Equation of Progressive Wave

The wave which transmit energy from one point to another point of a medium is called progressive wave. In such wave,disturbance travels forward and handover to next particle after a certain time. During the propagation of the wave, all the particles of the medium vibrates with the same amplitude and wavelength, but vibration begins a little later than the other particle immediately before it. So, phase lag gets increased along the direction of propagation.

Equation of progressive wave:

Let us consider a wave travelling from left to right along the x- axis. Consider a particle at the origin O vibrate simple harmonically and its displacement (y) at time t is given by:

$$y = aSin  \omega T…..(i)$$

If alpha be the phase difference of the particle at P then,

$$y = aSin  (\omega T –  \alpha) $$

The phase difference alpha at p at a distance x from origin is given by:

$$y = aSin  (\omega T –   \frac{2  \pi x}{ \lambda } ) $$

Since, K is equal to 2 π/λ  called the propagation constant. Then,

$$y = aSin  (\omega T –   kx)…..(ii)$$

Equation (ii) is the equation of progressive wave.

When the wave is travelling from left to right, the phase difference has to be added. So, the equation becomes,

$$y = aSin  (\omega T +   kx)…..(iii)$$

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