# Progressive Wave Equation

**Introduction:**

The propagation of a wave in a medium, the particles of medium execute simple harmonic motion, then the wave is said to be a simple harmonic progressive wave equation and moreover if the amplitude of such a wave remains unchanged, then it is said to be simple harmonic plane equation of progressive wave. When we throw a stone in the calm water of a pond, we note that a disturbance is produced at the place where stone strikes the water.

## Equation for a Progressive Wave

Suppose a plane progressive wave is propagating in a medium along positive X-axis. The positions of particles O, A, B, C, D …. are see diagram. As the wave propagates, all the particles of the medium begin to vibrate to and fro about their mean positions. The instantaneous positions of these particles are shown. The curve joining these positions represents the progressive wave.

Let the particle begin to vibrate from origin O at time t=0. If y is the displacement of the particle at time t, then progressive wave equation of particle executing simple harmonic motion about O is

**Y = a sin ωt**

Where a is amplitude and ω is angular velocity. If n is frequency of wave, the ω=2πn. As the advancing wave reaches the other particles A, B, C, …. These particles begin to vibrate. If v is the speed of wave and can C is a particle at a distance x from O, then the time taken by wave to reach point C is x/v seconds, therefore the particle will start vibrating x/v seconds after particle O.

Therefore, the equation of progressive wave displacement of particle C at any time t will be the same which was of particle O at time

The displacement of particle O at time can be obtained by substituting

in place of t in equation.