Question

Two periodic waves of intensities $I_{1}$ and $I_{2} \, $ pass through a region at the same time in the same direction. The sum of the maximum and minimum intensities is

• A. $\left(\sqrt{\left(I ⁡\right)_{1}} - \sqrt{\left(I ⁡\right)_{2}}\right)^{2}$
• B. $2 \left(\left( I \right)_{1} + \left( I ⁡\right)_{2}\right)$
• C. $\textit{I}_{1} + \textit{I}_{2}$
• D. $\left(\sqrt{\left(I ⁡\right)_{1}} + \sqrt{\left(I ⁡\right)_{2}}\right)^{2}$

Answer 1

Answer: (B)

Other factors such as ω and v remaining the same,
$ I = A ⁡^{2} \times \text{constant K,} \text{or} A ⁡ = \sqrt{\frac{ I ⁡}{ K ⁡}}$
On superposition
A_max = A₁ + A₂ and A_min = A₁ – A₂
$\therefore A _{\text{max}}^{2}=A⁡_{1}^{2}+A⁡_{2}^{2}+2A⁡_{1}A⁡_{2}\Rightarrow \frac{I ⁡_{\text{max}}}{K ⁡}=\frac{I ⁡_{1}}{K ⁡}+\frac{I ⁡_{2}}{K ⁡}+\frac{2 \sqrt{I ⁡_{1} I ⁡_{2}}}{K ⁡}$
$A _{\text{min}}^{2}=A⁡_{1}^{2}+A⁡_{2}^{2}-2A⁡_{1}A⁡_{2}\Rightarrow \frac{I ⁡_{\text{min}}}{K ⁡}=\frac{I ⁡_{1}}{K ⁡}+\frac{I ⁡_{2}}{K ⁡}-\frac{2 \sqrt{I ⁡_{1} I ⁡_{2}}}{K ⁡}$
$\text{∴ } \textit{I}_{\text{max}} + \textit{I}_{\text{min}} = 2 \textit{I}_{1} + 2 \textit{I}_{2}$