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Chapter 17: Problem 24

Suppose that the sound level of a conversation is initially at an angry \(70 \mathrm{~dB}\) and then drops to a soothing \(50 \mathrm{~dB}\). Assuming that the frequency of the sound is \(500 \mathrm{~Hz}\), determine the (a) initial and (b) final sound intensities and the (c) initial and (d) final sound wave amplitudes.

Short Answer

Expert verified
The initial and final intensities are \(10^{-5}\) and \(10^{-7}\) W/m虏; initial and final amplitudes are approximately \(1.26 \times 10^{-5}\) m and \(1.26 \times 10^{-6}\) m.

Step by step solution

01

Understanding the Decibel Scale

Decibels (dB) is a logarithmic scale used to measure sound intensity. The formula relating sound level in decibels to intensity is given by: \[ L = 10 \times \log_{10}\left( \frac{I}{I_0} \right) \] where \( L \) is the sound level in decibels, \( I \) is the sound intensity in watts per square meter, and \( I_0 = 10^{-12} \ \text{W/m}^2 \) is the reference intensity.
02

Calculate Initial Sound Intensity

For the initial sound level of \(70 \ \mathrm{dB}\), use the formula: \[ 70 = 10 \times \log_{10}\left( \frac{I_1}{10^{-12}} \right) \] Rearrange to solve for \( I_1 \): \[ \log_{10}\left( \frac{I_1}{10^{-12}} \right) = 7 \] \[ \frac{I_1}{10^{-12}} = 10^7 \] \[ I_1 = 10^{-12} \times 10^7 = 10^{-5} \ \text{W/m}^2 \] The initial sound intensity is \( 10^{-5} \ \text{W/m}^2 \).
03

Calculate Final Sound Intensity

For the final sound level of \(50 \ \mathrm{dB}\), use the formula: \[ 50 = 10 \times \log_{10}\left( \frac{I_2}{10^{-12}} \right) \] Rearrange to solve for \( I_2 \): \[ \log_{10}\left( \frac{I_2}{10^{-12}} \right) = 5 \] \[ \frac{I_2}{10^{-12}} = 10^5 \] \[ I_2 = 10^{-12} \times 10^5 = 10^{-7} \ \text{W/m}^2 \] The final sound intensity is \( 10^{-7} \ \text{W/m}^2 \).
04

Find Relationship Between Intensity and Amplitude

The sound intensity \( I \) is related to the amplitude \( A \) and frequency \( f \) of the sound wave by the equation: \[ I = \frac{1}{2} \rho v (2\pi f A)^2 \] where \( \rho \) is the air density (approximately \( 1.2 \ \text{kg/m}^3 \)) and \( v \) is the speed of sound in air (approximately \( 343 \ \text{m/s} \)).
05

Calculate Initial Sound Amplitude

To find the initial amplitude \(A_1\), rearrange the intensity equation: \[ 10^{-5} = \frac{1}{2} \times 1.2 \times 343 \times (2\pi \times 500 \times A_1)^2 \] Solve for \( A_1 \): \[ A_1 = \frac{\sqrt{10^{-5}}}{\sqrt{\frac{1}{2} \times 1.2 \times 343 \times (2\pi \times 500)^2}} \approx 1.26 \times 10^{-5} \ \text{m} \] The initial amplitude is approximately \(1.26 \times 10^{-5} \ \text{m}\).
06

Calculate Final Sound Amplitude

To find the final amplitude \(A_2\), rearrange the intensity equation for the final intensity: \[ 10^{-7} = \frac{1}{2} \times 1.2 \times 343 \times (2\pi \times 500 \times A_2)^2 \]Solve for \( A_2 \): \[ A_2 = \frac{\sqrt{10^{-7}}}{\sqrt{\frac{1}{2} \times 1.2 \times 343 \times (2\pi \times 500)^2}} \approx 1.26 \times 10^{-6} \ \text{m} \] The final amplitude is approximately \(1.26 \times 10^{-6} \ \text{m}\).

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Key Concepts

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

Decibel Scale
Sound is often described using the decibel (dB) scale, which is a logarithmic way of expressing sound intensity levels. Why logarithmic? Because human hearing spans a very large range of intensities and a linear scale isn鈥檛 practical.
  • This scale uses a reference intensity of \( I_0 = 10^{-12} \ \text{W/m}^2 \), which is considered the threshold of hearing for the average human ear.
  • The decibel level \( L \) can be calculated using the formula \( L = 10 \times \log_{10}\left( \frac{I}{I_0} \right) \), where \( I \) is the actual sound intensity.
Because of the logarithmic nature:- An increase of 10 dB represents a sound that鈥檚 10 times more intense.- This means that a change from 70 dB to 50 dB, as noted in the exercise, indicates a significant decrease in intensity, specifically a factor of 100 times quieter.
Sound Wave Amplitude
Sound amplitude refers to the maximum extent of a vibration or oscillation, measured from the position of equilibrium. In simpler terms, it's about how "big" the waves are.
  • The greater the amplitude, the louder we perceive the sound.
  • Amplitude is related to sound intensity: if the amplitude increases, the sound intensity and volume increase.
Calculating amplitude involves delving into physical equations that tie intensity \( I \) with amplitude \( A \) using factors such as air density \( \rho \), speed of sound \( v \), and frequency \( f \). For example: \[ I = \frac{1}{2} \rho v (2\pi f A)^2\]Using this equation, amplitude can be derived:- Rearrange and solve for \( A \) when you have the intensity and frequency available, as in the exercise example.
Frequency
Frequency, in the context of sound, refers to the number of times a sound wave oscillates per second and is measured in Hertz (Hz). One might think of frequency as the "pitch" of the sound:
  • Higher frequency means higher pitch (like the sound of a whistle), and lower frequency means lower pitch (like a drum).
  • In the exercise, the sound frequency is given as \(500 \ \text{Hz}\).
Frequency is a crucial component in understanding how sound behaves and how we perceive it. It鈥檚 important to note that frequency alone doesn鈥檛 affect the sound鈥檚 intensity, but it factors significantly in equations that relate to intensity and amplitude.
Intensity-Amplitude Relationship
Sound intensity and amplitude are related but are not the same thing. Intensity measures the power per unit area, while amplitude describes the height of the sound wave.
  • The formula connecting these involves factors like medium properties (e.g., air density) and is expressed in the equation: \( I = \frac{1}{2} \rho v (2\pi f A)^2 \).
  • This relationship allows us to calculate amplitude when other variables such as frequency and intensity are known, as demonstrated in the previous calculations for initial and final amplitudes.
The intensity-amplitude relationship illustrates that while amplitude alone gives us an idea of how "loud" a sound might be, combining it with properties like frequency and medium density provides a full picture of sound behavior and intensity.

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Most popular questions from this chapter

(a) Find the speed of waves on a violin string of mass \(800 \mathrm{mg}\) and length \(22.0 \mathrm{~cm}\) if the fundamental frequency is \(920 \mathrm{~Hz}\) (b) What is the tension in the string? For the fundamental, what is the wavelength of (c) the waves on the string and (d) the sound waves emitted by the string?0

A sound source sends a sinusoidal sound wave of angular frequency \(3000 \mathrm{rad} / \mathrm{s}\) and amplitude \(12.0 \mathrm{~nm}\) through a tube of air. The internal radius of the tube is \(2.00 \mathrm{~cm} .\) (a) What is the average rate at which energy (the sum of the kinetic and potential energies) is transported to the opposite end of the tube? (b) If, simultaneously, an identical wave travels along an adjacent, identical tube, what is the total average rate at which energy is transported to the opposite ends of the two tubes by the waves? If, instead, those two waves are sent along the same tube simultaneously, what is the total average rate at which they transport energy when their phase difference is (c) 0, (d) \(0.40 \pi\) rad, and (e) \(\pi\) rad?

A sinusoidal sound wave moves at \(343 \mathrm{~m} / \mathrm{s}\) through air in the positive direction of an \(x\) axis. At one instant during the oscillations, air molecule \(A\) is at its maximum displacement in the negative direction of the axis while air molecule \(B\) is at its equilibrium position. The separation between those molecules is \(15.0 \mathrm{~cm}\), and the molecules between \(A\) and \(B\) have intermediate displacements in the negative direction of the axis. (a) What is the frequency of the sound wave? In a similar arrangement but for a different sinusoidal sound wave, at one instant air molecule \(C\) is at its maximum displacement in the positive direction while molecule \(D\) is at its maximum displacement in the negative direction. The separation between the molecules is again \(15.0 \mathrm{~cm}\), and the molecules between \(C\) and \(D\) have intermediate displacements. (b) What is the frequency of the sound wave?

You have five tuning forks that oscillate at close but different resonant frequencies. What are the (a) maximum and (b) minimum number of different beat frequencies you can produce by sounding the forks two at a time, depending on how the resonant frequencies differ?

Two sound waves, from two different sources with the same frequency, \(540 \mathrm{~Hz}\), travel in the same direction at \(330 \mathrm{~m} / \mathrm{s}\). The sources are in phase. What is the phase difference of the waves at a point that is \(4.40 \mathrm{~m}\) from one source and \(4.00 \mathrm{~m}\) from the other?
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