The equations of two waves are given as $\begin{aligned} & \mathrm{y}_1=\mathrm{asin}\left(\omega…

The equations of two waves are given as $\begin{aligned} & \mathrm{y}_1=\mathrm{asin}\left(\omega \mathrm{t}+\phi_1\right) \\ & \mathrm{y}_2=\operatorname{asin}\left(\omega \mathrm{t}+\phi_2\right) \end{aligned}$ If amplitude and time period of resultant wave is same as the individual waves, then $\left(\phi_1-\phi_2\right)$ is
  1. $\cos ^{-1}\left(\frac{-1}{2}\right)$
  2. $\cos ^{-1}\left(\frac{-1}{4}\right)$
  3. $\cos ^{-1}\left(-\frac{1}{6}\right)$
  4. $\quad \cos ^{-1}\left(-\frac{1}{8}\right)$

Solution

$\begin{aligned} & \mathrm{y}_1=\mathrm{a} \sin \left(\omega \mathrm{t}+\phi_1\right) \\ & \mathrm{y}_2=\operatorname{asin}\left(\omega \mathrm{t}+\phi_2\right) ...(i)\end{aligned}$
Superposition of two waves is given by $a^2=a_1^2+a_2^2+2 a_1 a_2 \cos \phi$ ...(ii) Here, $\phi=\phi_1-\phi_2$ and $a_1=a_2=a$ Substituting these values in equation (i), we get, $\begin{aligned} & \cos \left(\phi_1-\phi_2\right)=-\frac{1}{2} \\ \therefore \quad & \left(\phi_1-\phi_2\right)=\cos ^{-1}\left(\frac{-1}{2}\right) \end{aligned}$

Asked in: MHT CET 2024 (09 May Shift 2)

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