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Chapter 16: Problem 17

A sound wave in air at \(20^{\circ} \mathrm{C}\) has a frequency of 150 \(\mathrm{Hz}\) and a displacement amplitude of \(5.00 \times 10^{-3} \mathrm{mm} .\) For this sound wave calculate the (a) pressure amplitude (in \(\mathrm{Pa} ) ;\) (b) intensity (in \(\mathrm{W} / \mathrm{m}^{2} ) ;(\mathrm{c})\) sound intensity level (in decibels).

Short Answer

Expert verified
(a) 1.95 Pa; (b) 0.0046 W/m虏; (c) 96.6 dB

Step by step solution

01

Calculate Speed of Sound

The speed of sound in air at a given temperature can be calculated using the formula:\[ v = 331.4 + 0.6 imes T \]where \( T \) is the temperature in degrees Celsius. For \( T = 20^{\circ} \),\[ v = 331.4 + 0.6 \times 20 = 343.4 \text{ m/s} \]
02

Calculate Pressure Amplitude

Pressure amplitude \( P_0 \) can be calculated using the formula:\[ P_0 = B k A \]where:- \( A = 5.00 \times 10^{-3} \times 10^{-3} \text{ m} = 5.00 \times 10^{-6} \text{ m} \) (convert mm to m)- Bulk modulus \( B = 1.42 \times 10^5 \text{ Pa} \) for dry air at \( 20^{\circ}C \).Wave number \( k \) is:\[ k = \frac{2\pi f}{v} = \frac{2\pi \times 150}{343.4} \approx 2.74 \text{ rad/m} \]Calculate \( P_0 \):\[ P_0 = 1.42 \times 10^5 \times 2.74 \times 5.00 \times 10^{-6} \approx 1.95 \text{ Pa} \]
03

Calculate Intensity

Intensity \( I \) is given by:\[ I = \frac{(P_0)^2}{2 \rho v} \]For air, assume \( \rho = 1.21 \text{ kg/m}^3 \) (density of air).Substitute the values:\[ I = \frac{(1.95)^2}{2 \times 1.21 \times 343.4} \approx 0.0046 \text{ W/m}^2 \]
04

Calculate Sound Intensity Level

Sound intensity level \( L \) in decibels is given by:\[ L = 10 \log_{10} \left( \frac{I}{I_0} \right) \]where \( I_0 = 1.0 \times 10^{-12} \text{ W/m}^2 \) is the reference intensity. Substitute:\[ L = 10 \log_{10} \left( \frac{0.0046}{1.0 \times 10^{-12}} \right) \approx 96.6 \text{ dB} \]

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Key Concepts

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

Pressure Amplitude
Pressure amplitude is a crucial concept when discussing sound waves. It refers to the maximum change in pressure from its equilibrium value, caused by a sound wave through a medium.
  • The formula to find pressure amplitude is: \[ P_0 = B k A \] where:
    • \( B \) is the bulk modulus of the air, representing how the medium resists compression.
    • \( k \) represents the wave number, calculated as \( k = \frac{2\pi f}{v} \), with \( f \) being the frequency and \( v \) the speed of sound.
    • \( A \) stands for displacement amplitude, essentially the degree of motion within the medium.
Pressure amplitude is directly influenced by the wave properties (frequency and amplitude) and the medium's resistance (bulk modulus). In practice, calculating pressure amplitude helps in understanding the perceived loudness and energy of the sound.
Intensity of Sound
The intensity of sound is an important measure of the energy carried by a sound wave through a unit area per unit time. It's often referred to as the "loudness" of the sound, but technically offers a more precise measurement.
  • Intensity \( I \) can be calculated using: \[ I = \frac{(P_0)^2}{2 \rho v} \] where:
    • \( P_0 \) is the pressure amplitude.
    • \( \rho \) is the density of the air (typical value is 1.21 \( \text{ kg/m}^3 \) for dry air).
    • \( v \) is the speed of sound in the medium. For air at 20掳C, it's about 343.4 \( \text{ m/s} \).
Sound intensity is usually expressed in \( \text{ W/m}^2 \), which provides a good representation of how much power a sound wave emits in a particular area. It helps in distinguishing between sounds of different volumes and is vital in applications like soundproofing or speaker design.
Sound Intensity Level
Sound intensity level offers a way to quantify sound intensity using a logarithmic scale. This is expressed in decibels (dB), which gives a more intuitive understanding of sound volume differences.
  • The formula used is: \[ L = 10 \log_{10} \left( \frac{I}{I_0} \right) \] where:
    • \( I \) is the sound intensity.
    • \( I_0 \) is a reference intensity, typically \( 1.0 \times 10^{-12} \text{ W/m}^2 \), which represents the threshold of human hearing.
Using sound intensity level, we can compare different sound sources or environments effectively. For instance, an increase of 10 dB represents a sound that is 10 times more intense. This logarithmic nature helps in managing large ranges of sound intensities in a practical and meaningful way.

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Most popular questions from this chapter

CP You have a stopped pipe of adjustable length close to a taut \(85.0-\mathrm{cm}, 7.25-\mathrm{g}\) wire under a tension of 4110 \(\mathrm{N}\) . You want to adjust the length of the pipe so that, when it produces sound at its fundamental frequency, this sound causes the wire to vibrate in its second overtone with very large amplitude. How long should the pipe be?

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?

cp Longitudinal Waves on a Spring. A long spring such as a Slinky"is is often used to demonstrate longitudinal waves. (a) Show that if a spring that obeys Hooke's law has mass \(m,\) length \(L,\) and force constant \(k^{\prime},\) the speed of longitudinal waves on the spring is \(v=L \sqrt{k^{\prime} / m}\) . (see Section 16.2\()\) (b) Evaluate \(v\) for a spring with \(m=0.250 \mathrm{kg}, L=2.00 \mathrm{m},\) and \(k^{\prime}=1.50 \mathrm{N} / \mathrm{m}\)

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?

Wagnerian Opera. A man marries a great Wagnerian soprano but, alas, he discovers he cannot stand Wagnerian opera. In order to save his eardrums, the unhappy man decides he must silence his larklike wife for good. His plan is to tie her to the front of his car and send car and soprano speeding toward a brick wall. This soprano is quite shrewd, however, having studied physics in her student days at the music conservatory. She realizes that this wall has a resonant frequency of 600 \(\mathrm{Hz}\) , which means that if a continuous sound wave of this frequency hits the wall, it will fall down, and she will be saved to sing more Isoldes. The car is heading toward the wall at a high speed of 30 \(\mathrm{m} / \mathrm{s}\) . (a) At what frequency must the soprano sing so that the will will crumble? (b) What frequency will the soprano hear reflected from the wall just before it crumbles?
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