Progressive Wave | Characteristics and Equation

A wave which travels forward undamped and unobstructed is known as progressive wave. It may be either longitudinal or transverse. Light wave, sound wave, etc are some of the examples of progressive waves.

Characteristics

The disturbance produced in the medium travels forward, undamped and unobstructed.
Each particle in the medium executes same type of vibration. They vibrate about their mean position simple harmonically with the same amplitude and frequency. But the phase varies continuously along with the medium.
No particle remains permanently at rest. But each particle comes to rest momentarily while at the extreme position of its vibration. Different particles attain this position of rest at different times.
The wave travel in the form of crests and troughs or compressions and rarefactions.
The displacement, velocity and acceleration at any instant will have the same value for the particles which are separated by an integral multiple of the wavelength.
Energy is transferred across every plane along the direction of propagation.
The wave travels in a medium with a particular velocity depending upon the nature of the medium.
The wave motion repeats itself after every $T$ (time period) seconds or it repeats after every $λ$ distance at a particular instant. Thus, a plane progressive harmonic wave has double periodicity.
The equation of a progressive wave propagating through a medium is given by $y = A \sin (ω t \pm k x)$

Equation of a Progressive Wave

Let a progressive wave starts from origin $O$ and travels along the positive $X -$ axis with velocity $(v)$ as shown in the displacement-time graph.

The particle at the origin $O$ at any time $t$ vibrates according to the equation $y = a \sin ω t$ Where, $ω$ is the angular velocity and $a$ is amplitude. Consider a particle at point $P$ which is at $x$ distance apart from the origin. The particle at $P$ starts vibrating after definite time interval with respect to the particle at $O$ starts vibrating. Hence, the phase lag goes on increasing in this direction.

If $Φ$ be the phase difference, then the particle at $P$ vibrates according to the equation $y = a \sin (ω t - Φ)___(1)$

The phase difference of the particle at $P$ with respect to the origin is given by $Φ = \frac{2 π}{λ} x$

Putting this value in equation $(1)$ , $y = a \sin (ω t - \frac{2 π}{λ} x)$ $y = a \sin (ω t - k x)___(2)$ Where, $k = \frac{2 π}{λ}$ is propagation constant or wave number or wave vector.

If the wave travels in opposite direction i.e. along negative $X -$ axis, then equation $(2)$ becomes $y = a \sin (ω t + k x)___(3)$ From $(2)$ and $(3)$ , $y = a \sin (ω t \pm k x)$ This is the general equation of progressive wave.

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