Question

A wave in a string has amplitude of 2 cm. The wave travels in the +ve direction of x axis with a speed of $128{ms}^{-1}$ and it is noted that 5 complete waves fit in 4 m length of the string. The equation describing the wave is:

• (A) $y=(0.02)m\sin (7.85x+1005t)$
• (B) $y=(0.02)\mathrm{m}\sin (15.7x-2010t)$
• (C) $y=(0.02)\mathrm{m}\sin (15.7x+2010t)$
• (D) $y=(0.02)m\sin (7.85x-1005t)$

Answer 1

The Correct Option is D

Explanation:
It is given that the amplitude of the wave in a string is, i.e., A = 2 cmAnd also, the wave is travelling in the positive x-direction.And the velocity $v=128{ms}^{-1}$According to the question 5 complete waves fit in a 4m length of the string, which means that:$5\lambda =4$ ----(1)From (1) equation,$\lambda =\frac{4}{5}$And the propagation constant,$k=\frac{2\pi }{\lambda }=2\frac{\pi 5}{4}=7.85$Also, the phase velocity of the wave is given by:$v=\frac{\omega }{k}=128{ms}^{-1}$$\Rightarrow \omega =v\times k=128\times 7.85$$=1004.8$$\approx 1005rad/sec$The equation of wave can be represented as:$y=A\sin (kx-\omega t)$By putting the given values in the above equation we get:$y=2\sin (7.85x-1005t)$$\Rightarrow y=(0.02)\sin (7.85x-1005t)$ (amplitude in meter $=0.02m)$$\Rightarrow y=(0.02)\mathrm{m}\sin (7.85x-1005t)$Hence the correct option is (D).

Answer 2

Displacement Relation For a Progressive Wave

A traveling wave is also referred to as a progressive wave. If during the propagation of a wave in a medium, the particle of the medium performs simple harmonic motion, then the wave is called a simple harmonic progressive wave.

If a wave is traveling along + x direction, then the displacement y of the element located at distance x in time t is given by

y = a sin (kx - ωt)

If a wave is traveling along the -x direction, then

y = a sin (kx + ωt)

Where

• a is the maximum amplitude
• k is the propagation constant or angular wave number
• ω is the angular frequency