Displacement Relation in Progressive wave
A wave which travels continuously in a medium in the same direction without any change in its amplitude is called a progressive wave or a traveling wave.
A progressive wave may be transverse or longitudinal in nature.Suppose that a plane simple harmonic wave travels from origin O along the positive direction of X-axis.
Figure-Displacement curve of a |
If time is counted from the instant, the particle at the origin just passes through mean position in positive direction, then the displacement of a particle at O and at any time t is given by
y = r sin Ï‰ t
where r is amplitude of the S.H.M. executed by the particle and Ï‰ is its angular frequency. Let us find the displacement of the particle at point P situated at a distance x from the origin at any time t.
A particle on positive X-axis receives disturbance a definite time later than that preceding it. Therefore, the phase lag of the particles w.r.t. particle at point O goes on increasing as we move more and more away from the point O. If Ï† is the phase lag of the particle at point P w.r.t. that at point O, then displacement of particle at point P at any time t is given by
y = r sin (Ï‰ t - Ï† )
Since for a distance Î» , phase changes by 2Ï€ ; the change in phase for a distance x will be x i.e.
Ï† =
Therefore, y = r sin (Ï‰ t - )
If T is period of vibration, then
Ï‰ =
âˆ´y = r sin
y = r sin 2Ï€ ---------(i)
This is equation of a plane progressive simple harmonic wave traveling along positive X-axis. Let us express it in the form, it was introduced in the earlier discussion.
Now, for a wave motion, v = vÎ»
Or v = or
Therefore equation (i) becomes
y = r sin 2Ï€
or y = r sin ---------(ii)
If the wave is traveling along negative X-axis, the equations (i) and (ii)
respectively become
y= r sin 2Ï€
and y = r sin
Discussion
The following points may be noted about the equation representing a traveling wave:- It gives the displacement at any position x and at any time t.
- At any position (x fixed), the displacement again becomes same after time T.
Substituting (t + T) in the place of t in equation (ii), we have
y = r sin = r sin
= r sin = r sin( [._{.}. vT = Î»]
y = r sin
It tells that time period of a vibrating particle is T. - At any time (t fixed), the displacement again becomes same at a distance Î» . Substituting for x in equation (ii), we have
y = r sin - The equation tells that the disturbance at any point is same after a time Î” t but at a point v Î” t. In other words,
y (x, t) = y (x + v Î” t + Î” t)
y (x, t) = y (x + v Î” t, t + Î” t)
y (x+ v Î” t, t + Î” t) = r sin = r sin
y (x + v Î” t, t + Î” t) = y (x, t)
It tells that wave travels in the medium with velocity v without any change in its amplitude.