Three harmonic waves having equal frequency v and same intensity I_ 0 , have phase angles 0 , 4 and - 4 respectively. When they are superimposed the intensity of the resultant wave is close to

Three harmonic waves having equal frequency v and same intensity …

MCQ

Three harmonic waves having equal frequency $\mathrm{v}$ 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

✓
$5.8 \mathrm{I}_{0}$
B
$0.2 \mathrm{I}_{0}$
C
$\mathrm{I}_{0}$
D
$3 \mathrm{I}_{0}$

✓

Answer

Correct option: A.

$5.8 \mathrm{I}_{0}$

a
Let amplitude of each wave is $A.$

Resultant wave equation

$=A \sin \omega t+A \sin \left(\omega t-\frac{\pi}{4}\right)+A \sin \left(\omega t+\frac{\pi}{4}\right)$

$=\mathrm{A} \sin \omega \mathrm{t}+\sqrt{2} \mathrm{A} \sin \omega \mathrm{t}$

$=(\sqrt{2}+1) \mathrm{A} \sin \omega \mathrm{t}$

Resultant wave amplitude $=(\sqrt{2}+1) \mathrm{A}$

as $I \propto A ^{2}$

so $\frac{\mathrm{I}}{\mathrm{I}_{0}}=(\sqrt{2}+1)^{2}$

$I=5.8 \mathrm{I}_{0}$

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