(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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You have learnt that a travelling wave in one dimension is represented by a function y = f (x, t) where x and t must appear in the combination x – v t or x + v t, i.e. y = f (x ± v t). Is the converse true? Examine if the following functions for y can possibly represent a travelling wave :
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The transverse displacement of a string (clamped at its both ends) is given by
y(x, t) = 0.06 sin \((\frac{2π}{3 }x)\) cos (120 πt)
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 ×10–2 kg.
Answer the following :
(a) Does the 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
Given below are some functions of x and t to represent the displacement (transverse or longitudinal) of an elastic wave. State which of these represent (i) a travelling wave, (ii) a stationary wave or (iii) none at all:
(a) y = 2 cos (3x) sin (10t)
(b) \(y=2\sqrt{x-vt}\)
(c) y = 3 sin (5x – 0.5t) + 4 cos (5x – 0.5t)
(d) y = cos x sin t + cos 2x sin 2t
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