Question

Of the following, the equation of plane progressive wave is:

A.
$\begin{align}
  & y=rsin\omega t \\
 & \\
\end{align}$
B.
$\begin{align}
  & y=rsin\left( \omega t-kx \right) \\
 & \\
\end{align}$
C.
$\begin{align}
  & y=\dfrac{a}{\sqrt{r}}sin\left( \omega t-kx \right) \\
 & \\
\end{align}$
D.
 $\begin{align}
  & y=\dfrac{a}{r}sin\left( \omega t-kt \right) \\
 & \\
\end{align}$

Answer 1

Hint: The vibration of each particle in a medium will be about its mean position itself. The disturbance moves forward from one particle to another. Another characteristic is when the particles of the medium vibrate, they all will be with the same amplitude about its mean positions.

Complete step by step answer:
First of all let us look at what a plane progressive wave is.
 When a wave is propagating through a specific medium, if the particles in that medium are oscillating simple harmonically about their mean positions, such waves are said to be plane progressive waves. For simple harmonic motion, the displacement at any instant of time is given by
$y=A\sin \omega t$$y=A\sin (\omega t-kx)$ ………… (1)
Where ω is the angular frequency of the wave and A is the amplitude.
Now let us assume something,
Suppose a particle P is at a distance x from the particle O which is on its right. From left to right, let the wave travel with a velocity v. As per this conditions, its displacement after the time it takes to reach is given as
$y=A\sin (\omega t-\phi )$ …………… (2)
Here is the phase difference between the o and p.
We all know, the phase difference is given as,
 $\phi =\dfrac{2\pi }{\lambda }x$………… (3)
Propagation constant k is given as
 $k=\dfrac{2\pi }{\lambda }$………….. (4)
Now substitute the equation (4) in equation (1)
 $y=A\sin (\omega t-kx)$
This is the required solution or the equation of the plane progressive wave.

Note:
The equation for propagation constant and that of phase difference should be known before. Otherwise solving this question will be difficult. When a wave is propagating through a specific medium, if the particles in that medium are oscillating simple harmonically about their mean positions, such waves are said to be plane progressive waves.