Two waves are simultaneously passing through a string and their equations are :
$\mathrm{y}_{1}=\mathrm{A}_{1} \sin \mathrm{k}(\mathrm{x}-\mathrm{vt}), \mathrm{y}_{2}=\mathrm{A}_{2} \sin \mathrm{k}\left(\mathrm{x}-\mathrm{vt}+\mathrm{x}_{0}\right)$. Given amplitudes $\mathrm{A}_{1}=12 \mathrm{~mm}$ and $\mathrm{A}_{2}=5 \mathrm{~mm}$, $\mathrm{x}_{0}=3.5 \mathrm{~cm}$ and wave number $\mathrm{k}=6.28 \mathrm{~cm}^{-1}$. The amplitude of resulting wave will be ........ $\mathrm{mm}$.
$\mathrm{y}_{1}=\mathrm{A}_{1} \sin \mathrm{k}(\mathrm{x}-\mathrm{vt})$
$\mathrm{y}_{1}=12 \sin 6.28(\mathrm{x}-\mathrm{vt})$
$\mathrm{y}_{2}=5 \sin 6.28(\mathrm{x}-\mathrm{vt}+3.5)$
$\Delta \phi=\frac{2 \pi}{\lambda}(\Delta x)$
$=\mathrm{K}(\Delta \mathrm{x})$
$=6.28 \times 3.5=\frac{7}{2} \times 2 \pi=7 \pi$
$\mathrm{A}_{\text {net }}=\sqrt{\mathrm{A}_{1}^{2}+\mathrm{A}_{2}^{2}+2 \mathrm{~A}_{1} \mathrm{~A}_{2} \cos \phi}$
$\mathrm{A}_{\text {net }}=\sqrt{(12)^{2}+(5)^{2}+2(12)(5) \cos (7 \pi)}$
$=\sqrt{144+25-120}$
Ans. 7