MHT CET · Physics · Oscillations
Two simple harmonic progressive waves have displacements \(\quad \rightarrow \quad \mathrm{y}_1=\mathrm{a}_1 \sin \left(\frac{2 \pi \mathrm{x}}{\lambda}-\omega \mathrm{t}\right)\) and \(\mathrm{y}_2=\mathrm{a}_2 \cos \left(\frac{2 \pi \mathrm{x}}{\lambda}-\omega \mathrm{t}+\phi\right)\)
What is the phase difference between two waves?
- A \(\left(\phi+\frac{\pi}{2}\right)\)
- B \(\phi\)
- C \(\left(\phi-\frac{\pi}{2}\right)\)
- D \(\quad(\phi+\pi)\)
Answer & Solution
Correct Answer
(A) \(\left(\phi+\frac{\pi}{2}\right)\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned}
\mathrm{y}_1 & =\mathrm{a}_1 \sin \left(\frac{2 \pi \mathrm{x}}{\lambda}-\omega \mathrm{t}\right) ...(i)\\
\mathrm{y}_2 & =\mathrm{a}_2 \cos \left(\frac{2 \pi \mathrm{x}}{\lambda}-\omega \mathrm{t}+\phi\right) \\
& =\mathrm{a}_2 \sin \left(\frac{2 \pi \mathrm{x}}{\lambda}-\omega \mathrm{t}+\phi+\frac{\pi}{2}\right) \ldots(ii)
\end{aligned}\)
From (ii) and (i)
\(\therefore \quad\) Phase difference \(=\phi+\frac{\pi}{2}\)
\mathrm{y}_1 & =\mathrm{a}_1 \sin \left(\frac{2 \pi \mathrm{x}}{\lambda}-\omega \mathrm{t}\right) ...(i)\\
\mathrm{y}_2 & =\mathrm{a}_2 \cos \left(\frac{2 \pi \mathrm{x}}{\lambda}-\omega \mathrm{t}+\phi\right) \\
& =\mathrm{a}_2 \sin \left(\frac{2 \pi \mathrm{x}}{\lambda}-\omega \mathrm{t}+\phi+\frac{\pi}{2}\right) \ldots(ii)
\end{aligned}\)
From (ii) and (i)
\(\therefore \quad\) Phase difference \(=\phi+\frac{\pi}{2}\)
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