Name: Waves and Their Phases
Uploaded: 2019-07-19
Duration: 6 min 51 s
Description: In this video, we will learn how two or more waves can be studied in terms of their phases. Equations of two waves and graphs of two waves. Learn more!

Question

Write down the equations of progressive harmonic wave in as many different forms as you can and explain the different symbols.

Accepted Answer

Consider a simple harmonic progressive wave travelling with a speed v along the positive x-axis. Let y denote the displacement of a particle of the medium from its mean position. We assume that the initial phase of the oscillatory motion is zero. At time t, the displacement of the particle of the medium situated at the origin is

$y = A \sin ω t . . . (1)$

Where A is the amplitude of the wave and ω is related to the frequency n of the wave motion by the equation $ω = 2 π n$ .

Simple harmonic progressive wave
Consider a particle of the medium situated at point P at a distance x from the origin O as shown in the figure. This particle also performs SHM with the same amplitude A and the frequency n. However, the disturbance at O reaches P only after some time, i.e., the particle at P displacement of the particle at P is

$y = A \sin (ω t - α) . . . (2)$

Since a path difference of λ (wavelength) corresponds to a phase difference of 2π radians, a path difference of x corresponds to a phase difference of α such that

$\frac{λ}{2 π} = \frac{x}{α} ∴ α = \frac{2 π}{λ} x ∴ y = A \sin (ω t - \frac{2 π}{λ} x) . . . . (3)$

If n is the frequency of vibration,

$ω = 2 π n = \frac{2 π v}{λ} . . . (4)$

Where v is the speed of the wave,

$∴ y = A \sin 2 π (n t - \frac{x}{λ}) . . . (5)$

Equation (5) is the equation of a simple harmonic progressive wave travelling in the positive direction of the x - axis.
Also, from Eq. (4),

$y = A \sin 2 π (\frac{v}{λ} t - \frac{x}{λ}) ∴ y = A \frac{\sin (2 π)}{λ} (v t - x) . . . . (6)$

in terms of v and λ, as required.

Write down the equations of progressive harmonic wave in as many different forms as you can and explain the different symbols.

Important Questions on Waves

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