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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
wave at any time t.

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 ω t

where r is amplitude of the S.H.M. executed by the particle and ω 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 φ 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 (ω t - φ )

Since for a distance λ , phase changes by 2π ; the change in phase for a distance x will be x i.e.
φ =

Therefore, y = r sin (ω t - )
If T is period of vibration, then
ω =

y = r sin
y = r sin 2π ---------(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 = vλ
Or                               v =        or       

Therefore equation (i) becomes
y = r sin 2π
or    y = r sin          ---------(ii)
If the wave is traveling along negative X-axis, the equations (i) and (ii)
respectively become
y= r sin 2π
and y = r sin

Discussion

The following points may be noted about the equation representing a traveling wave:
  1. It gives the displacement at any position x and at any time t.
  2. 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 = λ]
    y = r sin
    It tells that time period of a vibrating particle is T.
  3. At any time (t fixed), the displacement again becomes same at a distance λ . Substituting for x in equation (ii), we have
    y = r sin
     
    It tell that wavelength of the wave is λ.
  4. The equation tells that the disturbance at any point is same after a time Δ t but at a point v Δ t. In other words,
    y (x, t) = y (x + v Δ t + Δ t)
     
    Replacing x by x + v Δ t and t by t + Δ t in equation (ii), we obtain
    y (x, t) = y (x + v Δ t, t + Δ t)
     
    Replacing x by x + v Δ t and t by t + Δ t in equation (ii), we obtain
    y (x+ v Δ t, t + Δ t) = r sin = r sin
    y (x + v Δ t, t + Δ t) = y (x, t)
    It tells that wave travels in the medium with velocity v without any change in its amplitude.




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