12.3 Sound intensity and sound level  (Page 3/7)

Calculating sound intensity levels: sound waves

Calculate the sound intensity level in decibels for a sound wave traveling in air at $\mathrm{0ºC}$ and having a pressure amplitude of 0.656 Pa.

Strategy

We are given $\mathrm{\Delta }p$ , so we can calculate $I$ using the equation $I={\left(\mathrm{\Delta }p\right)}^2/{\left(2{\text{pv}}_{\mathrm{w}}\right)}^2$ . Using $I$ , we can calculate $\beta$ straight from its definition in $\beta \left(\text{dB}\right)=\text{10}{\text{log}}_{\text{10}}(I/I_0)$ .

Solution

(1) Identify knowns:

Sound travels at 331 m/s in air at $\mathrm{0ºC}$ .

Air has a density of $\mathrm{1.29\; kg}{\mathrm{/m}}^3$ at atmospheric pressure and $\mathrm{0ºC}$ .

(2) Enter these values and the pressure amplitude into $I={\left(\mathrm{\Delta }p\right)}^2/\left({2\text{ρv}}_{\mathrm{w}}\right)$ :

(3) Enter the value for $I$ and the known value for $I_0$ into $\beta \left(\text{dB}\right)=\text{10}{\text{log}}_{\text{10}}(I/I_0)$ . Calculate to find the sound intensity level in decibels:

Discussion

This 87 dB sound has an intensity five times as great as an 80 dB sound. So a factor of five in intensity corresponds to a difference of 7 dB in sound intensity level. This value is true for any intensities differing by a factor of five.

Change intensity levels of a sound: what happens to the decibel level?

Show that if one sound is twice as intense as another, it has a sound level about 3 dB higher.

Strategy

You are given that the ratio of two intensities is 2 to 1, and are then asked to find the difference in their sound levels in decibels. You can solve this problem using of the properties of logarithms.

Solution

(1) Identify knowns:

The ratio of the two intensities is 2 to 1, or:

We wish to show that the difference in sound levels is about 3 dB. That is, we want to show:

Note that:

(2) Use the definition of $\beta$ to get:

Thus,

Discussion

This means that the two sound intensity levels differ by 3.01 dB, or about 3 dB, as advertised. Note that because only the ratio $I_2/I_1$ is given (and not the actual intensities), this result is true for any intensities that differ by a factor of two. For example, a 56.0 dB sound is twice as intense as a 53.0 dB sound, a 97.0 dB sound is half as intense as a 100 dB sound, and so on.