Residents of Hawaii are warned of the approach of a tsunami...

Residents of Hawaii are warned of the approach of a tsunami by sirens mounted on the tops of towers. Suppose a siren produces a sound that has an intensity level of 120 $\mathrm{dB}$ at a distance of 2.0 $\mathrm{m}$ . Treating the siren as a point source of sound, and ignoring reflections and absorption, find the intensity level heard by an observer at a distance of (a) 12 $\mathrm{m}$ and (b) 21 $\mathrm{m}$ from the siren. (c) How far away can the siren be heard?

Residents of Hawaii are warned of the approach of a tsunami by
sirens mounted on the tops of towers. Suppose a siren produces a
sound that has an intensity level of 120 $\mathrm{dB}$ at a distance of 2.0 $\mathrm{m}$ .
Treating the siren as a point source of sound, and ignoring reflections and absorption, find the intensity level heard by an observer
at a distance of (a) 12 $\mathrm{m}$ and (b) 21 $\mathrm{m}$ from the siren. (c) How far
away can the siren be heard?

Key Concepts

Point Source Approximation

Treating a sound source as a point source simplifies the analysis by assuming that the sound radiates uniformly in all directions. With this approximation, the mathematical modeling of sound propagation using the inverse square law becomes valid, allowing calculations to predict how sound intensity decreases as distance from the source increases.

Sound Intensity Level

This concept refers to measuring the power carried by sound waves per unit area, usually expressed in decibels (dB). The decibel scale is logarithmic, meaning that a small change in dB corresponds to a large change in actual sound intensity. It quantifies how loud a sound is, relative to a reference intensity, and is crucial for understanding how sound diminishes over distance.

Inverse Square Law for Sound Propagation

The inverse square law describes how sound intensity decreases as one moves away from a point source. Specifically, the intensity is inversely proportional to the square of the distance from the source, because the sound energy is spread out over the surface of a sphere whose area increases with the square of the radius. This principle is essential for calculating the change in intensity level at different distances.

Transcript

00:02 In problem 39, there's a siren, and when you're two meters away from the siren, it has an intensity level 120 decibels.

00:11 And i may ask, what is the intensity level when it's 12 meters away? and what's the intensity level when it's 21 meters away? and how far away can you still hear the siren? so the power of the siren doesn't change, no matter how far away you are from it.

00:26 So that's what i want to find first.

00:28 I want to find the power of the siren.

00:29 And then there's no equation that directly relates my distance to my intensity level.

00:34 I have this equation that does intensity level to intensity, and this one it does intensity to distance.

00:40 So i'm going to combine these two equations.

00:42 You can see here, if i just put this part in over here for the i, i can combine those two equations.

00:50 So i'm going to do that first, because that's what i'll need for all of this.

00:53 So the intensity level will equal 10 times the log of all of this.

01:01 That power 4 pi times the radius squared over intensity knot and i don't like this fraction of a fraction thing so i'm just going to bring this to the top you know how dividing by a number is the same as multiplying by the reciprocal so i'm just going to do that to simplify this a little bit so it'll be power and on the bottom will be 4 pi radius squared times this intensity not.

01:46 So there.

01:47 So now this combines these two equations.

01:54 So now i can use that to find the power of this siren.

01:58 So i'm just going to plug the numbers i have into that.

02:02 So i know my intensity is 120.

02:07 So 10 times the log.

02:11 I'm looking for the power.

02:15 On the bottom i have 4 pi.

02:18 My radius, which was 2 meters, so 2 squared, and the intensity knot, which is the threshold, 10 to negative 12.

02:33 I'm going to divide both sides by this, 10.

02:44 So now i have 12 .0 will equal log of the power over.

02:57 Here i have 2 squared, 2 will be 4 times another 4, so 16 pi times 10, to the negative 12...