Question

A dog while barking delivers 1mW of power. If this power is uniformly distributed over a hemispherical area, the sound level at a distance of 5m is $(Given:10{\log _{10}}6.337 = 0.8)$
A) 50 dB
B) 76 dB
C) 68 dB
D) 48 dB

Answer 1

The power carried by sound waves per unit area in a direction perpendicular to that area is known as sound intensity or acoustic intensity. Whereas, sound intensity level (SIL) or acoustic intensity level is the level (a logarithmic quantity) of the intensity of a sound relative to a reference value.

Complete step by step answer:
In the given question the power (P) delivered by the dog while barking is,
$P = 1mW = 1 \times {10^{ - 3}}W$
The radius of the hemispherical area would be, r = 5m.
Hence, the perpendicular surface area (A) is given by,$A = 2\pi {r^2} = 2\pi {5^2} = 157{m^2}$
Intensity of the barking sound of the dog is,
$I = \dfrac{P}{A}$
$\Rightarrow I = \dfrac{{1 \times {{10}^{ - 3}}}}{{157}}$
$\Rightarrow I = 6.37 \times {10^{ - 6}}W/{m^2}$
Now, the sound intensity level (SIL) is given by.
$SIL = 10{\log _{10}}\left( {\dfrac{I}{{{I_o}}}} \right)$.................... (1)
where, ${{I_o}}$is the threshold of human hearing and its value is,${I_o} = 1 \times {10^{ - 12}}W/{m^2}$
Now, put all the values of I and I0 in equation no. 1, and solve it
$\Rightarrow SIL = 10{\log _{10}}\left( {\dfrac{{6.37 \times {{10}^{ - 6}}}}{{{{10}^{ - 12}}}}} \right)$
$\Rightarrow SIL = 10{\log _{10}}(6.37) + 10{\log _{10}}({10^6})$
It is given in the problem that logarithmic value of 6.37 is 0.8 hence putting the same in the above equation and solving it we get,
$SIL = 68dB$

Therefore, the sound intensity level over a hemispherical level caused due to barking of a dog at a distance of 5m is found to be 68 dB.

Note: There is a difference between sound intensity and sound intensity level. This difference must be always kept in mind while solving such problems. The unit of sound intensity is W/m2, whereas the unit of sound intensity level is dB or decibel [it is a relative unit of measurement corresponding to one tenth of a bel (B)]. So, one should carefully read the question to find what is the unit of expected answer so that even if it is not mentioned in the problem if we have to calculate SI or SIL.