Question 504: Waves

Question

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

Accepted Answer

The intensity of a wave is proportional to the square of its amplitude i.e., I
0
= cA
2
, where c is a constant. The amplitudes of four harmonic waves are equal as their intensities are equal. Let these waves be travelling along the x direction with wave vector k and angular frequency $\omega$. The resultant displacement of these waves is given by
$y = {y_1} + {y_2} + {y_3} + {y_4}$
$ = A\sin (\omega t - kx + 0) + A\sin (\omega t - kx + \pi /3) + Asin(\omega t - kx + 2\pi /3) + A\sin (\omega t - kx + \pi )$
$ = A\sin (\omega t - kx + \pi /3) + Asin(\omega t - kx + 2\pi /3)$
$ = 2A\sin (\omega t - kx + \pi /2)\cos (\pi /6)$
$ = \sqrt 3 A\cos (\omega t - kx)$.
The amplitude of the resultant wave is ${A_r} = \sqrt 3 A$ and its intensity is ${I_r} = cA_r^2 = 3c{A^2} = 3{I_0}$.

Waves

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

Integer / Numeric Answer Type