\mathrm{x}\right)$ and because of different amplitudes, they are having different energies.