Question

Let's compare the loudness of sound at different distances from two types of speakers: a home speaker operating at 1.00 Watts, and a concert venue speaker operating at 1000 Watts. Recall that normal room conversation is in the 60-80 dB range, while long-term damage to hearing can occur with chronic exposure in the 90-110 dB range. Sound greater than 120 dB exceeds the average threshold for pain, while even brief exposures to 140 dB sound can cause permanent hearing loss.

A small home speaker produces 0.200 W of acoustical power. (This is achieved, for example, by a 10-watt speaker operating at 2% electrical efficiency). If the speaker projects sound uniformly in all directions, what is the loudness of the sound (in dB) at a distance of 1.47 m from the speaker?

Answer 1

98.67dB

Step-by-step explanation:

To find the loudness of the sound you use the following formula:

`L_s(dB)=10log((I)/(I_o))` ( 1 )

I_o: threshold of hearing = 10^-12W/m^2

I: intensity of the sound = P/A=0.200W/(4pi r^2) (it is assumed that the sound wave is a spherical wave)

r: distance to the source of the sound

Then, by calculating I and replace it in (1) you obtain:

`I=(P)/(4\pi r^2)=(0.200W)/(4\pi(1.47m)^2)=7.365*10^(-3)W/m^2\\\\L_s(dB)=10log((7.365*10^(-3)W/m^2)/(10^(-12)W/m^2))=98.67dB`

hence, the loudness of the sound is 98.67dB