(i) For the wave on a string described in Exercise 15.11, 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?
Solution and Explanation
(i) Yes, except at the nodes
Yes, except at the nodes
No, 0.042 m
Explanation:
(i) All the points on the string oscillate with the same frequency, except at the nodes which have zero frequency.
All the points in any vibrating loop have the same phase, except at the nodes.
All the points in any vibrating loop have different amplitudes of vibration.
The given equation is:
y(x,t)=0.06 sin \((\frac{2\pi}{3}x)\) cos (120 πt)
For x = 0.375 m and t = 0
Amplitude= Displacement= 0.06 sin \((\frac{2\pi}{3}x)\) cos 0
= 0.06 sin \((\frac{2\pi}{3}× 0,375)×1\)
= 0.06 sin (0.25 π)=0.06 sin \((\frac{\pi}{4})\)
= 0.06× \(\frac{1}{\sqrt2}=0.042\, m\)
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