Three harmonic waves having equal frequency v and same intensity $\mathrm{I}_{0}$, have phase angles $0, \frac{\pi}{4}$ and $-\frac{\pi}{4}$ respectively. When they are superimposed the intensity of the resultant wave is close to:

JEE Main · 2020 · 09 Jan 2020 · Shift 1 · Waves & Sound

Three harmonic waves having equal frequency v and same intensity

I_{0}

, have phase angles

0, \frac{π}{4}

and

- \frac{π}{4}

respectively. When they are superimposed the intensity of the resultant wave is close to:

Answer: (D)

Treat each wave as a phasor of the same length

A

, with

I_{0} \propto A^{2}

. The three phasors point at

0

+ \frac{π}{4}

and

- \frac{π}{4}

; the two tilted ones have equal and opposite vertical parts, so they add along the horizontal to

2 A cos \frac{π}{4} = A 2

. The resultant amplitude is therefore

A (1 + 2)

, and the resultant intensity is

(1 + 2)^{2} I_{0} = (3 + 22) I_{0} \approx 5.8 I_{0}

JEE Main · 2020 · 09 Jan 2020 · Shift 1 · Single correct option