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Chapter 18: Problem 22

Assume that a noisy freight train on a straight track emits a cylindrical, expanding sound wave, and that the air absorbs no energy. How does the amplitude \(\Delta P^{\max }\) of the wave depend on the perpendicular distance \(r\) from the source?

Short Answer

Expert verified
The amplitude \(\text{\textDelta} P^{\text{max}}\) is proportional to \(\frac{1}{\text{\textsqrt}{r}}\).

Step by step solution

01

Understand the problem

We need to determine how the amplitude \(\text{\textDelta} P^{\text{max}} \) of a sound wave depends on the distance \( r \) from a source emitting a cylindrical sound wave.
02

Recall the inverse proportionality for cylindrical waves

For a cylindrical wave, the amplitude of the wave is inversely proportional to the square root of the distance from the source. This relationship can be expressed as \(\text{\textDelta} P^{\text{max}} \propto \frac{1}{\text{\textsqrt}{r}}\text{.}\)
03

Formulate the mathematical relationship

Since the amplitude is inversely proportional to the square root of the distance, we can write the relationship as \(\text{\textDelta} P^{\text{max}} = \frac{C}{\text{\textsqrt}{r}}\), where C is a constant that depends on the properties of the wave and the source.
04

Summarize the dependency

The amplitude \(\text{\textDelta} P^{\text{max}}\) decreases as the perpendicular distance \( r \) from the source increases, following an inverse square root relationship.

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Key Concepts

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

Cylindrical Sound Wave
When we talk about a cylindrical sound wave, we are referring to the way the sound energy spreads out from the source. Imagine sound waves coming from a train on straight tracks. These waves move outwards in a cylindrical shape, like expanding rings around the source.
For cylindrical waves, the energy is distributed over the surface of a cylinder, which grows larger as you move away from the source.
This type of wave pattern is different from spherical waves, where the energy spreads out over a growing sphere.
Inverse Proportionality
Inverse proportionality means that as one quantity increases, another quantity decreases in a specific way. For cylindrical sound waves, the amplitude of the wave and the distance from the source have an inverse proportional relationship.
The key relationship here is that the amplitude is inversely proportional to the square root of the distance from the source. Mathematically, this is expressed as:
\[\text{螖} P^{\text{max}} \propto \frac{1}{\sqrt{r}}\].
This means that if you double the distance from the source, the amplitude decreases by a factor of about 1.4 (which is the square root of 2).
Amplitude Dependency on Distance
The amplitude of a sound wave indicates how strong or loud the sound is at a given point. For a source emitting cylindrical waves, the amplitude \(\text{螖} P^{\text{max}}\) depends on the distance \(r\) you are from the source.
The relationship is described by the equation:
\[\text{螖} P^{\text{max}} = \frac{C}{\sqrt{r}}\],
where C is a constant, representing factors like the energy of the source and medium properties.
This means:
  • As you move farther from the source, the amplitude decreases.
  • The amplitude follows the inverse square root law, so it drops off more slowly than, say, an inverse square law.

Understanding this dependency is crucial in fields like acoustics and engineering, where predicting how sound levels diminish over distance is important for designing quieter environments or better sound systems.

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Most popular questions from this chapter

A certain sound source is increased in sound level by \(30 \mathrm{~dB}\). By what multiple is (a) its intensity increased and (b) its pressure amplitude increased?

Fig. \(18-32, S\) is a small loudspeaker driven by an audio oscillator and amplifier, adjustable in frequency from 1000 to \(2000 \mathrm{~Hz}\) only. Tube \(D\) is a piece of cylindrical sheet-metal pipe \(45.7 \mathrm{~cm}\) long and open at both ends. (a) If the speed of sound in air is \(344 \mathrm{~m} / \mathrm{s}\) at the existing temperature, at what frequencies will resonance occur in the pipe when the frequency emitted by the speaker is varied from \(1000 \mathrm{~Hz}\) to \(2000 \mathrm{~Hz}\) ? (b) Sketch the standing wave (using the style of Fig. \(18-16 b\) ) for each resonant frequency.

A violin string \(30.0 \mathrm{~cm}\) long with linear density \(0.650 \mathrm{~g} / \mathrm{m}\) is placed near a loudspeaker that is fed by an audio oscillator of variable frequency. It is found that the string is set into oscillation only at the frequencies 880 and \(1320 \mathrm{~Hz}\) as the frequency of the oscillator is varied over the range \(500-1500 \mathrm{~Hz}\). What is the tension in the string?

Pipe \(A\), which is \(1.2 \mathrm{~m}\) long and open at both ends, oscillates at its third lowest harmonic frequency. It is filled with air for which the speed of sound is \(343 \mathrm{~m} / \mathrm{s}\). Pipe \(B\), which is closed at one end, oscillates at its second lowest harmonic frequency. These frequencies of pipes \(A\) and \(B\) happen to match. (a) If an \(x\) axis extends along the interior of pipe \(A\), with \(x=0\) at one end, where along the axis are the displacement nodes? (b) How long is pipe \(B ?\) (c) What is the lowest harmonic frequency of pipe \(A\) ?

You are at a large outdoor concert, seated \(300 \mathrm{~m}\) from the speaker system. The concert is also being broadcast live via satellite (at the speed of light, \(3.0 \times 10^{8} \mathrm{~m} / \mathrm{s}\) ). Consider a listener \(5000 \mathrm{~km}\) away who receives the broadcast. Who hears the music first, you or the listener and by what time difference?
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