Question

The equation of progressive wave in a wire is y = 4 sin [π/2(8t - x/8)] if x and y are measured in cm then velocity of wave is:
1. 64 cm/s along -x direction
2. 32 cm/s along -x direction
3. 32 cm/s along +x direction
4. 64 cm/s along +x direction

Answer 1

Correct Answer - Option 4 : 64 cm/s along +x direction

CONCEPT:

The general equation of wave motion is given by:

y =  Asin (kx - ωt)

where A = amplitude, ω = angular frequency, k = angular wavenumber, t = time, and x is position

• Wavelength (λ): Distance between two nearest particles vibrating in the same phase.
• Frequency (f): Number of vibrations complete by a particle in one second.
• Time Period (T): Time taken by the wave to travel a distance equal to one wavelength.
• Amplitude (A): Maximum displacement of vibrating particles from the mean position.

The relationship between frequency (f), wavelength (λ), and velocity (v) of a wave is given by:

f × λ = v 

CALCULATION:

y = 4 sin [π/2(8t - x/8)]

We know that ω = 2πf and f = v/λ

ω = 2πv/λ  ... eq [i]

We also know that 

k = 2π/λ

λ = 2π/k

Now put the value of λ in eq[i] we get

ω = vk

π × 4 = v × π/16

v = 64 cm/s in positive x-axis

• Hence option 4 is correct