Explain what is Plane progressive harmonic wave?

During the propagation of a wave through a medium, if the particles of the medium vibrate simple harmonically about their mean positions, then the wave is said to be plane progressive harmonic wave.

Displacement relation for a wave travelling in +X direction is given by

\(y=Asin left{ 2pi left( frac{t}{T}-frac{x}{lambda } right)pm phi right}\)

While displacement relation for wave travelling a in –X direction is given by

\(y=Asin left{ 2pi left( frac{t}{T}-frac{x}{lambda } right)pm phi right}\)

In terms of speed of wave (v), above relation can be written as

For +X-direction, \(y=Asin left{ frac{2pi }{lambda }left( vt-x right)pm phi right}\)

For –X-direction, \(y=Asin left{ frac{2pi }{lambda }left( vt+x right)pm phi right}\)

In the above equation v represent speed of wave and not that of the particle.

Other forms of displacement relation for progressive waves are

\(y=Asin left{ left( omega tpm kx right)pm phi right}\) \(=Asin left{ kleft( vtmp x right)pm phi right}\) \(=Asin left{ omega left( tmp frac{x}{v} right)pm phi right}\)

Here, \(omega =frac{2pi }{T}\) is the angular frequencies.

\(k=frac{2pi }{lambda }=frac{omega }{v},\) (λ is wavelength and v is wave velocity)

k is called angular wave number or propagation constant.

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