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Expression for Plane Progressive Harmonic Wave

Let us consider a wave originating at 0 (Figure 2.1) is travelling along +X -axis. If y be the  Simple harmonic displacement of a particle at position P, distant x from 0 at 0, then y is expressed as

 

y =a sin (wt- φ),

 

where a is the amplitude, w is the angular velocity and φ  the phase lag of the particle. Since the

successive particles to the right of point 0 receive and repeat the harmonic motion after a definite interval of time resulting in a phase lag. This phase lag goes on increasing with  increasing distance from 0.

If A be the wavelength, then we know a distance A corresponds to a phase difference of 2π. Thus, a path difference of x corresponds to a phase difference φ = 2πx /λ

 

Substituting  w = 2π T and  φ = 2πx /λ

Y =a sin (2πt/t-2πtλ)

 

 

If v is the wave or phase velocity, then v = λv = λ/t

T=λ/v

 

Substituting this value of T in the above equation, we have

Equation (3) can also be expressed in terms of propagation constant k. We know

 

2π/λ = k, therefore,

 

Y =a sin (wt- kx)

When the wave is travelling towards the left, i.e. -x-axis, the wave equation expressed

by equation (3), can be written as

The quantity (rot± kx) represents the phase of wave motion. We have assumed that at t = 0, y = 0 i.e., motion starts at 0. If this is not the case and the particle has some initial phase

 

80′ the wave equation is represented as