 # 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.