Question

Sound waves are emitted uniformly in all directions from a point source. The dependence of sound level $\beta$ in decibels on the distance $r$ can be expressed as ($a$ and $b$ are positive constants)

• A. $\beta =a-\frac{b}{r^2}$
• B. $\beta =a-b(logr{)}^2$
• C. $\beta =a-blogr$
• D. $\beta =-blog{r}^2$

Answer 1

The correct option is C $\beta =a-blogr$

As we know, sound level in decibels,

$\beta =10log\frac{I}{I_0}$

If $P$ is power of sound source, then intensity $I=\frac{P}{4\pi r^2}$

$\beta =10log\frac{\left(\frac{P}{4\pi r^2}\right)}{I_0}$

$\beta =10log\frac{P}{4\pi r^2.I_0}$

$\beta =10[logP-log(4\pi I_0.r^2\left)\right]$

$\beta =10[logP-log(4\pi I_0)-2logr]$

$\beta =a-blogr$

where $a=10logP-10log4\pi I_0=$ constant

And $b=20=$ constant

Final answer: (c)