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Chapter 16: Problem 22

The Sacramento City Council adopted a law to reduce the allowed sound intensity level of the much-despised leaf blowers from their current level of about 95 \(\mathrm{dB}\) to 70 \(\mathrm{dB}\) . With the new law, what is the ratio of the new allowed intensity to the previously allowed intensity?

Short Answer

Expert verified
The ratio is approximately 0.00316.

Step by step solution

01

Understand Sound Intensity Level

The sound intensity level is measured in decibels (dB), a logarithmic scale. An increase of 10 dB corresponds to a tenfold increase in sound intensity.
02

Apply the Decibel Formula

The formula to calculate the intensity level in decibels is: \( L = 10 \log_{10}\left(\frac{I}{I_0}\right) \), where \( I \) is the sound intensity and \( I_0 \) is the reference intensity. We can use this formula to understand how changes in decibels relate to changes in intensity.
03

Calculate the Intensity Ratio

When changing from 95 dB to 70 dB, we use the formula for the difference in decibels: \( \Delta L = 10 \log_{10}\left(\frac{I_2}{I_1}\right) \). Plugging in the values, we get 25 dB = \( 10 \log_{10}\left(\frac{I_2}{I_1}\right) \).
04

Solve for the Intensity Ratio

Rearrange and solve the equation for the intensity ratio: \( \frac{I_2}{I_1} = 10^{\frac{25}{10}} = 10^{2.5} \). Calculate \( 10^{2.5} \), which equals approximately 316.23.
05

Compute the New to Old Intensity Ratio

The ratio of the new allowed intensity to the previously allowed intensity is \( \frac{1}{316.23} \) since the sound is being reduced, not increased. Compute this ratio to get approximately 0.00316.

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Key Concepts

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

Decibels
Decibels (dB) are a unit used to measure the intensity of sound. A decibel does not measure sound directly. Instead, it measures the intensity level on a logarithmic scale. This scale takes a wide range of sounds and compresses it into a manageable number. This way of measuring is very effective because human hearing perceives sound intensity logarithmically rather than linearly.

This means each increase of 10 decibels is experienced as a doubling of the perceived volume to the human ear. If a sound increases from 60 dB to 70 dB, it is perceived to be twice as loud, though its actual intensity has increased by ten times.
  • 0 dB is near-total silence, the faintest sound the human ear can hear.
  • Each 10 dB increment means that a sound has become ten times more intense.
Intensity Ratio
The intensity ratio is a way to express how one sound intensity compares to another. In context of sound, it shows how much a sound's power level changes. This is especially important when laws or regulations limit sound levels, like in the case of leaf blower noise limits set by the Sacramento City Council.

To find the intensity ratio between two different sound levels, you can compare their intensities using the decibel scale. The formula for the decibel difference between two sound levels is: \[ \Delta L = 10 \log_{10}\left(\frac{I_2}{I_1}\right) \] where \(I_2\) is the new intensity and \(I_1\) is the original intensity. A positive difference indicates an increase in intensity, while a negative difference indicates a decrease.

In our case, moving from 95 dB to 70 dB, the intensity ratio \(\frac{I_2}{I_1}\) is approximately \(10^{-2.5}\), which equals roughly 0.00316. This indicates the new intensity allowed is only a fraction of the original intensity.
Logarithmic Scale
A logarithmic scale is a nonlinear scale used for a large range of quantities, where each step on the scale is a consistent multiple of the previous one. This kind of scale is used when dealing with quantities that can cover a huge range of values, such as sound intensity.

The need for a logarithmic scale arises because our hearing is extremely sensitive and can detect a very wide range of sound intensities. Measuring sound intensity in direct units would be unwieldy. Therefore, the logarithmic decibel scale simplifies these measurements.
  • On a logarithmic scale, every 10-fold increase in intensity adds 10 dB to the sound level.
  • This is why the decibel formula includes a logarithm: \[ L = 10 \log_{10}\left(\frac{I}{I_0}\right) \]which compares the sound intensity \(I\) to a reference intensity \(I_0\).
By compressing large ranges into manageable numbers, the logarithmic scale makes things easier to comprehend, similar to how the Richter scale measures seismic activity.

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Most popular questions from this chapter

Standing sound waves are produced in a pipe that is 1.20 \(\mathrm{m}\) long. For the fundamental and first two overtones, determine the locations along the pipe (measured from the left end) of the dis- placement nodes and the pressure nodes if (a) the pipe is open at both ends and (b) the pipe is closed at the left end and open at the right end.

Longitudinal Waves in Different Fluids. (a) A longitudinal wave propagating in a water-filled pipe has intensity \(3.00 \times 10^{-6} \mathrm{W} / \mathrm{m}^{2}\) and frequency 3400 \(\mathrm{Hz}\) . Find the amplitude \(A\) and wavelength \(\lambda\) of the wave. Water has density 1000 \(\mathrm{kg} / \mathrm{m}^{3}\) and bulk modulus \(2.18 \times 10^{9} \mathrm{Pa}\) . (b) If the pipe is filled with air at pressure \(1.00 \times 10^{5} \mathrm{Pa}\) and density \(1.20 \mathrm{kg} / \mathrm{m}^{3},\) what will be the amplitude \(A\) and wavelength \(\lambda\) of a longitudinal wave with the same intensity and frequency as in part (a)? (c) In which fluid is the amplitude larger, water or air? What is the ratio of the two amplitudes? Why is this ratio so different from 1.00\(?\)

CP Two identical taut strings under the same tension \(F\) produce a note of the same fundamental frequency \(f_{0}\) . The tension in one of them is now increased by a very small amount \(\Delta F .\) (a) If they are played together in their fundamental, show that the frequency of the beat produced is \(f_{\text { beat }}=f_{0}(\Delta F / 2 F)\) . (b) Two identical violin strings, when in tune and stretched with the same tension, have a fundamental frequency of 440.0 Hz. One of the strings is retuned by increasing its tension. When this is done, 1.5 beats per second are heard when both strings are plucked simultaneously at their centers. By what percentage was the string tension changed?

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?

A swimming duck paddles the water with its feet once every 1.6 \(\mathrm{s}\) , producing surface waves with this period. The duck is moving at constant speed in a pond where the speed of surface waves is \(0.32 \mathrm{m} / \mathrm{s},\) and the crests of the waves ahead of the duck are spaced 0.12 \(\mathrm{m}\) apart. (a) What is the duck's speed? (b) How far apart are the crests behind the duck's.
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