Waves II: Sound Waves

1
Waves II Sound Waves

• Sound Waves Sound waves are longitudinal waves
  (i.e. involve oscillations parallel to the
  direction of the wave travel) that propagate
  through a medium (e.g. air, water, iron).

• Speed of Sound The speed of any mechanical wave
  depends on both the inertial property of the
  medium (stores kinetic energy) and the elastic
  property (stores potential energy).

(wave speed)

• Stretched String (last chapter) The speed of the
  transverse wave along a stretched string is

(string, t tension, m linear mass density)

• Sound The speed of the longitudinal sound wave
  is

(sound, B bulk modulus -DP/(DV/V), r volume
mass density)
2
Waves II Sound Waves

• Traveling Sound Waves in Air A traveling sound
  waves consist of a moving periodic pattern of
  expansions and compressions of the air. As the
  wave passes the air elements oscillate
  longitudinally in simple harmonic motion.

(1000 Hz sound wave with DPmax at the threshold
of pain!)
Longitudinal displacement
Pressure variation
(The pressure amplitude is related to the
displacement amplitude!)
3
Sound Waves Intensity and Level

• Intensity Traveling sound waves transport energy
  (kinetic and potential) from one point to
  another. The intensity, I, of a sound wave at a
  surface is the average energy per unit time per
  unit area transmitted by the wave to the surface
  (i.e. average power per unit area). It is also
  equal to the average energy per unit volume in
  the wave times the speed of propagation of the
  wave.

(the intensity is proportional to the square of
the amplitude!)
Watts/m2

• Variation with Distance If sound is emitted
  isotropically (i.e. equal intensity in all
  directions) from a point source with power Ps and
  if the mechanical energy of the wave is conserved
  then

(intensity from isotropic point source)

• The Decibel Scale Instead of speaking about the
  intensity I of sound, it is more convenient to
  speak of the sound level b, where

(sound level, dB decibel, I0 10-12 W/m2)
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Sound Waves Examples

• The maximum pressure amplitude DPmax that the
  human ear can tolerate in loud sounds is about 28
  Pa (which is much less that atmospheric pressure
  of about 105 Pa). What is the displacement
  amplitude smax and the intensity I (in dB) for
  such a sound in air of density 1.21 kg/m3, at a
  frequency of 1,000 Hz, and a speed of 343 m/s?

• The pressure amplitude DPmax for the faintest
  detectable sound at 1,000 Hz is 2.810-5 Pa.
  What is the displacement amplitude smax and the
  intensity I (in dB) for such a sound in air of
  density 1.21 kg/m3 and a speed of 343 m/s?

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Sound Waves Doppler Effect
(Source and detector at rest)

• Doppler Shift If either the detector or the
  source of sound is moving, or if both are moving,
  then the emitted frequency, fs, of the source and
  the detected frequency, fobs, are different. If
  both the source and the detector are at rest with
  respect to the air and v lsfs is the speed of
  sound in air then

(source stationary, detector stationary)

• Detector Moving, Source Stationary If source is
  at rest and the detector is moving toward the
  source with speed vD with respect to the air and
  v lsfs is the speed of sound in air then

(Source at rest, Detector moving toward the
source)
(source stationary, detector moving toward the
source)
(source stationary, detector moving toward () or
away (-) from the source)
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Sound Waves Doppler Effect
(detector stationary, source moving toward the
detector)

• Detector Stationary, Source Moving If detector
  is at rest and the source is moving toward the
  detector with speed vs with respect to the air
  and v lsfs is the speed of sound in air and Ts
  1/fs is the time between emitted pulses then

(detector stationary, source moving toward the
detector)
(detector stationary, source moving toward (-) or
away () from the source)

• Detector and Source Moving If detector is moving
  with speed vD and the source is moving with speed
  vs with respect to the air then there are the
  following four possibilities

Detector
Detector
Detector
Detector
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Sound Waves Examples

• A low flying airplane skims the ground at a speed
  of 200 m/s as it approaches a stationary
  observer. A loud horn whose fundamental
  frequency is 400 Hz is carried on the plane.
  What frequency does the ground observer hear?
  (Assume that the speed of sound in the air is 340
  m/s.)

• If instead the horn were on the ground, what
  frequency would the airplane pilot hear as she
  approached?

8
Sound Waves Shock Waves

• Source Moving at the Speed of Sound If a source
  is moving at a stationary detector at speed vs
  v then

(source moving at speed vs v)
(source keeping up with its own spherical
wavefronts!)

• Supersonic Speeds If the source moves faster
  than sound (i.e. vs gt v) then the Doppler Shift
  formula no longer apply. The favefronts extend
  in three dimensions and bunch up along a cone
  called the Mach cone and a shock wave is said
  to exist along the surface of the cone, because
  the bunching causes an abrupt rise and and fall
  of the air pressure as the cone passes through
  any point. The burst of sound caused by the
  Mach cone is called a sonic boom.

(source moving at speed vs gt v)
(Mach cone angle, vs/v Mach number)