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Question
The displacement of a string is given by y (x, t) = 0.06 sin (2πx/3) 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.
- It represents a progressive wave of frequency 60 Hz.
- It represents a stationary wave of frequency 60 Hz.
- It is the result of superposition of two waves of wavelength 3 m, frequency 60 Hz each travelling with a speed of 180 m/s in opposite direction.
- Amplitude of this wave is constant.
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Solution
b and c
Explanation:
We know that the standard equation of stationary wave is `y(x, t) = a sin(kx) cos(ωt)`
Given equation is `y(x, t) = 0.06 sin((2πx)/3) cos(120 πt)`
a. Comparing with a standard equation of stationary wave `y(x, t) = a sin(kx) cos(ωt)` Clearly, the given equation belongs to stationary wave. Hence, option (a) is not correct.
b. By comparing,
ω = 120 π
⇒ 2πf = 120 π
⇒ f = 60 Hz
c. k = `(2π)/3 = (2π)/λ`
⇒ λ = wavelength = 3m
Frequency = f = 60 Hz
Speed = v = fλ = (60 Hz) (3m) = 180 m/s
d. Since in stationary waves, all particles of the medium execute SHM with varying amplitude nodes.
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