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Chapter 16: Problem 20

The intensity due to a number of independent sound sources is the sum of the individual intensities. (a) When four quadruplets cry simultaneously, how many decibels greater is the sound intensity level than when a single one cries? (b) To increase the sound intensity level again by the same number of decibels as in part (a), how many more crying babies are required?

Short Answer

Expert verified
(a) 6.02 dB; (b) 12 more babies are needed.

Step by step solution

01

Understand the Relationship Between Intensity and Decibels

The formula for the sound intensity level in decibels is given by:\[ L = 10 \log_{10} \left(\frac{I}{I_0}\right) \]where \( I \) is the intensity of the sound, and \( I_0 \) is the reference intensity. We need to find the increase in decibels, \( \Delta L \), when the number of sources increases.
02

Calculate Increase in Decibels for Quadruplets

When one baby cries, the intensity is \( I \), so the level is \( L_1 = 10 \log_{10}\left(\frac{I}{I_0}\right) \). When four cry, the intensity is \( 4I \), so the new level is \( L_4 = 10 \log_{10}\left(\frac{4I}{I_0}\right) \). The change in decibels, \( \Delta L \), is calculated as:\[ \Delta L = L_4 - L_1 = 10 \log_{10}(4) \approx 6.02 \text{ dB} \]
03

Determine New Intensity to Increase Level by Another 6.02 dB

We need to find the total number of babies causing a 6.02 dB increase again. The initial intensity was \( 4I \). If we denote \( n \) as the new total number of babies, the problem becomes to find \( n \) such that:\[ 10 \log_{10}\left(\frac{nI}{I_0}\right) - 10 \log_{10}\left(\frac{4I}{I_0}\right) = 6.02 \]Simplifying, we get:\[ 10 \log_{10}\left(\frac{n}{4}\right) = 6.02 \]\[ \log_{10}\left(\frac{n}{4}\right) = 0.602 \]\[ \frac{n}{4} = 10^{0.602} \approx 4 \]\[ n \approx 16 \]
04

Calculate Additional Babies Needed

Since we had 4 babies initially, to reach the new count of approximately 16 babies, we need additional:\[ n - 4 = 16 - 4 = 12 \] extra babies to increase the sound intensity level again by approximately 6.02 dB.

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Key Concepts

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

Decibels
Sound intensity is often measured in decibels (dB), which is a logarithmic unit. Decibels quantify how intense a sound is compared to a reference level.
Understanding decibels is crucial when discussing sound intensity because it allows us to express very large or small ratios in a more manageable way.
  • One of the key points about decibels is that they express ratios, not absolute values. That means a specific decibel level indicates how much more intense one sound is relative to another.
  • For example, a 10 dB increase represents a tenfold increase in intensity, a 20 dB increase means a hundredfold increase, and so forth.
In the original exercise, the change of 6.02 dB was calculated when multiple sound sources鈥攃rying babies in this case鈥攃ontributed to overall intensity. Understanding this concept helps grasp how intensity levels can vary significantly even with small changes in decibel values.
Logarithms
Logarithms play a pivotal role in the calculation of sound intensity levels because these calculations heavily rely on logarithmic expressions.
The formula for sound intensity level is given as:\[ L = 10 \log_{10}\left(\frac{I}{I_0}\right) \]where \( L \) represents the level in decibels, \( I \) is the measured intensity, and \( I_0 \) is a reference value.
  • Using a logarithm to calculate sound intensity level allows us to compare greatly different intensities in a systematic way. It compresses the scale of loudness, making it easier to manage.
  • The logarithmic nature means each increase by 10 dB translates to a tenfold increase in the actual intensity. For the calculations in the solution, understanding how to manipulate logarithms was key, especially in finding the change from multiple sound sources.
Reference Intensity
Reference intensity, often denoted as \( I_0 \), is a crucial component when calculating sound intensity levels in decibels.
This reference value represents the baseline level of sound intensity, typically \( 10^{-12} \text{ W/m}^2 \) for air, which is the quietest sound that can be typically heard by humans.
  • By using a standard reference intensity, it ensures our calculations for sound levels are consistent and comparable.
  • In the context of the exercise, reference intensity is used to establish the sound level of both single and multiple jeopardizing sound sources, enabling the calculation of increases in intensity level by comparison.
As a result, when you see or hear that something is measured in decibels, it's often being measured against this standard level or another contextual reference value.

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Most popular questions from this chapter

The sound source of a ship's sonar system operates at a frequency of 22.0 \(\mathrm{kHz}\) . The speed of sound in water (assumed to be at a uniform \(20^{\circ} \mathrm{C}\) ) is 1482 \(\mathrm{m} / \mathrm{s}\) . (a) What is the wavelength of the waves emitted by the source? (b) What is the difference in frequency between the directly radiated waves and the waves reflected from a whale traveling directly toward the ship at 4.95 \(\mathrm{m} / \mathrm{s}^{\prime \prime}\) The ship is at rest in the water.

CP At point \(A, 3.0\) m from a small source of sound that is emitting uniformly in all directions, the sound intensity level is 53 dB. (a) What is the intensity of the sound at \(A ?\) (b) How far from the source must you go so that the intensity is one-fourth of what it was at \(A ?\) (c) How far must you go so that the sound intensity level is one-fourth of what it was at \(A ?\) (d) Does intensity obey the inverse-square law? What about sound intensity level?

A long tube contains air at a pressure of 1.00 atm and a temperature of \(77.0^{\circ} \mathrm{C}\) . The tube is open at one end and closed at the other by a movable piston. A tuning fork near the open end is vibrating with a frequency of 500 \(\mathrm{Hz}\) . Resonance is produced when the piston is at distances \(18.0,55.5,\) and 93.0 \(\mathrm{cm}\) from the open end. (a) From these measurements, what is the speed of sound in air at \(77.0^{\circ} \mathrm{C} ?\) (b) From the result of part (a), what is the value of \(\gamma ?\) (c) These data show that a displacement antinode is slightly outside of the open end of the tube. How far outside is it?

A certain pipe produces a fundamental frequency of 262 \(\mathrm{Hz}\) in air. (a) If the pipe is filled with helium at the same temperature. what fundamental frequency does it produce? (The molar mass of air is \(28.8 \mathrm{g} / \mathrm{mol},\) and the molar mass of helium is 4.00 \(\mathrm{g} / \mathrm{mol.}\) (b) Does your answer to part (a) depend on whether the pipe is open or stopped? Why or why not?

Two train whistles, \(A\) and \(B\) , each have a frequency of 392 \(\mathrm{Hz} . A\) is stationary and \(B\) is moving toward the right (away from \(A\) ) at a speed of 35.0 \(\mathrm{m} / \mathrm{s}\) . A listener is between the two whistles and is moving toward the right with a speed of 15.0 \(\mathrm{m} / \mathrm{s}\) (Fig. E16.45). No wind is blowing. (a) What is the frequency from A as heard by the listener? (b) What is the frequency from \(B\) as heard by the listener? (c) What is the beat frequency detected by the listener?
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