Question

The difference of loudness in decibels $\left(dB\right)$ between maximum and minimum intensities of two coherent sound sources as they interfere in space is $20log3$. Find the ratio of intensities of two coherent sound sources.

• A. $4$
• B. $2$
• C. $8$
• D. $4\sqrt{2}$

Answer 1

The correct option is A $4$

The difference of loudness in decibels $\left(dB\right)$ between maximum and minimum intensities of two coherent sound sources as they interfere in space,

$L_1-L_2=20log3$

This can be written as,

$L_1-L_2=20log3=10log9$

$=10log\cdot \frac{I_{\text{max}}}{I_{\text{min}}}$

So, $\frac{I_{\text{max}}}{I_{\text{min}}}=9$

Now, $I_{\text{max}}$ and $I_{\text{min}}$ are expressed using formula as given below,
$\frac{I_{\text{max}}}{I_{\text{min}}}=\frac{{[{\left(\frac{I_1}{I_2}\right)}^{\frac{1}{2}}+1]}^2}{{[{\left(\frac{I_1}{I_2}\right)}^{\frac{1}{2}}-1]}^2}$

$9=\frac{(x+1{)}^2}{(x-1{)}^2}$

Here, $x=\sqrt{I_1/I_2}=2$

$\frac{I_1}{I_2}=4$

Final Answer: (b)