Waves

JEE Main level, numerical answer, medium

11 of 142

Question 11 of 142

Two waves are simultaneously passing through a string and their equations are : y1=A1sin⁡k(x−vt),y2=A2sin⁡k(x−vt+x0).{y}_{1}={A}_{1}\sin k(x-vt),{y}_{2}={A}_{2}\sin k\left(x-vt+{x}_{0}\right). Given amplitudes A1=12mm{A}_{1}=12mm and A2=5mm{A}_{2}=5mm, x0=3.5cm{x}_{0}=3.5cm and wave number k=6.28cm−1.k=6.28{cm}^{-1}. The amplitude of resulting wave will be _____ mm.mm.

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Answer: 7

Δϕ=kx0=2πλx0=2π×3.5(λ=2πk=1)\Delta \phi ={kx}_{0}=\dfrac{2\pi }{\lambda }{x}_{0}=2\pi \times 3.5\left(\lambda =\dfrac{2\pi }{k}=1\right)
=2π×3510=7π=2\pi \times \dfrac{35}{10}=7\pi
AR=122+52+2×12×5×cos⁡(7π)=(12−5)2=7mm{A}_{R}=\sqrt{{12}^{2}+{5}^{2}+2\times 12\times 5\times \cos (7\pi )}=\sqrt{(12-5{)}^{2}}=7mm

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