A jet plane at takeoff can produce sound of intensity 10.0 W / m^2 at 30.0 m away. But you pre

step 1 is taking a look at the idea of sound. And so the idea of intensity, I, is that it's basically a

A jet plane at takeoff can produce sound of intensity 10.0...

$\mathrm{A}$ jet plane at takeoff can produce sound of intensity $10.0 \mathrm{~W} / \mathrm{m}^{2}$ at $30.0 \mathrm{~m}$ away. But you prefer the tranquil sound of normal conversation, which is $1.0 \mu \mathrm{W} / \mathrm{m}^{2}$. Assume that the plane behaves like a point source of sound. (a) What is the closest distance you should live from the airport runway to preserve your peace of mind? (b) What intensity from the jet does your friend experience if she lives twice as far from the runway as you do? (c) What power of sound does the jet produce at takeoff?

Key Concepts

Inverse Square Law

This fundamental principle dictates that a physical quantity, such as sound intensity, decreases proportionally to the square of the distance from the source. For sound radiating from a point source, the intensity spreads out evenly over the surface of a sphere, causing it to drop off with the inverse square of the distance. This concept is critical for understanding how sound diminishes with increasing distance from the source.

Sound Intensity

Sound intensity is defined as the power per unit area carried by a sound wave, typically measured in watts per square meter. It quantifies how much sound energy is passing through a given area and is used to compare the loudness of different sound sources, such as a jet engine versus a normal conversation. Understanding sound intensity is crucial for evaluating noise levels at various distances.

Point Source and Spherical Spreading

A point source is an idealized sound emitter that radiates energy uniformly in all directions. When sound propagates from a point source, it spreads out spherically, with the total energy being distributed over the surface area of a sphere. This spherical spreading is the reason why the intensity decreases as the distance from the source increases, following the inverse square law.

Sound Power

Sound power is the total acoustic energy emitted by a source per unit time, measured in watts. Unlike sound intensity, sound power is an intrinsic property of the source and does not depend on the distance from the listener. By using the measured intensity at a known distance and applying the inverse square law, one can calculate the total power output of the sound source.

Transcript

00:01 Is taking a look at the idea of sound.

00:03 And so the idea of intensity, i, is that it's basically a measure of how power, p, is spread out uniformly over the surface area of a sphere.

00:13 And basically that says if you have a point source of sound and it's spreading out in all directions, we know that energy is conserved.

00:20 So the power at all points is going to be the same, but the way it's spread out over the surface area of a sphere is decreasing.

00:32 So it gets more spread out as you get further away from the sound source.

00:36 So here you have a, we have a jet plane that's taking off with a power, i'm sorry, actually with an intensity of 10 watts per meter squared, and it's 30 meters away.

00:54 And i'm actually just going to call this r1 and i1 for reasons that will become clear.

00:59 So we prefer that we want to buy a house, but we don't want it to be too close to the jet, wherever the jets take off, because we prefer our intensity to be no greater than one micro watt per meter squared.

01:16 So we want to know how far do we have to move away for that to be the case.

01:21 So let's figure out how we can manipulate these equations.

01:25 So if i look at i1 over i2, and that's p over 4 pi, p over 4 pi r1 squared divided by p over 4 pi r2 squared and the power isn't changing it's just dissipating the 4 pi is cancel and then this you flip the denominators here and that's going to be r2 squared over r1 squared so now we have a relationship between the ratio of the intensities and the ratio of the radii which is what we're looking for.

02:06 And that tells us that r2 is going to be equal to r1 times a square root of i1 over i2, which is 30 meters times a square root of 10 watts per meter squared over 10 of the minus 6 watts per meter squared, because that's a micro watt.

02:32 And that's going to be 95 kilometers.

02:37 So now, let's see, that was part a.

02:39 Part b, the question says, you have a friend, and she decides to move twice as far away from wherever this jet is taking off as you.

02:46 And we want to know what intensity she experiences...

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