First slide

Superposition of waves

Question

Two waves travelling in a medium are given by $y_{1} = 2 \sin (3 π t - \frac{π}{2} x)$ and $y_{2} = 4 \sin (6 π t - \frac{3 π}{2} x)$ , where $y_{1}, y_{2}$ and $x$ are in cm and t in sec the displacement of a particle at $x = 1 c m$ and $t = 1 \sec$ is

Moderate

A

-2 cm
B

-3 cm
C

2 cm
D

6 cm

Solution

$F o r F i r s t W a v e y_{1} = 2 \sin (3 π \times 1 - π / 2 \times 1) \Rightarrow y_{1} = 2 \sin (3 π - π / 2) \Rightarrow y_{1} = 2 \sin (5 π / 2) = 2 \sin (2 π + π / 2) = 2 \sin π / 2 = 2 c m F o r S e c o n d W a v e y_{2} = 4 \sin (6 π \times 1 - 3 π / 2 \times 1) = 4 \sin (9 π / 2) = 4 \sin (4 π + π / 2) = 4 \sin π / 2 = 4 c m ∴ a c c o r d i n g t o s u p e r p o s i t i o n p r i n c i p l e y = y_{1} + y_{2} = 2 + 4 = 6 c m$

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