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Chapter 16: Problem 66

Using an intensity of \(1 \times 10^{-12} \mathrm{W} / \mathrm{m}^{2}\) as a reference, the threshold of hearing for an average young person is 0 dB. Person 1 and person 2 , who are not average, have thresholds of hearing that are \(\beta_{1}=-8.00 \mathrm{dB}\) and \(\beta_{2}=+12.0 \mathrm{dB}\). What is the ratio \(I_{1} / I_{2}\) of the sound intensity \(I_{1}\) when person 1 hears the sound at his own threshold of hearing compared to the sound intensity \(I_{2}\) when person 2 hears the sound at his own threshold of hearing?

Short Answer

Expert verified
The ratio \( I_1/I_2 \) is \( 10^{-2} \).

Step by step solution

01

Understanding the Decibel Formula

The intensity level in decibels (dB) is given by the formula: \( \beta = 10 \log_{10}\left(\frac{I}{I_0}\right) \), where \(I_0 = 1 \times 10^{-12} \mathrm{W/m^2}\) is the reference intensity. We use this formula to compare the intensity levels for persons 1 and 2.
02

Calculate Intensity for Person 1

For person 1 with a threshold of \( \beta_1 = -8.00 \mathrm{dB}\), we use the formula: \( \beta_1 = 10 \log_{10}\left(\frac{I_1}{I_0}\right) \). Solving for \(I_1\), we find: \[ I_1 = I_0 \times 10^{\beta_1 / 10} = 1 \times 10^{-12} \times 10^{-8/10} = 1 \times 10^{-12} \times 10^{-0.8} \].
03

Calculate Intensity for Person 2

For person 2 with a threshold of \( \beta_2 = 12.0 \mathrm{dB}\), the formula is: \( \beta_2 = 10 \log_{10}\left(\frac{I_2}{I_0}\right) \). Solving for \(I_2\), we find: \[ I_2 = I_0 \times 10^{\beta_2 / 10} = 1 \times 10^{-12} \times 10^{12/10} = 1 \times 10^{-12} \times 10^{1.2} \].
04

Calculate the Intensity Ratio

Now, we calculate the ratio \( \frac{I_1}{I_2} \). Substitute the expressions we derived: \[ \frac{I_1}{I_2} = \frac{1 \times 10^{-12} \times 10^{-0.8}}{1 \times 10^{-12} \times 10^{1.2}} = \frac{10^{-0.8}}{10^{1.2}} = 10^{-0.8-1.2} = 10^{-2} \].
05

Conclusion

The ratio of the sound intensity \(I_1\) to \(I_2\) is \(10^{-2}\), meaning the intensity for person 1 is 0.01 times that for person 2 at their respective thresholds of hearing.

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Key Concepts

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

Decibel Formula
The decibel (dB) is a unit used to measure the intensity of sound. It's based on a logarithmic scale, which compares the intensity of a sound to a reference intensity. The formula for calculating the intensity level of a sound in decibels is:\[\beta = 10 \log_{10}\left(\frac{I}{I_0}\right)\]where \( \beta \) represents the intensity level in decibels, \( I \) is the intensity of the sound, and \( I_0 \) is the reference intensity, commonly the threshold of hearing, which is \(1 \times 10^{-12} \mathrm{W/m^2}\).
This formula allows us to easily compare different sound intensities through a logarithmic transformation, making it manageable to express very large or very small numbers.
Threshold of Hearing
The threshold of hearing refers to the quietest sound that the average young human ear can detect. This level is denoted as 0 dB, which corresponds to an intensity of \(1 \times 10^{-12} \mathrm{W/m^2}\). However, individual hearing thresholds can vary.
  • For instance, some individuals might detect sounds quieter than \(0 \mathrm{dB}\), resulting in a negative decibel value, such as \(-8.00 \mathrm{dB}\).
  • Conversely, those who require louder sounds to reach their hearing threshold might have positive decibel values, like \(+12.0 \mathrm{dB}\).

Understanding this concept is crucial in audiology and the study of sound perception.
Intensity Ratio
The intensity ratio is a comparison of two sound intensities, reflecting how much louder or quieter one sound is compared to another. The calculation of the intensity ratio between two sounds at different decibel levels involves subtracting and converting these levels back to their respective intensities.
  • In our exercise, the calculated ratio \( \frac{I_1}{I_2} \) shows how the first person's threshold intensity compares to the second person's.
  • This ratio is simple to compute when using the decibel formula, as it mainly involves finding the anti-logarithm of the decibel difference.

Essentially, an intensity ratio highlights the relative loudness as perceived by different individuals or under varying conditions.
Logarithmic Scale
Sound intensity measurements use a logarithmic scale because of the vast range of intensities our ears can detect. A logarithmic scale compresses this wide range into a manageable format, enabling more accessible computation and comparison of sound levels.
  • The logarithmic relationship means that each 10-decibel increase represents a sound ten times more intense.
  • This scaling is crucial when dealing with very high or low sound intensity values since direct comparison via linear scales would be impractical.

By using logarithmic calculations, the decibel scale makes it feasible to express and work with sound intensity changes effectively.

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Most popular questions from this chapter

As a prank, someone drops a water-filled balloon out of a window. The balloon is released from rest at a height of \(10.0 \mathrm{m}\) above the ears of a man who is the target. Then, because of a guilty conscience, the prankster shouts a warning after the balloon is released. The warning will do no good, however, if shouted after the balloon reaches a certain point, even if the man could react infinitely quickly. Assuming that the air temperature is \(20^{\circ} \mathrm{C}\) and ignoring the effect of air resistance on the balloon, determine how far above the man's ears this point is.

A speedboat, starting from rest, moves along a straight line away from a dock. The boat has a constant acceleration of \(+3.00 \mathrm{m} / \mathrm{s}^{2}\) (see the figure). Attached to the dock is a siren that is producing a \(755-\mathrm{Hz}\) tone. If the air temperature is \(20^{\circ} \mathrm{C},\) what is the frequency of the sound heard by a person on the boat when the boat's displacement from the dock is \(+45.0 \mathrm{m} ?\)

When an earthquake occurs, two types of sound waves are generated and travel through the earth. The primary, or \(P\), wave has a speed of about \(8.0 \mathrm{km} / \mathrm{s}\) and the secondary, or \(\mathrm{S}\), wave has a speed of about \(4.5 \mathrm{km} / \mathrm{s}\). A seismograph, located some distance away, records the arrival of the P wave and then, 78 s later, records the arrival of the \(S\) wave. Assuming that the waves travel in a straight line, how far is the seismograph from the earthquake?

The middle \(C\) string on a piano is under a tension of 944 N. The period and wavelength of a wave on this string are \(3.82 \mathrm{ms}\) and \(1.26 \mathrm{m}\), respectively. Find the linear density of the string.

You are flying in an ultralight aircraft at a speed of \(39 \mathrm{m} / \mathrm{s}\). An eagle, whose speed is \(18 \mathrm{m} / \mathrm{s},\) is flying directly toward you. Each of the given speeds is relative to the ground. The eagle emits a shrill cry whose frequency is \(3400 \mathrm{Hz}\). The speed of sound is \(330 \mathrm{m} / \mathrm{s}\). What frequency do you hear?
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