First slide

Superposition of waves

Question

The amplitude of a wave represented by displacement equation $y = \frac{1}{\sqrt{a}} sinωt \pm \frac{1}{\sqrt{b}} cosωt$ will be

Moderate

A

$\frac{a + b}{ab}$
B

$\frac{\sqrt{a} + \sqrt{b}}{ab}$
C

$\frac{\sqrt{a} \pm \sqrt{b}}{ab}$
D

$\sqrt{\frac{a + b}{ab}}$

Solution

$y = \frac{1}{\sqrt{a}} sinωt \pm \frac{1}{\sqrt{b}} \sin (ωt + \frac{π}{2})$
Here phase difference = $\frac{π}{2} ∴$ The resultant amplitude
$= \sqrt{{(\frac{1}{\sqrt{a}})}^{2} + {(\frac{1}{\sqrt{b}})}^{2}} = \sqrt{\frac{1}{a} + \frac{1}{b}} = \sqrt{\frac{a + b}{ab}}$

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