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17.1 Sound waves  (Page 2/5)

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Models describing sound

Sound can be modeled as a pressure wave by considering the change in pressure from average pressure,

Δ P = Δ P max sin ( k x ∓ ω t + ϕ ) .

This equation is similar to the periodic wave equations seen in Waves , where Δ P is the change in pressure, Δ P max is the maximum change in pressure, k = 2 π λ is the wave number, ω = 2 π T = 2 π f is the angular frequency, and ϕ is the initial phase. The wave speed can be determined from v = ω k = λ T . Sound waves can also be modeled in terms of the displacement of the air molecules. The displacement of the air molecules can be modeled using a cosine function:

s ( x , t ) = s max cos ( k x ∓ ω t + ϕ ) .

In this equation, s is the displacement and s max is the maximum displacement.

Not shown in the figure is the amplitude of a sound wave as it decreases with distance from its source, because the energy of the wave is spread over a larger and larger area. The intensity decreases as it moves away from the speaker, as discussed in Waves . The energy is also absorbed by objects and converted into thermal energy by the viscosity of the air. In addition, during each compression, a little heat transfers to the air; during each rarefaction, even less heat transfers from the air, and these heat transfers reduce the organized disturbance into random thermal motions. Whether the heat transfer from compression to rarefaction is significant depends on how far apart they are—that is, it depends on wavelength. Wavelength, frequency, amplitude, and speed of propagation are important characteristics for sound, as they are for all waves.

Summary

  • Sound is a disturbance of matter (a pressure wave) that is transmitted from its source outward. Hearing is the perception of sound.
  • Sound can be modeled in terms of pressure or in terms of displacement of molecules.
  • The human ear is sensitive to frequencies between 20 Hz and 20 kHz.

Conceptual questions

What is the difference between sound and hearing?

Sound is a disturbance of matter (a pressure wave) that is transmitted from its source outward. Hearing is the human perception of sound.

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You will learn that light is an electromagnetic wave that can travel through a vacuum. Can sound waves travel through a vacuum?

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Sound waves can be modeled as a change in pressure. Why is the change in pressure used and not the actual pressure?

Consider a sound wave moving through air. The pressure of the air is the equilibrium condition, it is the change in pressure that produces the sound wave.

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Problems

Consider a sound wave modeled with the equation s ( x , t ) = 4.00 nm cos ( 3.66 m −1 x − 1256 s −1 t ) . What is the maximum displacement, the wavelength, the frequency, and the speed of the sound wave?

s max = 4.00 nm , λ = 1.72 m , f = 200 Hz , v = 343.17 m/s

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Consider a sound wave moving through the air modeled with the equation s ( x , t ) = 6.00 nm cos ( 54.93 m −1 x − 18.84 × 10 3 s −1 t ) . What is the shortest time required for an air molecule to move between 3.00 nm and –3.00 nm?

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Consider a diagnostic ultrasound of frequency 5.00 MHz that is used to examine an irregularity in soft tissue. (a) What is the wavelength in air of such a sound wave if the speed of sound is 343 m/s? (b) If the speed of sound in tissue is 1800 m/s, what is the wavelength of this wave in tissue?

a. λ = 68.60 μ m; b. λ = 360.00 μ m

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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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