Question

The ratio of intensity between two coherent sound sources is $4:1$. When they interfere in space, the difference of loudness between points of maximum and minimum intensity in decibel ($dB$) is

• A. $10log2$
• B. $20log3$
• C. $10log3$
• D. $20log2$

Answer 1

The correct option is B $20log3$

Given

$\frac{I_1}{I_2}=\frac{4}{1}$ or $\sqrt{\frac{I_1}{I_2}}=\frac{2}{1}$

$\therefore \frac{I_{max}}{I_{min}}={\left(\frac{\sqrt{\frac{I_1}{I_2}}+1}{\sqrt{\frac{I_1}{I_2}}-1}\right)}^2={\left(\frac{2+1}{2-1}\right)}^2=9$

Loudness is measured by $\beta =10log\frac{I}{I_0}$

Now, change in loudness is

$\therefore {\beta }_{max}-{\beta }_{min}=10log\frac{I_{max}}{I_0}-10log\frac{I_{min}}{I_0}$

$=10log\frac{I_{max}}{I_{min}}=10log\left(9\right)=20log3$