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Question: A progressive wave whose displacement equation is given by: \({\text{y = A sin(2}}\omega {\text{t - ...

A progressive wave whose displacement equation is given by: y = A sin(2ωt - kx/2){\text{y = A sin(2}}\omega {\text{t - kx/2)}} falls on a wall normal to its surface and gets reflected. At what minimum distance in front of the wall the particles of air will be vibrating with maximum amplitude?
A. πk\dfrac{\pi }{k}
B. 2πk\dfrac{{2\pi }}{k}
C. π2k\dfrac{\pi }{{2k}}
D. π4k\dfrac{\pi }{{4k}}

Explanation

Solution

The wave which travels continuously in a medium in the same direction without any change in its amplitude is known as a travelling wave or a progressive wave. Amplitude can be defined as the magnitude of maximum displacement of a particle in a wave from the equilibrium position.

Complete step by step answer:
A progressive wave can be transverse or longitudinal. During the propagation of a wave through a medium, if the particles of the medium vibrate simply harmonically about their mean positions, then the wave is called a plane progressive wave.

Generally, the displacement of a sinusoidal wave propagating in the positive direction of x-axis is given by: y(x,t)=asin(kxωt+ϕ)y(x,t) = a\sin (kx - \omega t + \phi )
Where a=a = amplitude of the wave, k=k = angular wave number and ω=\omega = angular frequency.
(kxωt+ϕ)=(kx - \omega t + \phi ) = phase
ϕ=\phi = phase angle
The equation given in the question is y = A sin(2ωt - kx/2){\text{y = A sin(2}}\omega {\text{t - kx/2)}}.
At time t=0t = 0:
±A = A sin( - kx/2)\pm {\text{A = A sin( - kx/2)}}
sin( - kx/2) = ±1\Rightarrow {\text{sin( - kx/2) = }} \pm {\text{1}}
kx2=π2\Rightarrow \dfrac{{kx}}{2} = \dfrac{\pi }{2}
x=πk\therefore x = \dfrac{\pi }{k}

Therefore, option A is the correct answer.

Note: The phase describes the state of motion of the wave. The points on a wave that travel in the same direction and rise and fall together are said to be in phase with each other. The points on a wave which travel in opposite directions to each other such that one is falling and another one is rising, are said to be in anti-phase with each other.