Physics of Sound

1
Physics of Sound

• Wave equation Part. diff. equation relating
  pressure and velocity as a function of time and
  space
• Nonlinear contributions are not considered.
• Valid under small partical Velocity.
• Three steps
• Lossless uniform tube model
• Nonuniform, losses due to voval tract walls.
• Boundary effects (lip radiation)

2
Sound

• Sound is vibration of particles in a medium.
• Particle velocity
• Pressure
• Sound wave is the propagation of disturbance of
  particles through a medium.
• c ? f
• ? 2? f
• ?/c 2? / ? wave number
• At sea level c 344 m/s (70º F)
• At f 50 Hz ? c / f 6.88 m

3

• Isothermal processes
• Slow variation (of pressure), temp. stays
  constant (no time for heat transfer)
• Adiabatic processes
• Fast variation (of pressure), temp. changes (time
  for heat transfer)
• Example Bicycle pump
• Typical usage of the terms Isothermal/adiabatic
  compression of a gas
• For most frequencies (except very low
  frequencies) sound is adiabatic.

4
Wave Equation

• Atmospheric pressure P0 105 N/m2
• Pressure P0 p(x,t)
• p(x,t)
• 0 dB, threshold of hearing 2(10-5) N/m2 at 1000
  Hz.
• threshold of pain20 N/m2.
• Particle velocity v(x,t), m/s, (around zero
  average)
• Density of air particles ?(x,t), kg/ m3 (around
  an average of ?0 --gt, ?0 ?(x,t) )

5
Wave Equation

• 3 laws of physics, to be applied on the cubic
  volume of air.
• F ma
• P V? Const P total pressure, V volume, ?
  1.4
• Conservation of mass The cube may be deformed if
  pressure changes but the of particles inside
  remains the same.
• 
  F - (?p/ ?x) ?x (?y ?z)
• 
  net press. vol.
• (no
  frictional pressure, zero viscosity)
• 
  m ? ?x ?y ?z

8
6
Wave Equation

• F m a ? - (?p/?x) ?x (?y ?z) ? ?x ?y ?z
  (dv/dt)
• dv (?v/?x) dx v (?v/?t) dt
• dv/dt v (?v/?x) (?v/?t)
• nonlinear can be
  neglected in speech production since particle
  velocity is small
• ? - (?p/ ?x) ? (?v/?t)
• Gas law and cons. of mass yields
  coupled wave equation pair
• ? - (?p/ ?t) ?c2 (?v/?x)
• The two can be combined as
• (?2p/?x2) (1/c2) (?2p/?t2) wave equation for
  p
• or
• (?2v/?x2) (1/c2) (?2v/?t2) wave equation v

7
Uniform Tube Model (lossless)

• No air friction along the walls
• For convenience volume velocity is defined
• u (x,t) A v (x,t) m3/s

• 2nd order wave equations are the same in this
  case except the replacement v(x,t) ? u(x,t)
• Coupled pair becomes
• The solutions are of the form

8
Uniform Tube Model (lossless)

• To find the particular solution let,
• at x 0, ug(t) u(0,t) Ug(O) ej O t
  (glottal flow)
• at x l, p(l,t) 0 (no radiation at the
  lips)
• The general solution is
• To solve for unknown constants k and k-, apply
  the boudary conditions above.
• and

9
Uniform Tube Model (lossless)

• These are standing waves.
• The envelopes are orthogonal in space and in time

l
0
x
10
Uniform Tube Model (lossless)

• The frequency response for vol. Velocity, Va(O)
• The resonances occur at
• Ex Consider a uniform tube of length l 35 cm.
  For c 350 m/s, the roots, resonances, are at f
  O / (2?) 2000 k / 8 250, 750, 1250,...

As l decrease resonance frequencies increase
Volume velocity
O
11
Uniform Tube Model (lossless)

• Acoustic impedance The ratio of presure to
  volume velocity.
• The frequency response can be changed to transfer
  function O ? s / j
• Under some restrictions it can be written as
• The poles are the resonant frequencies of the tube

12
Energy Loss Due to Wall Vibration

• Let the crosssection of the tube be A(x,t), then
• Now consider the model

Wave eqns.
13
Energy Loss Due to Wall Vibration

• Assuming A(x,t) A0(x,t) ?A(x,t), an equaton
  can be written for ?A(x,t)
• Then, the three equations can be written (under
  some simplifications, AA0?A)

14
Energy Loss Due to Wall Vibration

• Assuming again ug(t) u (0, t) Ug(O) ej O t
  yields solutions of the form
• These forms eliminate time dependence and the
  equations become
• They are solved by numerical techniques

15
Formants would be at 500, 1500, 2500,..., in the
lossless case.
l17.5 cm, A05cm2, mw 0.4gr/cm2, bw
6500dyne-sec/cm3 , kw0 Bandwidth is not zero!
Viscosity (friction of air with walls) and
thermal loss included.