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17.S: Sound (Summary)

Key Terms

beat frequency frequency of beats produced by sound waves that differ in frequency
beats constructive and destructive interference of two or more frequencies of sound
bow wake v-shaped disturbance created when the wave source moves faster than the wave propagation speed
Doppler effect alteration in the observed frequency of a sound due to motion of either the source or the observer
Doppler shift actual change in frequency due to relative motion of source and observer
fundamental the lowest-frequency resonance
harmonics the term used to refer collectively to the fundamental and its overtones
hearing perception of sound
loudness perception of sound intensity
notes basic unit of music with specific names, combined to generate tunes
overtones all resonant frequencies higher than the fundamental
phon numerical unit of loudness
pitch perception of the frequency of a sound
shock wave wave front that is produced when a sound source moves faster than the speed of sound
sonic boom loud noise that occurs as a shock wave as it sweeps along the ground
sound traveling pressure wave that may be periodic; the wave can be modeled as a pressure wave or as an oscillation of molecules
sound intensity level unitless quantity telling you the level of the sound relative to a fixed standard
sound pressure level ratio of the pressure amplitude to a reference pressure
timbre number and relative intensity of multiple sound frequencies
transducer device that converts energy of a signal into measurable energy form, for example, a microphone converts sound waves into an electrical signal

Key Equations

Pressure of a sound wave $$\Delta P = \Delta P_{max} \sin (kx \mp \omega t + \phi)$$
Displacement of the oscillating molecules of a sound wave $$s(x,t) = s_{max} \cos (kx \mp \omega t + \phi)$$
Velocity of a wave $$v = f \lambda$$
Speed of sound in a fluid $$v = \sqrt{\frac{\beta}{\rho}}$$
Speed of sound in a solid $$v = \sqrt{\frac{Y}{\rho}}$$
Speed of sound in an ideal gas $$v = \sqrt{\frac{\gamma RT}{M}}$$
Speed of sound in air as a function of temperature $$v = 331\; m/s \sqrt{\frac{T_{K}}{273\; K}} = 331\; m/s \sqrt{1 + \frac{T_{C}}{273 \;^{o} C}}$$
Decrease in intensity as a spherical wave expands $$I_{2} = I_{1} \left(\dfrac{r_{1}}{r_{2}}\right)^{2}$$
Intensity averaged over a period $$I = \frac{\langle P \rangle}{A}$$
Intensity of sound $$I = \frac{(\Delta p_{max})^{2}}{2 \rho v}$$
Sound intensity level $$\beta (dB) = 10\; \log_{10} \left(\dfrac{I}{I_{0}}\right)$$
Resonant wavelengths of a tube closed at one end $$\lambda_{n} = \frac{4}{n} L,\; n = 1, 3, 5, \ldots$$
Resonant frequencies of a tube closed at one end $$f_{n} = n \frac{v}{4L},\; n = 1, 3, 5, \ldots$$
Resonant wavelengths of a tube open at both ends $$\lambda_{n} = \frac{2}{n} L,\; n = 1, 2, 3, \ldots$$
Resonant frequencies of a tube open at both ends $$f_{n} = n \frac{v}{2L},\; n = 1, 2, 3, \ldots$$
Beat frequency produced by two waves that differ in frequency $$f_{beat} = |f_{2} - f_{1}|$$
Observed frequency for a stationary observer and a moving source $$f_{o} = f_{s} \left(\dfrac{v}{v \mp v_{s}}\right)$$
Observed frequency for a moving observer and a stationary source $$f_{o} = f_{s} \left(\dfrac{v \pm v_{o}}{v}\right)$$
Doppler shift for the observed frequency $$f_{o} = f_{s} \left(\dfrac{v \pm v_{o}}{v \mp v_{s}}\right)$$
Mach number $$M = \frac{v_{s}}{v}$$
Sine of angle formed by shock wave $$\sin \theta = \frac{v}{v_{s}} = \frac{1}{M}$$

Summary

17.1 Sound Waves

  • 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.

17.2 Speed of Sound

  • The speed of sound depends on the medium and the state of the medium.
  • In a fluid, because the absence of shear forces, sound waves are longitudinal. A solid can support both longitudinal and transverse sound waves.
  • In air, the speed of sound is related to air temperature T by \(v = 331\; m/s \sqrt{\frac{T_{K}}{273\; K}} = 331\; m/s \sqrt{1 + \frac{T_{C}}{273 \;^{o} C}} \ldotp\)
  • v is the same for all frequencies and wavelengths of sound in air.

17.3 Sound Intensity

  • Intensity I = \(\frac{P}{A}\) is the same for a sound wave as was defined for all waves, where P is the power crossing area A. The SI unit for I is watts per meter squared. The intensity of a sound wave is also related to the pressure amplitude \(\Delta\)p:$$I = \frac{(\Delta p)^{2}}{2 \rho v}$$where \(\rho\) is the density of the medium in which the sound wave travels and vw is the speed of sound in the medium.
  • Sound intensity level in units of decibels (dB) is $$\beta (dB) = 10\; \log_{10} \left(\dfrac{I}{I_{0}}\right)$$where I0 = 10−12 W/m2 is the threshold intensity of hearing.
  • The perception of frequency is pitch. The perception of intensity is loudness and loudness has units of phons.

17.4 Normal Modes of a Standing Sound Wave

  • Unwanted sound can be reduced using destructive interference.
  • Sound has the same properties of interference and resonance as defined for all waves.
  • In air columns, the lowest-frequency resonance is called the fundamental, whereas all higher resonant frequencies are called overtones. Collectively, they are called harmonics.

17.5 Sources of Musical Sound

  • Some musical instruments can be modeled as pipes that have symmetrical boundary conditions: open at both ends or closed at both ends. Other musical instruments can be modeled as pipes that have anti-symmetrical boundary conditions: closed at one end and open at the other.
  • Some instruments, such as the pipe organ, have several tubes with different lengths. Instruments such as the flute vary the length of the tube by closing the holes along the tube. The trombone varies the length of the tube using a sliding bar.
  • String instruments produce sound using a vibrating string with nodes at each end. The air around the string oscillates at the frequency of the string. The relationship for the frequencies for the string is the same as for the symmetrical boundary conditions of the pipe, with the length of the pipe replaced by the length of the string and the velocity replaced by v = \(\sqrt{\frac{F_{T}}{\mu}}\).

17.6 Beats

  • When two sound waves that differ in frequency interfere, beats are created with a beat frequency that is equal to the absolute value of the difference in the frequencies.

17.7 The Doppler Effect

  • The Doppler effect is an alteration in the observed frequency of a sound due to motion of either the source or the observer.
  • The actual change in frequency is called the Doppler shift.

17.8 Shock Waves

  • The Mach number is the velocity of a source divided by the speed of sound, M = \(\frac{v_{s}}{v}\).
  • When a sound source moves faster than the speed of sound, a shock wave is produced as the sound waves interfere.
  • A sonic boom is the intense sound that occurs as the shock wave moves along the ground.
  • The angle the shock wave produces can be found as \(\sin \theta = \frac{v}{v_{s}} = \frac{1}{M}\)
  • A bow wake is produced when an object moves faster than the speed of a mechanical wave in the medium, such as a boat moving through the water.

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