Simple Harmonic Progressive Wave

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Simple Harmonic Progressive Wave:

• A simple harmonic progressive wave is a wave which continuously advances in a given direction without change of form and the particles of the medium perform simple harmonic motion about their mean position with the same amplitude and period, when the waves pass over them.

Expression for One Dimensional Simple Harmonic Progressive Wave:

• Consider a simple, harmonic progressive wave travelling along the positive direction of the x-axis.  Let the vibration of the particles of the medium be parallel or perpendicular to the x-axis.  At any instant t, the displacement of the particles of the medium situated at the origin is given by,

y    =    a sin ωt

where a is the amplitude and 2π/ω is the period of oscillation.

• Now consider a particle of the medium situated at a point P at a distance of x  from O. This particle also performs S.H.M. with the same amplitude and period, However, the disturbance from O will reach P only after some time.  In other words, the particle at P lags behind the particle at O in phase.  Hence the displacement of the particle at P at the instant t is given by

y    =    a sin ( ωt – ∅ ).

Where ∅ is the phase difference between O and P.

• The angle ∅ depends on the distance OP. In wave motion, two particles separated by a distance λ differ in phase by 2π radians.  In other words, the path difference λ corresponds to a phase difference of 2p radians.  Therefore, the path difference x will correspond to a phase difference of ∅ = 2πx / λ. Hence, the displacement of the particle at P is given by,

This is the equation of a one dimensional simple harmonic progressive wave

travelling along the positive x-axis in terms of the period of the wave.

• If n is the frequency of oscillation of the particle, then  T = 1/n, substituting in equation (1) we get

This is an expression for a simple harmonic progressive wave in terms of frequency of a wave.

• If v is the velocity of the wave in the medium,

This is an expression for a simple harmonic progressive wave in terms of velocity of a wave.

• Equation (2) and (3) are two different forms of equation of a one dimensional simple harmonic progressive wave travelling along positive x-axis.

Characteristics of a Simple Harmonic Progressive Wave:

• All particles of medium perform S.H.M. when the wave passes through the medium.
• All particles vibrate with the same amplitude.
• All particles vibrate with the same frequency.
• The propagation of the simple harmonic progressive wave is a doubly periodic phenomenon, periodic in time and periodic in space.
• Energy is transmitted through the medium.
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