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Chapter 18: Problem 17

A source emits sound waves isotropically. The intensity of the waves \(2.50 \mathrm{~m}\) from the source is \(1.91 \times 10^{-4} \mathrm{~W} / \mathrm{m}^{2}\) Assuming that the energy of the waves is conserved, find the power of the source.

Short Answer

Expert verified
The power of the source is approximately 0.015 W.

Step by step solution

01

Understand intensity and relationship to power

Intensity (\text{I}) is defined as the power (\text{P}) per unit area (\text{A}). The formula can be expressed as: \[ I = \frac{P}{A} \]. Therefore, the power can be found using the relation \[ P = I \times A \].
02

Determine the area for isotropic emission

For an isotropic source, the sound waves spread out uniformly in all directions, forming a sphere around the source. The area (\text{A}) of a sphere is given by \[ A = 4\pi r^2 \], where \text{r} is the radius of the sphere.
03

Calculate the area at the given distance

Given the distance \text{r} from the source is 2.50 m, calculate the area as: \[ A = 4\pi (2.50 \text{ m})^2 = 4\pi \times 6.25 \text{ m}^2 = 25\pi \text{ m}^2 \]
04

Calculate the power of the source

Using the intensity provided (1.91 \times 10^{-4} \text{ W/m}^2) and the area calculated, find the power as: \[ P = I \times A = 1.91 \times 10^{-4} \text{ W/m}^2 \times 25\pi \text{ m}^2 \approx 1.50 \times 10^{-2} \text{ W} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

isotropic sound emission
Isotropic sound emission refers to a source emitting sound waves uniformly in all directions.
Think of it like a light bulb radiating light equally around its circumference.
This means the intensity and power of the sound are evenly distributed across a spherical surface centered on the source.

This concept is crucial for calculating sound wave intensity at various distances.
Since the sound spreads out equally, the surface area over which the power and intensity are distributed increases with the square of the distance from the source.
Imagine blowing up a balloon; as it gets larger, the surface area increases rapidly.
power and intensity relationship
Understanding the relationship between power (P) and intensity (I) is essential.
Intensity is defined as the power per unit area.

Mathematically, this can be expressed as: \[ I = \frac{P}{A} \]
This formula shows that the intensity depends on both the power of the source and the area over which the sound is distributed.

Hence, if you need to determine the power, you can rearrange it to: \[ P = I \times A \]
This relationship helps when you know the intensity at a certain distance and the area of the surface at that distance.
surface area of a sphere
When sound waves are emitted isotropically, they form a sphere around the source.
The surface area (A) of this sphere is crucial for intensity calculations.

The formula for the surface area of a sphere is: \[ A = 4\text{蟺}r^2 \]
Here, \text{r}\text{ represents the radius, or the distance from the source.

For example, if the radius is 2.50 m, the area is: \[ A = 4\text{蟺}(2.50 \text{ m})^2 = 4\text{蟺}(6.25 \text{ m}^2 = 25\text{蟺 m}^2 \].
Understanding this helps you calculate how widely the sound's power is distributed.
energy conservation in waves
Energy conservation plays a crucial role in understanding wave phenomena.
When sound waves travel through a medium, the total energy does not change but spreads out over a larger area.

This concept is especially important for isotropic sound emission.
As the distance from the source increases, the energy conserves but distributes over a larger surface area, reducing the intensity.

This conservation can be seen in the formula for intensity: \[ I = \frac{P}{A} \]
When the area increases, the intensity decreases if the power stays constant.
Understanding energy conservation helps explain why sound becomes quieter as you move away from the source.

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Most popular questions from this chapter

A tube \(1.20 \mathrm{~m}\) long is closed at one end. A stretched wire is placed near the open end. The wire is \(0.330 \mathrm{~m}\) long and has a mass of \(9.60 \mathrm{~g} .\) It is fixed at both ends and oscillates in its fundamental mode. By resonance, it sets the air column in the tube into oscillation at that column's fundamental frequency. Find (a) that frequency and (b) the tension in the wire.

The source of a sound wave has a power of \(1.00 \mu \mathrm{W}\). If it is a point source, (a) what is the intensity \(3.00 \mathrm{~m}\) away and (b) what is the sound level in decibels at that distance?

The speed of sound in a certain metal is \(v^{\text {metal }}\) One end of a long pipe of that metal of length \(L\) is struck a hard blow. A listener at the other end hears two sounds, one from the wave that travels along the pipe and the other from the wave that travels through the air. (a) If \(v^{\text {air }}\) is the speed of sound in air, what time interval \(\Delta t\) elapses between the arrivals of the two sounds? (b) Suppose that \(\Delta t=1.00 \mathrm{~s}\) and the metal is steel. Find the length \(L\).

Two loudspeakers are located \(3.55 \mathrm{~m}\) apart on an outdoor stage. A listener is \(18.3 \mathrm{~m}\) from one and \(19.5 \mathrm{~m}\) from the other. During the sound check, a signal generator drives the two speakers in phase with the same amplitude and frequency. The transmitted frequency is swept through the audible range \((20 \mathrm{~Hz}\) to \(20 \mathrm{kHz})\). (a) What are the three lowest frequencies at which the listener will hear a minimum signal because of destructive interference? (b) What are the three lowest frequencies at which the listener will hear a maximum signal?

A certain violin string is \(30 \mathrm{~cm}\) long between its fixed ends and has a mass of \(2.0 \mathrm{~g}\). The "open" string (no applied finger) sounds an A note \((440 \mathrm{~Hz})\). (a) To play a \(\mathrm{C}\) note \((523\) \(\mathrm{Hz}\) ), how far down the string must one place a finger? (b) What is the ratio of the wavelength of the string waves required for an A note to that required for a C note? (c) What is the ratio of the wavelength of the sound wave for an A note to that for a \(\mathrm{C}\) note?
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