Lecture 14: Sound Waves


Readings: Textbook pages 527-532
Sound waves are first of all waves. Remarkable is that sound, physical phenomena simingly very different from oscillation of a pendulum or wave on a tightened rope has same fundamental properties. So Chapter 16 is very much a rehash of what we have already learned - speed of waves, wave functions, intensity, interference, superposition etc. But there are things specific to sound, and also one can use sound to discuss some wave properties in more detail.
  • As any wave, sound wave is described by velocity that is determined by the medium, frequency , wavelength and amplitude ( and intensity )
  • Frequency f is especially important since human ear is sensitive to a specific range of frequencies, 20-20000 Hz (typically losing high frequencies with age). Frequency is perceived as a pitch
  • Amplitude A contribute to loudness of the sound
  • In contrast, the wavelength λ is not directly registered by a hearing appartus. It is not what you hear !

Sound Waves as 3D displacement waves

  • Sound wave is a longitudinal wave in the medium - e.g. gas, fluid, solid.
  • In longitudinal wave oscillatory displacements of medium particles occur in the direction of wave propagation

  • What is being displaced ? It is average position of atoms/molecules. In the example above, the properties of the medium are the same, at any time moment, on average, along the vertical planes. (red one is one example at the boundary). So we compute average x-position of atoms in each plane, and that is what oscillates
  • In 3D view, the structure of the wave is

This is what is called a plane wave

  • Sound in a tube is approximately described by plane wave (exact plane wave have inifinite planes)
  • 3D plane waves are most similar to 1D waves we studied before. The simplest plane wave propagating in x-direction is described by cosine wave function
    y(x,t) = A cos( kx x - ω t + φ0 )
    where y(x,t) is a displacement in x of the plane which equilibrium position is x. Namely, instant position of this plane is x(t) = y(x,t)+x .
  • Mathematically, in a plane wave surfaces of constant phase, k x - ω t = const are planes. surfaces of constant phase are called wavefronts . So here we have planar wavefronts One may speak about wavefront propagation

Other sound waves

  • Many sound waves are not planar. Sound tend to propagate in all 3 directions, plane wave need special arrangements (like a tube). Here are few examples of different waves from localized sound sources. Animations are courtesy of this site
Example Animation Wavefronts shape
Monopole, boxed loudspeaker
Dipole, unboxed loudspeaker

Next

Sound as pressure wave

  • Another view on sound is to treat it as a pressure wave
  • This view is "macroscopic" and simplier, since pressure is already a quantity averaged over motions of atoms in some volume. So wavefronts are surfaces of constant pressure, etc.
  • Note that atoms oscillating around equilibrium do not move with the same displacement as neighbours. It would not be a wave then - just solid body displacement back and forth !. Hence, the distance with neighbours change. There is a local compression and rarification. With that comes changes in pressure. Pressure is a force acting on a surface of volume from surroundings (or visa versa).
  • Units of pressure are Pascales Pa = N/m2 = kg/(m s2)
  • Standard atmospheric pressure is 101.325 kPa ~ 105Pa

What pressure oscillation results from oscillating displacements of atoms ?

  • We shall consider plane waves for simplicity, and calculations are done on the board. Important formula for fluctuations of pressure in the wave around the ambient value Δ p = P - Pa is
  • is bulk modulus . It plays the role of spring constant in Hooke's Law but for volumetric deformations (strains). Units of bulk modulus are Pascales, as that for pressure.
  • Bulk modulus point is it is often constant in a given environment. For solids, B is essentially constant, depending on composition. It is typically in the range of 1011 Pa . For gases, bulk modulus depends on density and temperature of the gas. So it matters how gas is compressed in the wave - is it isothermal compression ? adiabatic compression ? Typical value in air at room temperature is ~ 105 Pa

Properties of pressure waves

  • Pressure wave has the same frequency, wavelength and velocity as displacement wave. It is description of the same sound wave !
  • Pressure wave is shifted relative to displacement in phase by ½π. Peak of displacement is at zeroes of pressure change, maximum/minimum pressure is at zero desplacement.
  • Pressure wave amplitude is Δ pmax = A B k Pressure fluctuation in the air that a human can hear is ~ 10-4 Pa which is 10-9 fraction of the atmospheric one ! Exact threshold of hearing depends on frequency and is lower at lower frequencies.