Name: Velocity and Acceleration of Particles in Wave Motion
Uploaded: 2019-07-19
Duration: 7 min 3 s
Description: We know that there is no net movement of particles in a wave, but they vibrate at their place. The video discusses the velocity and acceleration of the parti...

Question

What is a plane progressive simple harmonic wave? Establish equation for this wave in two different forms.

Accepted Answer

A plane progressive simple harmonic 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 wave passes over them.

Now let's check equation of plane progressive simple harmonic wave:

Let us consider that a plane progressive simple harmonic wave propagating in $+ x$ direction.

At any instant t, the displacement of a particle situated at origin is,

$y = a \sin (ω t)$

Where, $a =$ amplitude

$ω =$ angular frequency $= 2 πn = \frac{2 π}{T}$

$n =$ frequency

$T =$ Time period

Now for the particle situated at P at distance x from the origin, the displacement is

$y = a \sin (ω t - φ)$ ............ $(1)$

where, $φ$ is the phase difference between the points O and P.

Now, we know that

Phase difference $= \frac{2 π}{λ} \times$ path difference

So, $φ = \frac{2 πx}{λ}$

putting this in the equation $(1)$

$y = a \sin (ω t - \frac{2 πx}{λ})$

or, $y = a \sin (\frac{2 π t}{T} - \frac{2 πx}{λ})$

or, $y = a \sin 2 π (\frac{t}{T} - \frac{x}{λ})$

or, we can write it as,

$y = a \sin 2 π (n t - \frac{x}{λ})$

or, $y = a \sin 2 π (\frac{v t}{λ} - \frac{x}{λ})$

or, $y = a \sin \frac{2 π}{λ} (v t - x)$

What is a plane progressive simple harmonic wave? Establish equation for this wave in two different forms.

Important Questions on Progressive Harmonic Waves

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