Acoustics of Music

1
Acoustics of Music

• The Physical Modelling of Wind Instruments

2
Characterising Wind Instruments

• Types of excitation source
• Single reed sax, clarinet
• Double Reed oboe, bassoon
• Air reed flute, organ pipe
• Brass lips
• Air Column bounded by
• Conical Bore
• Trumpet Flare
• Tone holes, valves or slide to control pitch

3

• Player controls excitation with breath and
  embouchure
• Higher frequency component (gt1.5kHz) of pressure
  wave propagates away from bore
• Lower frequency component of pressure wave is
  inverted and reflects back from bore and tone
  holes.
• Reflected wave controls excitation source
• Source behaviour non-linear

4

• Delay lines for bore length
• Reflection and Radiation filters for bore
  termination
• Non-linear behaviour for excitation i.e. harder
  blowing does not necessarily mean proportionately
  more energy into system

5
Describing the waveguide

• Time for wave to travel
• Characteristic impedance (ratio of pressure in
  wave to volume of air that moves)
• Where ddiameter of bore, ?density of air,
  cspeed of sound
• Changes in characteristic impedance along the
  waveguide mean some of wave transmitted and some
  reflected

6

• Consider reflection at impedance discontinuity
• Pressure and volume velocity must be continuous
• The acoustic impedance is given by
• Hence

7
Scattering Coefficients (Reflection and
Transmission)

• At x0 ratio of pressures is ratio of pressure
  amplitudes which is a reflection coefficient R
• And because of pressure continuity transmitted
  pressure to incident and reflected pressure

8
Scattering Coefficients in terms of bore diameters

• Using

9
Scattering Junction

• Used to represent movement between impedance
  discontinuities in a digital waveguide model
• For example forward wave pressure enters right
  section
• Similarly backward wave entering left section
• Can be used for modelling excitation, flare and
  tone holes

10
Bernoulli Equation

• "Bernoulli effect" is the lowering of fluid
  pressure in regions where the flow velocity is
  increased.
• Think of pressure as energy density effect
  results from conservation of energy where flow
  relates to kinetic energy

11
About Reed

• When you blow into a single reed instrument there
  is a pressure drop in the mouthpiece.
• This pressure drop is caused by the difference
  between blowing pressure in the mouth and the
  time varying pressure in the bore.
• The pressure drop helps the flow of air u into
  the instrument.

12

• At a small blowing pressure there is a small
  pressure drop and a small flow of air into the
  instrument.

13

• Going up to a medium blowing pressure there is a
  progressively larger flow. However note reed will
  begin to close.

14

• If the blowing pressure is large the pressure
  drop (and some additional lift) will cause the
  reed to aperture to close to the point where flow
  is restricted.

15
Modelling The Reed Characteristic

• At low blowing pressures the flow increases
  according to Bernoulli's equation
• At higher blowing pressure the aperture closes.
  Aperture size is given by original reed opening H
  minus reed displacement (hooks law FKx)
• Hence volume flow

16

• Hence excitation source has non linear
  characteristic
• Pressure coming back from bore reduces pressure
  in mouth
• Hence more flow into instrument
• Hence pressure drop closes reed
• Hence less flow into bore
• Hence oscillator sends pressure pulses into
  waveguide at a frequency that corresponds to
  effective length of waveguide

17
Reed Modelling

• Can model as scattering junction between mouth
  pressure and pressure in bore
• We know blowing pressure and reflected pressure
  from bore hence we need to find pressure entering
  bore
• Where R is a function of the pressure in the
  mouthpiece, hence implicitly a function of
  pressure entering bore
• Look up tables, root finders, etc

18
Tone Hole Modelling

• Again we can employ scattering junctions
• Tone holes pass high frequencies and reflect low
  frequencies
• Tone holes can be modelled as mass spring systems
  air in tone hole acts as lump that oscillates.

19
Air Reeds

• Air stream directed at an edge
• Fluctuations in pressure (and vortices) either
  side of edge create edge tone
• With a pipe attached this acts as flow-controlled
  valve pressure waves reflected back from bore
  govern whether flow is into or out of pipe
  hence pulses of energy into pipe at a frequency
  relating to effective length of pipe.
• Observations show air stream lags behind changes
  from standing waves in pipe hence transit time
  between players lips and embouchure hole needs to
  be taken into account.

20
Perry Cook slide flute

• length of the flute bore delay one period
• length of the jet (embouchure) delay ½ a period
• changing jet delay allows for over blowing
  effects