MHT CET · Physics · Waves and Sound

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

A \(\cos ^{-1}\left(\frac{-1}{2}\right)\)
B \(\cos ^{-1}\left(\frac{-1}{4}\right)\)
C \(\cos ^{-1}\left(-\frac{1}{6}\right)\)
D \(\cos ^{-1}\left(-\frac{1}{8}\right)\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(\cos ^{-1}\left(\frac{-1}{2}\right)\)

Step-by-step Solution

Detailed explanation

\(\begin{aligned}
& \mathrm{y}_1=\mathrm{a} \sin \left(\omega \mathrm{t}+\phi_1\right) \\
& \mathrm{y}_2=\operatorname{asin}\left(\omega \mathrm{t}+\phi_2\right)
...(i)\end{aligned}\)
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,
\(\begin{aligned}
& \cos \left(\phi_1-\phi_2\right)=-\frac{1}{2} \\
\therefore \quad & \left(\phi_1-\phi_2\right)=\cos ^{-1}\left(\frac{-1}{2}\right)
\end{aligned}\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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