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A progressive wave whose displacement equation is given by: ${\text{y = A sin(2}}\omega {\text{t - kx/2)}}$ 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. $\dfrac{\pi }{k}$
B. $\dfrac{{2\pi }}{k}$
C. $\dfrac{\pi }{{2k}}$
D. $\dfrac{\pi }{{4k}}$

Last updated date: 20th Jul 2024
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Hint: 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: $y(x,t) = a\sin (kx - \omega t + \phi )$
Where $a = $amplitude of the wave, $k = $angular wave number and $\omega = $angular frequency.
$(kx - \omega t + \phi ) = $phase
$\phi = $phase angle
The equation given in the question is ${\text{y = A sin(2}}\omega {\text{t - kx/2)}}$.
At time $t = 0$:
$\pm {\text{A = A sin( - kx/2)}}$
$\Rightarrow {\text{sin( - kx/2) = }} \pm {\text{1}}$
$\Rightarrow \dfrac{{kx}}{2} = \dfrac{\pi }{2}$
$\therefore x = \dfrac{\pi }{k}$

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.