Question

Two identical sound $A$ and $B$ each a point in the same phase. The resultant sound is $C$. The loudness of $C$ is $n$ $dB$ higher than the loudness of $A$

• A. $2$
• B. $3$
• C. $4$
• D. $6$

Answer 1

Answer: (D)

$6$

Given, two identical sound waves that reach a point in phase.we have to find the difference in the loudness of resulting sound and original sound .

let us suppose that the intensity of each source is $I$ and Amplitude is $A$

Since, the waves are identical and in phase $A_{res}=2A$

As we know that $I\propto A^2\Rightarrow I_{res}=4I$

Now,

$L_{res}-L_{single}=10\log \left(\frac{I_{res}}{I_o}\right)-10\log \left(\frac{I}{I_o}\right)$ where, $L$ is loudness in dB.

$L_{res}-L_{single}=10\log \left(\frac{4I}{I_o}\right)-10\log \left(\frac{I}{I_o}\right)$

$L_{res}-L_{single}=10\log (\log 4I-\log I_o-\log I+\log I_o)$

$L_{res}-L_{single}=10\log (\log 4+\log I-\log I)$

$L_{res}-L_{single}=10\log \left(\log 4\right)\approx 6$

So D is the correct answer.