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Chapter 17: Problem 27

A certain sound source is increased in sound level by \(30.0 \mathrm{~dB}\). By what multiple is (a) its intensity increased and (b) its pressure amplitude increased?

Short Answer

Expert verified
The intensity is increased by a factor of 1000 and the pressure amplitude by approximately 31.62.

Step by step solution

01

Understanding Decibels and Intensity Formula

The sound level in decibels is given by the formula: \[ \beta = 10 \log \left( \frac{I}{I_0} \right) \] where \( \beta \) is the sound level in decibels, \( I \) is the intensity of the sound, and \( I_0 \) is the reference intensity, typically \( 1 \times 10^{-12} \text{ W/m}^2 \). An increase of 30 dB means \( \beta = 30 + \beta_0 \). We aim to find the factor by which the intensity increases.
02

Calculating Intensity Increase

First, isolate the ratio of the intensities in the formula: \[ \frac{I}{I_0} = 10^{\frac{\beta}{10}} \] If \( \beta \) increases by 30 dB, the new ratio becomes:\[ \frac{I_{new}}{I_0} = 10^{\frac{\beta_0 + 30}{10}} = 10^3 \times \frac{I_0}{I_0} \]Thus, \( I_{new} = 10^3 I \), which means the intensity is increased by a factor of 1000.
03

Understanding Pressure Amplitude and Its Relationship

The sound pressure level is related to sound intensity by the equation: \[ \beta = 20 \log \left( \frac{P}{P_0} \right) \] where \( P \) is the pressure amplitude. This means if intensity increases by a factor of 1000, the pressure amplitude relationship needs to be determined: \( P^2 \) is proportional to \( I \).
04

Calculating Pressure Amplitude Increase

If intensity is increased by a factor of 1000, then \[ \left( \frac{P_{new}}{P} \right)^2 = 1000 \] To find the factor increase of the pressure amplitude: \[ \frac{P_{new}}{P} = \sqrt{1000} = 10^{3/2} \approx 31.62 \] Thus, the pressure amplitude is increased by a factor of approximately 31.62.

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Key Concepts

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

Decibels
Decibels are a unit used to measure the intensity of sound. This measurement is logarithmic, meaning that an increase of 10 decibels (dB) represents a ten-fold increase in sound intensity. Sound intensity is the power per unit area carried by a sound wave. By using decibels, we can express large differences in intensity in a more manageable way.
For example, if a sound level is increased by 30 dB, it doesn't mean the intensity increases three times; rather, it grows 1000 times more intense. This is because each 10 dB increase multiplies the intensity by 10 (10 x 10 x 10 = 1000). This non-linear scale helps us understand the wide range of sounds we can hear, from the quietest whisper to a loud concert.
In simple terms, decibels help us compare sound intensities for different sources easily and effectively.
Pressure Amplitude
Pressure amplitude is related to the loudness of sound, specifically how much the pressure fluctuates with the sound waves. It is connected directly to the intensity of the sound. The greater the pressure amplitude, the louder the sound we perceive. Intensity, which increases by a factor of 1000 in our example, is mathematically proportional to the square of the pressure amplitude. If the intensity of a sound increases, its pressure amplitude will increase too. In our exercise, when intensity increased by a factor of 1000, the pressure amplitude increased by a factor derived from taking the square root of 1000. This is approximately 31.62. Therefore, the pressure changes we perceive in our ears have increased by a much larger amount than if we only considered linear growth.
Understanding pressure amplitude is crucial for fields like acoustics and audio engineering, where precise sound control is necessary.
Sound Level
The sound level is a way to represent the power of sound using decibels. It's calculated through the formula for decibels, which incorporates the intensity of the sound. The reference intensity, normally denoted as \( I_0 \), is considered the threshold of hearing at about \(1 \times 10^{-12} \text{ W/m}^2\). Changes in sound level help us recognize how sounds become louder or softer across various environments.
When we refer to increasing sound levels, we're discussing how much more powerful or intense a sound is perceived compared to this reference point. For example, a sound level increase by 30 dB means that the intensity now is 1000 times greater than it was, revealing a significant jump in the overall strength of the sound.
This concept is important because it helps manage sound in practical environments, from designing quieter machines to setting up sound systems in auditoriums, ensuring that sound levels are appropriate for the given context.

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Most popular questions from this chapter

The speed of sound in a certain metal is \(v_{m}\). 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's metal wall and the other from the wave that travels through the air inside the pipe. (a) If \(v\) is the speed of sound in air, what is the time interval \(\Delta t\) between the arrivals of the two sounds at the listener's ear? (b) If \(\Delta t=1.00 \mathrm{~s}\) and the metal is steel, what is the length \(L ?\)

Two sound waves with an amplitude of \(12 \mathrm{nm}\) and a wave- length 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?

A tuning fork of unknown frequency makes 3.00 beats per second with a standard fork of frequency \(384 \mathrm{~Hz}\). The beat frequency decreases when a small piece of wax is put on a prong of the first fork. What is the frequency of this fork?

At a certain point, two waves produce pressure variations given by \(\Delta p_{1}=\Delta p_{m} \sin \omega t\) and \(\Delta p_{2}=\Delta p_{m} \sin (\omega t-\phi) .\) At this point, what is the ratio \(\Delta p_{r} / \Delta p_{m},\) where \(\Delta p_{r}\) is the pressure amplitude of the resultant wave, if \(\phi\) is (a) \(0,\) (b) \(\pi / 2,\) (c) \(\pi / 3\), and (d) \(\pi / 4 ?\)

A plane flies at 1.25 times the speed of sound. Its sonic boom reaches a man on the ground 1.00 min after the plane passes directly overhead. What is the altitude of the plane? Assume the speed of sound to be \(330 \mathrm{~m} / \mathrm{s}\)
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