Question

Four harmonic waves of equal frequencies and equal intensities ​ have phase angles $0,\pi ,\dfrac{\pi }{3},\dfrac{2\pi }{3}$. When they are superposed, the intensity of the resulting wave is nl. The value of n is

Answer 1

The intensity of wave is directly proportional to the square of the amplitude of the wave. When two waves get superposed, the net amplitude will be the product of the sum of amplitudes and the phase angle between the two waves. As the waves with phase angle 0,$\pi$get cancelled, the only waves that survive after superposition are two. Calculate the net amplitude of those two waves and find the intensity.

Formula used:
${{A}_{net}}=({{A}_{1}}+{{A}_{2}})\cos \phi$

Complete step-by-step answer:
Let us assume the amplitude of each wave as A and the intensities as I. As the waves are having equal frequencies and equal intensities, the superposed wave containing phase angles 0,$\pi$gets cancelled out.
Therefore, there are two waves present after superposition. The resultant amplitude of the two waves will be,
$\begin{aligned} & {{A}_{net}}=2A\cos \dfrac{\pi }{3} \\\ & {{A}_{net}}=\sqrt{3}A \\\ & \\\ \end{aligned}$
As we know, the intensity of a wave is directly proportional to the square of the amplitude of the resultant wave,
Therefore, the intensity of the resultant wave will be of order,
$\begin{aligned} & I\alpha {{A}^{2}} \\\ & \Rightarrow I=3{{A}^{2}} \\\ \end{aligned}$
If we compare it with the equation given in the question, we can say that the value of n=3.

Additional Information: According to the principle of superposition. The resultant displacement of a number of waves in a medium at a particular point is the vector sum of the individual displacements produced by each of the waves at that point. The phenomena of formation of maximum intensity at some points and minimum intensity at some other point when two (or) more waves of equal frequency having constant phase difference arrive at a point simultaneously, superimpose with each other is known as interference. There are two types of interference, constructive and destructive interference. If the amplitudes of the two waves add up resulting in the maximum amplitude, it is constructive interference and, if the amplitude resultant is minimum, the interference pattern in destructive pattern.

Note: In the above question., the waves with phase angles zero and pie get cancelled with each other during the superposition, hence the resultant wave will not be formed when such waves superpose. In the case of the other two waves, the phase difference won’t cancel out, hence a resultant wave will be formed .