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BIOMEASUREMENT 2202
Sensing - hearing
S2
Biological acoustics and the sense of hearing I.
Objectives
• To understand the basic physics of sound generation and transmission through various media.
• To outline the reception and transduction of a pressure wave by the inner ear and cochlea.
• To review the non-linear response and feedback aspects of audition.
The nature of sound
From a physical point of view, sound is a travelling wave of compression and rarefaction through a
medium (solid, liquid, vapour or plasma). Consider a leaf's rustling being heard by a cat. Four stages
of this process can be identified:
- generation of sound at the source
- transfer of sound through the intervening media
- reception and encoding of sound at the hearing apparatus
- cerebral decoding, interpretation and response
In these lectures we shall look at all 4 aspects but with emphasis on the physical description of the
first 3. The 4th stage is of concern to neuroscientists as well as musicians. Physics brings
investigative tools and the theories of waves, sound and acoustics to aid in the understanding of
audition.
• speed of sound in different media:
in a gas
c =
c = f
RT
Mw
in a liquid
c =
B

where  is the ratio of specific heats, T temperature in kelvin and Mw the mean molecular weight .
 p 
B     ,
B is the isentropic bulk modulus of elasticity defined as
   s
 is the density, and  a factor to account for differences between adiabatic and isothermal processes.
in a solid
c =
Y

where Y is Young's modulus of elasticity.
fo
c ± vo
=
fs
c  vs
• doppler shift due to relative motion:
where f is frequency, v is velocity, o refers to the observer and s to the sound source.
• The human ear perceives sound level logarithmically. The usual measure is the sound
pressure level (SPL) defines as:
p
SPL = 20 log p
ref
where the reference pressure amplitude is
(dB)
pref = 2 x 10-5 Pa.
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Lumped mechanical elements and resonance - 1D case
In order to comprehend the mechanism of hearing we must first review the behaviour of mechanical
resonance. The ringing of a bell, the clicking of fingernails, the shudder in an accelerating vehicle,
the sound of a piano, violin, flute and even the human voice - all owe their existence and
characteristics to this phenomenon.
There are three basic mechanical idealisations.
• elasticity (spring)
F = - k (x - xo)
• resistance (friction)
dx
F = - r .v = - r dt
• inertia (mass)
d2x
F = m a = - m dt2
From Newton's second law of motion:
d2x
m dt2
dx
+ r dt
+ k (x - xo) = 0
where we define the damping coefficient
or:
+ o2 (x - xo) = 0
k
o2 = m .

x = C ept
solving the quadratic for p gives
dx
+  dt
r
=m ,
and square of the undamped resonant frequency
Look for a solution of the form
d2x
dt2
-±
p =
p2 +  p + o2 = 0
2 - 4o2
2
This will only be a real number if  ≥ 2o in which case x(t) describes an exponential decay to its
equilibrium value xo. Otherwise the system undergoes damped simple harmonic motion. In this case
we must allow p to be a complex number whose magnitude undergoes exponential decay, while the
argument provides oscillation on the x axis:
o
We define the mechanical quality factor
Q =

and rewrite for the  < 2o case

p = - 2 ± i o
Using Euler's formula:
1
1 - 4Q2
where i is the imaginary number i2 = -1
ei = cos  + i sin 
we can write the transient solution for x (choice of x origin such that xo = 0)

x(t) = C exp ( - 2 t ) cos (t + )
where the system now oscillates at the damped resonant frequency res = o
1
1 - 4Q2 .
The constants C and  are determined by the initial conditions of the system, x(0) and v(0).
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What is happening during this behaviour? Essentially we can describe the process as follows:
dx
• The equilibrium state is x = xo and v = dt = 0.
• When displaced from equilibrium to position x1 we do work against the elastic forces to
1
store energy in the compression of the spring PE = 2 k (x1 - xo)2
• As x moves towards xo the PE is lost to friction (damping) but also is converted to KE which
can later be reconverted to PE as in a simple pendulum.
• How long the oscillations last depends on the relative size of the losses compared to the
energy stored as either PE or KE.
• The total energy (PE + KE) of the system decays exponentially to its equilibrium value.
• The system Q can be interpreted as the energy stored divided by the energy lost to friction
per cycle. It also gives a rough approximation of the number of oscillations the system will
undergo before losing most of its amplitude.
• The system is said to be
underdamped if
 < 2o
Q > 0.5
critically damped
 = 2o
Q = 0.5
overdamped
 > 2o
Q < 0.5
• For the overdamped case there are no oscillations and the effective decay time constant is
 =
2
(1 +

1 - 4Q2 )
-1
1.0
unforced resonance
displacement
x
0.8
f o = 3 Hz
ov erdamped
critically damped
underdamped
0.6
Q = 0.2
Q = 0.5
Q = 2.0
0.4
0.2
0.0
-0.2
-0.4
0.0
0.2
0.4
0.6
0.8
1.0
time t
The underdamped (Q > 0.5) case can be thought of as a sinusoidal wave with an exponential decay
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envelope with time constant  = 
1.0
x
0.5
displacement
unforced resonance
f o = 15 Hz, Q = 10
0.0
env elope  = 212 ms
-0.5
0.0
0.2
0.4
0.6
time t
3
0.8
1.0
Forced resonance
Assume
• xo = 0
• sinusoidal driving force
h(t) = a sin t
• solution of the form
x(t) = A() sin t + B() cos t
2
dx
dx
m dt2 + r dt + k (x - xo) = h
The solution is valid provided the following conditions are met:
a (o2 - 2)
a
A =
B =
2
2
2
2
2
2
(o -  ) +  
(o - 2)2 + 2 2
x(t) = C() cos(t - )
An alternate form of this solution is:
with
and
a2
C2 = A2 + B2 =
(o2 - 2)2 + 2 2
o2 - 2
o  
A
tan  = B =
= Q.  - 

  o
This shows us that a resonant system can oscillate at the same frequency as the driving force but
with a frequency dependent amplitude and phase. The amplitude reaches a peak at the natural
(unforced) resonant frequency o at which the phase switches sign.
Example 1: Forced amplitude response (C) for Q = 3 and fo = 200 Hz.
90
amplitude
phase
amplitude
60
30
1.0
0
-30
0.5
-60
0
200
400
f (Hz)
4
600
800
phase (degrees)
1.5
Example 2: Forced amplitude response (C) for Q = 20 and fo = 400 Hz.
90
3.0
amplitude
phase
60
30
2.0
1.5
0
1.0
-30
0.5
-60
0
200
400
600
phase (degrees)
amplitude
2.5
800
f (Hz)
High Q resonances are often referred to in terms of their bandwidth which is the frequency range for
which the (amplitude)2 is more than half the maximum.
a2
C2 =  

Maximum response occurs on resonance  = o
π
Half-power points at  = ± 4
±   = o2 - 2
o  
tan  = Q .  -  = ±1
  o
Alternately:
High Q approximation:
Let
 = o + ∆
with
∆ << o
o
The bandwidth is then BW = 2 ∆ ≈ Q = 
(rad/s)
then
2Q
fo
or Q
∆
≈ 1
o
(Hz)
Acoustic impedance
We can generalise the concepts of springiness, friction or inertia by developing the idea of
impedance.
Recall from electrical circuit theory:
We now write in similar form
V=ZI
p=Zv
where p is the pressure difference and v is the volume velocity.
The units of Z as defined here are N.s m-1 or kg s-1 . Some texts prefer alternate definitions, but all
follow the same approach of relating the potential difference to the corresponding flow in a linear
equation.
Both p and v are considered as the real components of complex phasors of the form eit. Euler's
formula (above) allows us to treat sinusoidal waves and any combination thereof. The ratio of p to v
will generally have a phase component as well as a magnitude. Think of pushing someone in a
child's swing. Your forcing function (p) while periodic is 90˚ out of phase with the swing's motion
(v).
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The forced mechanical resonance equation rewritten in terms of v becomes:
dv
m dt
+ r v + k vdt
= h

k 

im + r + i  v = h


For a phasor at frequency  we have1:
mass
damper
spring
Z: magnitude
m
r
Z: phase
π
+2
0
k

π
-2
KE
dissipation
PE
inductor
L
resistor
R
capacitor
1
C
element
electrical analog
Z: magnitude
Units:
F = - kx  k in N m-1 ;
 in rad s-1
F = - r v  r in N.s m-1
 m in kg s-1
NB: N = kg m s-2
Other terms in common use:
admittance:
this is the reciprocal of impedance often written as Y:
v=Yp
compliance2: the acoustic compliance Ca of an enclosed gas is defined as:
V
[m5 N-1]
Po
V: volume of container, : the ratio of specific heats, Po: gas pressure
Ca =
with
resistance:
the acoustic resistance for sound travel through a duct depends on
diameter, temperature and gas composition and mean pressure
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inertance:
air has mass that must be accelerated in order to move or change v.
acoustic inertance Ma depends on the medium as well as the geometry
example: air of density o in a piston of cross-sectional area S. The pipe
contains mass m of air in length .
o  S a
o 
o 
F
ma
dv
dv
p=S = S =
= S S dt = S S dt
S
o 
m
Ma = S 2 = S
[kg m-4]
Mechanical resonance - Helmholtz resonator
Putting all three elements together we get a simple resonance at a frequency determined by r,Ca and
Ma. This is the common bottle-resonance found by blowing across the neck. The mass of air
'bounces' on the compliance of the air trapped in the flask. Acoustic resistance in the neck and
radiation of sound energy dampen the resonance.
c
fo = 2π
S
LV
c = speed of sound in the gas
S = neck-area
L = effective length of neck
V = body volume
Exercises:
• Estimate the resonant frequency of a 1 litre glass soft drink bottle when it is
• empty
• half full
• full
• If the bottle were plastic how would squeezing the neck affect the resonant frequency?
• What if the plastic body were squeezed?
• Try some simple experiments with bottles at home to confirm your predictions.
• Show that for negligible damping, the resonance equation reduces to Hooke's law.
• Find a solution for x(t) and v(t) for the undamped unforced resonance. Use these to derive an
expression for the total energy (PE+KE) and show that this is constant with time.
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References:
Beranek LL, Acoustics, McGraw-Hill, FIZ 621.3818.
Elmore WC & Heald MA, Physics of waves, Wiley, FIZ 531.33.
Kinsler LE & Frey AR, Fundamentals of acoustics, Wiley, FIZ 534.
Rossing TD, The science of sound, Addison-Wesley, FIZ 534.
Lord Rayleigh, Phil. Mag., 34 (1917), p.94. Also his Theory of sound Vols I and II.
Plesset Milton S. "Bubble dynamics" in Cavitation in real liquids, R.Davies (ed.), Elsevier, 1964,
p1-17.
Bogdanov K, Biology in physics - is life matter?, chapter 6, Academic Press, 1999.
Hopp SL, Owren MJ & Evans CS (eds.), Animal acoustic communication, Springer, 1998, BIOL
591.594.
Tavolga WN, Popper AN & Fay RR (eds.), Hearing and sound communication in fishes, SpringerVerlag, 1981.
Wood A, Acoustics, Blackie & Son Ltd 1940.
Notes:
1.
The complex exponential function satisfies
Let
z = eit.
Then
The chain rule applies:
2.
deZ
Z
dz = e
dz
dt = i z
and
1
=
z
zdt

i
y = ea sin t

dy
a sin t = a y cos t
dt = a cos t e
Adiabatic expansion of an ideal gas satisfies
PV = PoVo = constant.
Where Po is the ambient gas pressure and Vo the rest volume. Rearranging and taking the
time derivative gives:
P
1 dP
 Vo  

- (1+) dV

Po =  V 
Po dt = -  Vo V
dt
Po dV
dP
1 dV
So:
where C is termed the compliance.
dt ≈ - V dt = - C dt
Example of natural resonance: Spectrum of the Westminster bell. Higher harmonics are clearly
visible but die out faster than the fundamental.
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Supplementary notes on musical acoustics.
Musical sources of sound
• How does it arise?
- resonant vibrations of solids
bell, chime, cymbal
- excited modes of a system
guitar string, flute, blown bottle, rain, boiling kettle, human voice
- fluid in fluid
jet, wind, wheezing, 'shh!'
• Sources
- strings & pipes
sine wave
- bars & cantilevers
sine & hyperbolic sine (sinh)
- membranes & plates
Bessel functions
- human voice
vocal chords (fleshy folds)
vibrating at ~ 150 Hz
excite vocal tract resonances
In all cases the sound heard depends on
- physical dimensions
- material properties
- how it is excited
- immediate surroundings
(length, volume, ...)
(stiffness, elasticity, ...)
(where, how hard, with what, ...)
(end connections, enclosure, medium, ...)
Natural modes of vibration:
Mathematical solutions since 1700's.
(Laplace, Helmholtz, Lord Rayleigh)
Each point in a vibrating body tries to stay in
equilibrium with its nearest neighbours.
Source
75 pce orchestra @ fff
bass drum
pipe organ
snare drum
cymbals
trombone
piano
bass saxophone
bass tuba
double bass
Power (W)
70
25
13
12
10
6
0.4
0.3
0.2
0.16
Source
75 pce orchestra @ mf
piccolo
flute
clarinet
french horn
triangle
bass voice
alto voice (@ pp)
average speech
violin (@ pppp)
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Power (W)
0.09
0.08
0.06
0.05
0.05
0.05
0.03
0.001
0.000024
0.0000038
Example 1:
Supplementary notes on musical acoustics.
c
Vibrating string
fn = n 2L
Solution:
y(x) = A sin(2πfnx)
with
y(0) = y(L) = 0
displacement
n=1
0
x
displacement
n=2
0
x
displacement
n=7
0
Example 2:
Pipe
x
Vibrating air column
1
fn = n f1
c
f1 = 2 L
2
as for string, but antinodes at ends
3
c: speed of sound in the air
flute, organ, bagpipes, didgeridoo etc
NOTE: For accurate work: Correction for length due to the width at the opening:
Leff = L + 0.6 a
Further notes and animations are available at:
10
http://www.biophysics.uwa.edu.au/music/music.html
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Supplementary notes on musical acoustics.
Stopped pipe
fn = (2n - 1)f1
c
f1 = 4 L
2n + 1
fn ≈  3 


• General solution (clamped-free):
Example 3:
Cantilever
2
f1
with
9πc
f1 ≈ 8 L2
y(x) = A cos(bx) + B sin(bx) + C cosh(bx) + D sinh(bx)
with
A+C=0
Free at both ends
Solution values for bL
1.875

and
cos(bL) cosh(bL) + 1 = 0
cos(bL) cosh(bL) - 1 = 0
Resonant frequency for mode is:
4.694
4.694
displacement
clamped-free
free-free
First 3 modes:
, B+D=0
length
0
length
0
length
displacement
displacement
0
12
7.855
7.855
 = ckb2
10.996
10.996
Supplementary notes on musical acoustics.
Example 4: The human voice
Vocalisation in humans has its origin in the glottal interruption of expired air. The harmonic content
and fundamental frequency depend on muscle tension and to some extent on the air flow rate.
Mechanical resonances in the vocal tract modify the amplitudes of the resulting harmonics. In this
way our ears can distinguish between 2 notes sung at the same pitch, but heard as "ah" or "ee" etc.
Figure: Principle of generation of voiced sounds in the human voice. The vocal
fold vibrations convert the steady airstream to a pulsed airstream, the glottal voice
source., corresponding to a complex curve of the vocal tract, which is
characterised by resonances or formants corresponding to peaks. Hence the
partials closest to the formants dominate the radiated spectrum.
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Supplementary notes on musical acoustics.
Human vowels distinguished by the frequencies of the two vocal resonances R1 and R2.
14
Measurements of human hearing:
• Frequency response:
(Fletcher-Munsen curves)
Phon: unit of loudness level. phon = SPL (compared to pure tone at 1000 Hz)
• Age-related loss. The average shifts with age of the threshold of hearing for pure tones of persons
with "normal" hearing (Spoor 1967)
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• Effect of background noise: Rating chart for determining speech communication capability from
speech interference levels.
PSIL: average in dB of the sound pressure levels of a noise in the 3-octave bands of centre
frequency 500, 1000 and 2000 Hz. (1 foot = 305 mm).
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The hearing apparatus
(audiology)
• ear vs microphone
- eardrum
converts varying air pressure into movement of the membrane
- cochlea
frequency selective hairs in the inner ear respond to vibration
and trigger nerve cluster
- CNS
brain receives multiple nerve pulses from different parts of the cochlea
• logarithmic perception
- loudness & dB (volume level)
reference level:
20 µPa
200 µPa
2 mPa
20 mPa
=
=
=
=
0 dB
10 dB
20 dB
30 dB
quietest sound
faint whisper
65 dB
80 dB
100 dB
120 dB
conversation
loud music
'deafening'
pain & damage
- pitch & octaves (the musical scale)
equal distances along human cochlea correspond to equal frequency ratios
pitch: musical term for the fundamental frequency of a continuous sound; A  440 Hz
octave: frequency ratio of 2
secret maths explanation?
a
(**** logb = log(a) - log(b) *****)
 
256 Hz
440 Hz
880 Hz
220 Hz
32 Hz
=
=
=
=
=
C4
A4
A5
A3
C1
!!
middle C
A above
octave above
octave below
3 octaves below
• characterisation
- sensitivity & dynamic range
The human ear is a 'dynamic' sensor with only about 30 dB range but with a variable level
determined by bio-feedback.
- frequency response
This depends on sound level and age, but typically covers from
However, the sense of pitch or melody only extends from about
20 Hz to 18 kHz.
60 Hz to 4 kHz.
Notes by Graeme Yates
revised & extended by Ralph James.
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