Chapter 2.3, Problem 66E

Sound Intensity The loudness of a sound is measured in decibels. The decibel level of a sound is given by the formula

$D=10\log \frac{I}{I_0}$ where D is the decibel level $\left(dB\right)$, I is its intensity in watts per square meter $\left(W/m^2\right)$, and $I_0={10}^{-12}W/m^2$ is the intensity of a barely audible "threshold" sound. A sound intensity of 90 decibels or greater causes damage to the average human ear.

a. Find the decibel levels of each of the following, rounding to the nearest decibel:

Whisper: $115\times {10}^{-12}W/m^2$

TV (average volume from 10 feet): $320\times {10}^{-7}W/m^2$

Loud music: $900\times {10}^{-3}W/m^2$

Jet aircraft (from 500 feet): $100W/m^2$

b. Which of the sounds above damages the average human ear?

c. Solve the given equation to express I in terms of D.

d. Use the answer to part (c) to show that if two sounds of intensity $I_1$ and $I_2$ register decibel levels of $D_1$ and $D_2$, respectively, then

$\frac{I_2}{I_1}={10}^{0.1\left(D_2-D_1\right)}$

e. Fill in the blank: If one sound registers 1 decibel more than another, then it is _____ times as intense.