Two simple harmonic progressive waves have displacements $\quad \rightarrow \quad \mathrm{y}_1=\mathrm{a}_1…

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?
  1. $\left(\phi+\frac{\pi}{2}\right)$
  2. $\phi$
  3. $\left(\phi-\frac{\pi}{2}\right)$
  4. $\quad(\phi+\pi)$

Solution

$\begin{aligned} \mathrm{y}_1 & =\mathrm{a}_1 \sin \left(\frac{2 \pi \mathrm{x}}{\lambda}-\omega \mathrm{t}\right) ...(i)\\ \mathrm{y}_2 & =\mathrm{a}_2 \cos \left(\frac{2 \pi \mathrm{x}}{\lambda}-\omega \mathrm{t}+\phi\right) \\ & =\mathrm{a}_2 \sin \left(\frac{2 \pi \mathrm{x}}{\lambda}-\omega \mathrm{t}+\phi+\frac{\pi}{2}\right) \ldots(ii) \end{aligned}$ From (ii) and (i) $\therefore \quad$ Phase difference $=\phi+\frac{\pi}{2}$ .

Asked in: MHT CET 2024 (02 May Shift 1)

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