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Chapter 17: Problem 104

Suppose a spherical loudspeaker emits sound isotropically at \(10 \mathrm{~W}\) into a room with completely absorbent walls, floor, and ceiling (an anechoic chamber). (a) What is the intensity of the sound at distance \(d=3.0 \mathrm{~m}\) from the center of the source? (b) What is the ratio of the wave amplitude at \(d=4.0 \mathrm{~m}\) to that at \(d=3.0 \mathrm{~m}\) ?

Short Answer

Expert verified
The intensity at 3.0 m is 0.0885 W/m虏. The amplitude ratio at 4.0 m to 3.0 m is 0.750.

Step by step solution

01

Understand Sound Intensity

Sound intensity is defined as the power per unit area. The intensity \( I \) of sound from a point source at a distance \( d \) is given by \( I = \frac{P}{4 \pi d^2} \), where \( P \) is the power of the source. This formula takes into account the spherical spread of the sound waves.
02

Calculate Intensity at 3.0 m

Using the intensity formula, substitute \( P = 10 \text{ W} \) and \( d = 3.0 \text{ m} \):\[ I = \frac{10}{4 \pi (3)^2} = \frac{10}{36 \pi} \]Calculating the value:\[ I \approx \frac{10}{113.097} = 0.0885 \text{ W/m}^2 \].
03

Calculate Intensity at 4.0 m

Using the same formula for \( d = 4.0 \text{ m} \):\[ I = \frac{10}{4 \pi (4)^2} = \frac{10}{64 \pi} \]Calculating the value:\[ I \approx \frac{10}{201.062} = 0.0498 \text{ W/m}^2 \].
04

Understand Wave Amplitude and Intensity Relationship

The wave amplitude \( A \) is related to the intensity \( I \) by the equation \( I \propto A^2 \). Therefore, the ratio of amplitudes \( \frac{A_2}{A_1} \) for different intensities \( I_2 \) and \( I_1 \) is given by \( \frac{A_2}{A_1} = \sqrt{\frac{I_2}{I_1}} \).
05

Calculate Amplitude Ratio at 4.0 m and 3.0 m

Using the formula for amplitude ratio, substitute \( I_2 = 0.0498 \text{ W/m}^2 \) and \( I_1 = 0.0885 \text{ W/m}^2 \):\[ \frac{A_2}{A_1} = \sqrt{\frac{0.0498}{0.0885}} \]Calculate the ratio:\[ \frac{A_2}{A_1} \approx \sqrt{0.5629} \approx 0.750 \].

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Key Concepts

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

Spherical Wave Propagation
Sound propagates in waves, and spherical wave propagation occurs when the wave fronts spread out uniformly in all directions from a point source, like ripples on a pond after tossing a stone.

This means that the sound energy is equally distributed in all directions, forming a spherical shape as it travels through space. As the wave travels farther from the source, the energy is spread over an increasingly larger area. This affects how loud the sound is perceived at different distances, as the same amount of energy is dispersed over a larger area.
  • The energy distribution follows a spherical pattern.
  • The distance from the source affects wave intensity.
This concept is crucial for understanding phenomena such as sound dispersion in open areas and how distance affects sound intensity.
Anechoic Chamber
An anechoic chamber is a specially designed room that absorbs sound completely, preventing echoes. It creates an ideal environment for sound testing by ensuring that all sounds originate only from the source being tested and are not affected by reflections.

The walls, floor, and ceiling of such a chamber are covered with materials that prevent sound from bouncing back, creating what is effectively an acoustically dead space. This means that any sound measured inside the chamber is not influenced by the room itself.
  • Sound absorption stops echoes and reflections.
  • Provides a controlled environment for accurate sound measurements.
Such a setup is essential for assessing the true characteristics of sound sources in research and development.
Wave Amplitude
Wave amplitude is the measure of the maximum displacement of points on a wave, from their resting positions. In sound terms, amplitude relates to the loudness of the sound, with higher amplitudes resulting in louder sounds.

The intensity of a sound wave and its amplitude are related; as intensity increases, the amplitude increases. This relationship can be expressed mathematically as the intensity being proportional to the square of the amplitude: \( I \propto A^2 \).
  • Amplitude affects the perceived loudness of sound.
  • Intensity is proportional to the square of the amplitude.
Understanding amplitude is fundamental in acoustics, as it helps explain why some sounds are louder than others at the same distance.
Inverse Square Law
The inverse square law is a principle that describes how a physical quantity, such as sound intensity, decreases with increasing distance from the source. Specifically, sound intensity follows the rule that it is inversely proportional to the square of the distance: \( I = \frac{P}{4 \pi d^2} \), where \( P \) is the power and \( d \) is the distance from the source.

This means that as you move away from the source, the intensity diminishes quickly. For example, doubling the distance from a sound source decreases its intensity to a quarter of its previous value.
  • Intensity decreases with the square of the distance.
  • Doubling distance reduces intensity to one-fourth.
This law is vital in understanding how sound travels and dissipates in an open environment, explaining why distant sounds are less intense.

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Most popular questions from this chapter

Two loud speakers are located \(3.35 \mathrm{~m}\) 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 is the lowest frequency \(f_{\min , 1}\) that gives minimum signal (destructive interference) at the listener's location? By what number must \(f_{\min , 1}\) be multiplied to get (b) the second lowest frequency \(f_{\min 2}\) that gives minimum signal and (c) the third lowest frequency \(f_{\min , 3}\) that gives minimum signal? (d) What is the lowest frequency \(f_{\max , 1}\) that gives maximum signal (constructive interference) at the listener's location? By what number must \(f_{\max , 1}\) be multiplied to get (e) the second lowest frequency \(f_{\max , 2}\) that gives maximum signal and (f) the third lowest frequency \(f_{\max , 3}\) that gives maximum signal?

An acoustic burglar alarm consists of a source emitting waves of frequency \(28.0 \mathrm{kHz}\). What is the beat frequency between the source waves and the waves reflected from an intruder walking at an average speed of \(0.950 \mathrm{~m} / \mathrm{s}\) directly away from the alarm?

Kundt's method for measuring the speed of sound. In Fig. \(17-51\), a rod \(R\) is clamped at its center; a disk \(D\) at its end projects into a glass tube that has cork filings spread over its interior. \(\mathrm{A}\) plunger \(P\) is provided at the other end of the tube, and the tube is filled with a gas. The rod is made to oscillate longitudinally at frequency \(f\) to produce sound waves inside the gas, and the location of the plunger is adjusted until a standing sound wave pattern is set up inside the tube. Once the standing wave is set up, the motion of the gas molecules causes the cork filings to collect in a pattern of ridges at the displacement nodes. If \(f=4.46 \times 10^{3} \mathrm{~Hz}\) and the separation between ridges is \(9.20 \mathrm{~cm}\), what is the speed of sound in the gas?

An avalanche of sand along some rare desert sand dunes can produce a booming that is loud enough to be heard \(10 \mathrm{~km}\) away. The booming apparently results from a periodic oscillation of the sliding layer of sand-the layer's thickness expands and contracts. If the emitted frequency is \(90 \mathrm{~Hz}\), what are (a) the period of the thickness oscillation and (b) the wavelength of the sound?

Two sound waves with an amplitude of \(12 \mathrm{~nm}\) and a wavelength of \(35 \mathrm{~cm}\) travel in the same direction through a long tube, with a phase difference of \(\pi / 3\) rad. What are the (a) amplitude and (b) wavelength of the net sound wave produced by their interference? If, instead, the sound waves travel through the tube in opposite directions, what are the (c) amplitude and (d) wavelength of the net wave?
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