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Question
A transverse harmonic wave on a string is described by y(x, t) = 3.0 sin (36t + 0.018x + π/4) where x and y are in cm and t is in s. The positive direction of x is from left to right.
- The wave is travelling from right to left.
- The speed of the wave is 20 m/s.
- Frequency of the wave is 5.7 Hz.
- The least distance between two successive crests in the wave is 2.5 cm.
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Solution
a, b and c
Explanation:
The general equation of a plane progressive wave with initial phase is
Various forms of the progressive wave function:
- `y = a sin (ωt - kx)`
- `y = a sin (ωt - (2π)/λ x)`
- `y = a sin 2π [t/T - x/λ]`
- `y = a sin (2π)/T (t - x T/λ)`
- `y = a sin (2π)/λ (vt - x)`
- `y = a sin ω(t - x/v)`
Given equation is `y(x, t) = 3.0 sin(36t + 0.018x + π/4)`
Option (a): Since there is +ve sign between wr and kx, the wave travels from right to left (the positive direction of x is from left to right). Hence it is correct.
Option (b): Speed of the wave, `v = ω/k = 36^-1/(0.018 cm)` = 2000 cm/s = 20 m/s. Hence it is correct.
Option (c): Frequency of the wave, `v = ω/(2π) = (36 s^-1)/(2π)` = 5.7 Hz. Hence it is correct.
Option (d): Least distance between two successive crests, `λ = (2π)/k = (2π)/(0.018 cm^-1)` = 349 cm. Hence it is wrong.
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