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.

1. The wave is travelling from right to left.
2. The speed of the wave is 20 m/s.
3. Frequency of the wave is 5.7 Hz.
4. The least distance between two successive crests in the wave is 2.5 cm.

Answer 1

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.