Two waves are given by $y_1 = \cos \, (4t - 2x)$ and $y_2 = sin \, \big(4t - 2x + \frac{\pi}{4} \big).$ The phase difference between the two waves is
- $\frac{\pi}{4}$
- $\frac{- \pi}{4}$
- $\frac{3 \pi}{4}$
- $\frac{\pi}{2}$
The Correct Option is B
Approach Solution - 1
$\hspace30mm y_1 = \cos (4t - 2x) = \sin \big(4t - 2x + \frac{\pi}{2} \big)$
and $\hspace20mm y_2 = \sin \big(4t - 2x + \frac{\pi}{4} \big)$
Therefore, phase difference between the two waves is
$\hspace40mm ? \phi = \big(4t - 2x + \frac{\pi}{4}\big) - \big(4t - 2x + \frac{\pi}{2}\big)$
$\hspace40mm = \frac{\pi}{4} - \frac{\pi}{2} = - \frac{\pi}{4}$
Approach Solution -2
Equations of waves
\(\hspace30mm y_1 = \cos (4t - 2x) = \sin \big(4t - 2x + \frac{\pi}{2} \big)\)
and \(\hspace20mm y_2 = \sin \big(4t - 2x + \frac{\pi}{4} \big)\)
$y_2 = cos(\frac{\pi}2 - 4t + 2x- \frac{\pi}4)$
$y_2 = cos(\frac{\pi}4 - 4t + 2x$
$y_2 = cos( 4t - 2x-\frac{\pi}4)$
$\Phi = -\frac{\pi}4$
Therefore, phase difference between the two waves is $\Phi = -\frac{\pi}4$
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Concepts Used:
Electromagnetic waves
The waves that are produced when an electric field comes into contact with a magnetic field are known as Electromagnetic Waves or EM waves. The constitution of an oscillating magnetic field and electric fields gives rise to electromagnetic waves.
Types of Electromagnetic Waves:
Electromagnetic waves can be grouped according to the direction of disturbance in them and according to the range of their frequency. Recall that a wave transfers energy from one point to another point in space. That means there are two things going on: the disturbance that defines a wave, and the propagation of wave. In this context the waves are grouped into the following two categories:
- Longitudinal waves: A wave is called a longitudinal wave when the disturbances in the wave are parallel to the direction of propagation of the wave. For example, sound waves are longitudinal waves because the change of pressure occurs parallel to the direction of wave propagation.
- Transverse waves: A wave is called a transverse wave when the disturbances in the wave are perpendicular (at right angles) to the direction of propagation of the wave.