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.

1. It represents a progressive wave of frequency 60 Hz.
2. It represents a stationary wave of frequency 60 Hz.
3. 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.
4. Amplitude of this wave is constant.

Answer 1

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.