Two waves are simultaneously passing through a string and their equations are: $y_{1}=A_{1} \sin k(x-v t), y_{2}=A_{2} \sin k\left(x-v t+x_{0}\right)$. Given amplitudes $A_{1}=12 \mathrm{~mm}$ and $A_{2}=5 \mathrm{~mm}, x_{0}=$ 3.5 cm and wave number $\mathrm{k}=6.28 \mathrm{~cm}^{-1}$. The amplitude of resulting wave will be _____ mm.

JEE Main · 2021 · 26 Aug 2021 · Shift 2 · Waves & Sound

Two waves are simultaneously passing through a string and their equations are:

y_{1} = A_{1} sin k (x - v t), y_{2} = A_{2} sin k (x - v t + x_{0})

. Given amplitudes

A_{1} = 12 mm

and

A_{2} = 5 mm, x_{0} =

3.5 cm and wave number

k = 6.28 cm^{- 1}

. The amplitude of resulting wave will be _____ mm.

Answer: 7

The two waves differ in phase by

δ = k x_{0} = (6.28 c m^{- 1}) (3.5 cm) = 21.98 rad

Since

k = 6.28 c m^{- 1} = 2 π c m^{- 1}

, this is

δ = 7 π

, an odd multiple of

π

, so

cos δ = - 1

and the waves arrive in opposition.

A = A_{1}^{2} + A_{2}^{2} + 2 A_{1} A_{2} cos δ = 1 2^{2} + 5^{2} - 2 (12) (5) = 49

The resultant amplitude is

7

mm.

JEE Main · 2021 · 26 Aug 2021 · Shift 2 · Integer answer