Question

The ratio of intensities between two coherent sound sources is $4 : 1$. The difference of loudness in decibels between maximum and minimum intensities, when they interfere in space, is

• A. $10 \log 2$
• B. $20 \log 3$
• C. $10 \log 3$
• D. $20 \log 2$

Answer 1

Answer: (B)

$20 \log 3$

Answer 2

Given $\frac{I_{1}}{I_{2}}=\frac{4}{1}$ We know $I \propto a^{2}$ $\therefore \frac{a_{1}^{2}}{a_{2}^{2}} =\frac{I_{1}}{I_{2}}=\frac{4}{1}$ $\frac{a_{1}}{a_{2}}=\frac{2}{1}$ $\therefore \frac{I_{\max }}{I_{\min }} =\frac{\left(a_{1}+a_{2}\right)^{2}}{\left(a_{1}-a_{2}\right)^{2}}$ $=\left(\frac{2+1}{2-1}\right)^{2}$ $=\left(\frac{3}{1}\right)^{2}=\frac{9}{1}$ Therefore, difference of loudness is given by $L_{1}-L_{2} =10 \log \frac{I_{\max }}{I_{\min }}=10 \log (9)$ $=10 \log 3^{2}=20 \log 3$