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Equation of a Plane Progressive Wave

Equation of a Plane Progressive Wave :- A plane progressive wave is a wave that travels through a medium in a fixed direction, transferring energy without permanently displacing the particles. In this wave, each particle of the medium executes simple harmonic motion (SHM) about its equilibrium position with the same amplitude and frequency, though their phases differ according to position.

Let us consider a simple harmonic progressive wave traveling along the positive direction of the x-axis (from left to right). In Fig. (a), particles 1, 2, 3, … are shown at their equilibrium positions along the medium.

When the wave passes through the medium, each particle begins to oscillate about its respective equilibrium position. Fig. (b) represents the instantaneous positions of these particles at a particular moment. The smooth curve joining these displaced positions gives the shape of the wave at that instant.

Let time be measured from the moment when particle 1, located at the origin, just begins to oscillate. If $y$ is the displacement of this particle after time $t$ , then

$\displaystyle y=Asin\omega t$ …..(1)

where $A$ is the amplitude of oscillation and , with ν being the frequency of oscillation.

As the wave travels forward, particles situated beyond particle 1 begin to oscillate one after another. If the wave travels with speed $v$ , then a particle located at a distance $x$ from particle 1 (say particle 8) will be reached by the wave after a time interval of seconds.

Therefore, particle 8 will start oscillating seconds later than particle 1. This means that the displacement of particle 8 at time $t$ will be equal to the displacement that particle 1 had at an earlier time (.

Hence, replacing in equation (1) by (, the displacement of particle 8 (at distance $x$ from the origin) at time $t$ is given by

$\displaystyle y=Asin\omega \left( t-\frac{x}{\upsilon } \right)$

$\displaystyle y=Asin\left( \omega t-\frac{\omega }{\upsilon }x \right)$

But $\displaystyle \frac{\omega }{\upsilon }=k$ , hence

$\displaystyle y=Asin\left( \omega t-kx \right)$ …..(2)

This expression represents the displacement of any particle in the medium located at a distance $x$ from the origin at time .

As $\displaystyle k=\frac{2\pi }{\lambda }$ also, hence two more alternative forms of the equation are,

$\displaystyle y=Asin\left[ \left( \frac{2\pi }{T} \right)t-\left( \frac{2\pi }{\lambda } \right)x \right]$ …..(3)

$\displaystyle y=Asin\left[ \frac{2\pi }{\lambda }\left( \upsilon t-x \right) \right]$ …..(4)

If a wave is travelling towards the negative x-axis, then x is replaced by (-x), i.e.,

$\displaystyle y=Asin\left( \omega t+kx \right)$

$\displaystyle y=Asin\left[ \left( \frac{2\pi }{T} \right)t+\left( \frac{2\pi }{\lambda } \right)x \right]$

$\displaystyle y=Asin\left[ \frac{2\pi }{\lambda }\left( \upsilon t+x \right) \right]$

If is the phase difference between the given wave traveling along the positive x-direction and another wave, then the equation of the second wave can be written as :

$\displaystyle y=Asin\left( \omega t-kx\pm \phi \right)$ …..(5)

Equation of a Plane Progressive Wave

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