First slide

Travelling wave

Question

The equation of displacement of two waves are given as $y_{1} = 10 \sin (3 πt + \frac{π}{3})$ ; $y_{2} = 5 (\sin 3 πt + \sqrt{3} \cos 3 πt)$ . Then what is the ratio of their amplitudes

Moderate

A

1 : 2
B

2 : 1
C

1 : 1
D

None of these

Solution

$y_{1} = 10 \sin (3 πt + \frac{π}{3})$ ...(i)
and $y_{2} = 5 [\sin 3 πt + \sqrt{3} \cos 3 πt]$
$= 5 \times 2 [\frac{1}{2} \times \sin 3 πt + \frac{\sqrt{3}}{2} \times \cos 3 πt]$
$= 10 [\cos \frac{π}{3} \sin 3 πt + \sin \frac{π}{3} cosπt]$
$= 10 [\sin (3 πt + \frac{π}{t})]$ ... (ii)
( $∵$ sin(A + B) = sinA cosB + cosA sinB)
Comparing equation (i) and (ii) we get ratio of amplitude 1 : 1.

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