Download Physics 193 Physics of Music The Ear

UIUC Physics 193 POM Physics of Music/Musical Instruments
Physics 193 Physics of Music
Lecture Notes Week V
[Chapter V ⎯ Acoustical Foundations of Music]
The Ear ⎯ Hearing and Loudness (Intensity) Levels
We’ve been discussing generation of sounds, so now we’ll discuss perception of sounds.
Human Perception of Sound:
⎯ 2 ears → stereo/3-D location of sounds
⎯ Pinpoint accuracy in 3-D for sounds in the f ∼ 100 Hz → ~ 1.5 KHz frequency range!
⎯ Mechanical & auditory sensory structure of ear preserves/is sensitive to/utilizes phase
information over this frequency range.
⎯ Our brains process/use frequency, amplitude and phase information.
Human Ear has 3 Basic Parts:
⎯ Outer ear – pinna concentrates sound waves into ear canal
⎯ Middle ear – eardrum, ossicles (hammer, anvil, stirrup)
⎯ Inner ear – cochlea (& semi-circular canals – needed only for balance/orientation)
The Outer Ear:
⎯ Pinna (ear flap) concentrates/focuses sound waves into ear canal.
⎯ We used to have moveable ears (like donkeys), not so much moveable, nowadays!!!
⎯ Auditory/ear canal is ~ 3cm long, closed at the inner ear at the eardrum – a membrane.
Auditory canal = organ pipe (~ 3cm long), with one end open, and one end closed
(Thus, there will be resonances in the ear canal!!!).
The Middle Ear:
⎯ Ossicular chain – hammer, anvil, stirrup; transmits sound vibrations on ear drum to
oval window of cochlea. Ossicles are a lever system, w/ 1.3× mechanical advantage.
⎯ Also has eustachian tube for pressure equalization (and drainage of fluids…).
⎯ Eardrum ~ 10× bigger than oval window → overall amplification factor is ~ 13×.
-1©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002 - 2013. All rights reserved.
UIUC Physics 193 POM Physics of Music/Musical Instruments
The Inner Ear:
⎯ Cochlea – coiled/rolled up (~ 2 ¾ turns), filled with perilymph fluid.
⎯ Cochlea is divided down its length by a soft partition known as the basilar membrane,
forming 2 long chambers connected together by an opening at the far end called the
helicotrema.
⎯ ~ 30,000 nerve endings – hair cells embedded in the basilar membrane sense acoustic
disturbances/vibrations in perilymph fluid and transmit information to brain via the
auditory nerve.
⎯ 3 semi-circular canal(s) attached to cochlea = 3-axis (x,y,z) accelerometers (!)
used for balance/orientation determination/inertial sensing…
⎯ Sound waves propagate on basilar membrane in perilymph fluid.
⎯ Amplitude of sound waves in perilymph fluid reaches a maximum at a particular point
along the cochlea for a particular frequency!
⎯ Sense of pitch (i.e. human perception of frequency) depends (in part) on location
along cochlea:
⎯ Spiral shape of cochlea enhances sensitivity to low-frequency sounds by ~ 20 dB!
D. Manoussaki, et al., Phys. Rev. Lett. 96, 088701 (2006)
-2©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002 - 2013. All rights reserved.
UIUC Physics 193 POM Physics of Music/Musical Instruments
Auditory signals transmitted to the brain undergo much additional processing there.
⎯ Speech sounds: mostly processed in the left hemisphere of the brain.
⎯ Music sounds: mostly processed in the right hemisphere of the brain.
Binaural Sound Perception:
dears
Ears are separated by dears ~ 6” (= 1/2 foot, or ~ 15 cm)
vair = f λ so if λ ∼ dears ∼ 1 foot
2
= 15 cm
= 0.15 m
and vair = 344 m s
v
344 m s
then f ~ air ~
λ
0.15 m
2300 Hz
For frequencies much higher than this it is hard for us to
localize a sound source (aka binaural fusion).
Also true when λ dear (i.e. very low frequencies) – due to diffraction of low
frequency sound waves.
Practically, studies (we and others have performed) show that humans can localize
sounds provided 100 Hz ≤ f ≤ 1500 Hz . Note that this is in air, with v = 344 m s .
Compare with our (poor) ability to localize sounds in water, where vH 2O
1500 m s
Arrival time differences of sound waves (left – right) ears is
L− R
Δt H
O
2
L− R
L− R
⇒ much harder for us humans to localize
~ 0.2Δtair
(344 1500)Δtair
sounds underwater! However, many fish & other marine creatures (e.g. dolphins) can
easily localize sounds underwater – because their hearing has been adapted/optimized to
propagation of sound waves in water with speed vH 2O 1500 m s !
Sound Intensity & Hearing:
The human ear is extremely sensitive to detecting sounds (n.b. dogs hear even better!)
Sound Intensity, I (Watts/m2) is proportional to the square of the pressure amplitude p2.
The (average) Intensity threshold of hearing (@ f = 1 KHz) is <Ithr> ~ 2.5×10−12 Watts/m2
(= 2.5 pico-Watts/m2)
Some people hear better/worse than others, and so the threshold of hearing from one
person to another can vary as much as 1/10 or 10× this!!!
We choose as the reference standard: I o = 10−12 Watts m2 (defined at f = 1 KHz) as the
official sound intensity threshold of hearing.
-3©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002 - 2013. All rights reserved.
UIUC Physics 193 POM Physics of Music/Musical Instruments
The human ear responds ≈ logarithmically to the intensity I of a sound.
∴ We define Loudness ≡ Sound Intensity Level SIL = LI which is proportional to the
{base-10} logarithm of the intensity, I {which in turn is proportional to the square of the
pressure amplitude p2}:
⎛ p 2 ⎞⎟
⎞⎟
⎛ p ⎟⎞2
⎛ p⎞
⎜
⎟
⎟⎟ = 10 log10 ⎜⎜ ⎟ = 10 log10 ⎜⎜ ⎟⎟ = 20 log10 ⎜⎜ ⎟⎟⎟
⎜⎜⎝ p ⎟⎠
⎜⎝ p 2 ⎟⎠⎟
⎝⎜⎜ po ⎟⎠
o ⎠⎟
o
o
⎛I
SIL = LI ≡ 10 log10 ⎜⎜⎜
⎜⎝ I
Unit of Loudness/Intensity Level is the deci-Bel (dB) { e.g. deci = 10}.
Use the fact that:
In honor of Alexander
Graham Bell
log10 A − log10 B = log10 ( A B )
L = L2 – L1 = Difference in two Loudnesses (= Difference in two Intensity Levels).
L1 = 10 log10 ( I1 I 0 )
L2 = 10 log10 ( I 2 I 0 )
L = L2 − L1 = 10 log10 ( I 2 I 0 ) −10 log10 ( I1 I 0 )
= 10 ⎡⎣ log10 ( I 2 I 0 ) − log10 ( I1 I 0 )⎤⎦
= 10 ⎡⎢(log10 I 2 − log10 I 0 ) − (log10 I1 − log10 I o )⎤⎥
⎣
⎦
= 10 [log10 I 2 − log10 I1 ]
= 10 log10 ( I 2 I1 )
If I2 = 2I1 then:
LI = L2 − L1 = 10 log10 (2) = 10*0.301 = 3.01
3 dB
i.e. there is only LI ≈ 3 dB difference in loudness/sound intensity levels for 2 sounds
which differ by a factor of 2× in intensity, I2 = 2I1.
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UIUC Physics 193 POM Physics of Music/Musical Instruments
Sound Pressure Level: (SPL = Lp = {Over-}Pressure Intensity Level)
SPL = L p = 20 log10 ( p po ) (in deci-Bels (dB))
p = rms (root-mean-square) of sound (over-)pressure amplitude (in Pascals = Newtons/m2)
p(rms) ≡
1
p (peak) { ⇒ p2(rms) = ½ p2(peak)}
2
po ≡ rms sound over-pressure amplitude corresponding to a sound intensity threshold
of I 0 = 10−12 Watts m 2
po = 2×10−5 Newtons m2 (= Pascals )
{1 Pascal = 1 Newton m2 }
For a (single) sound wave moving in one direction in free air (i.e. outdoors) SPL = Lp
and L are essentially the same numerical values in dB and typically are within 0.1 dB of
each other. However, e.g. inside an auditorium, due to sound reflection from the walls /
ceiling/floor (creating multiple sound waves), Lp and L will not necessarily be the same.
Most microphones are such that they are sensitive to pressure, (not displacement) and
hence can be used to measure the sound pressure level, SPL (not Sound Intensity Level, LI).
A device that measures SPL is known as a sound level meter ⎯ the results of SPL
measurement by this device are also weighted by the average frequency response of the
human ear. See/show Extech SPL meter…
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UIUC Physics 193 POM Physics of Music/Musical Instruments
The change in SPL that produces a Just Noticeable Difference (JND) is:
JND ≡ min ΔSPL = min ( SPL1 − SPL2 ) ~ 1 dB .
However, in fact the JND is frequency dependent, i.e. JND ( f ) ≡ min ΔSPL ( f )
The JND is also sound pressure level dependent! See graph below:
Musicians have defined six different loudness levels:
fortissimo
pianissimo
fff
ff
f
p
pp
ppp
I = 10−2 W/m2
10−3
10−4
10−6
10−7
10−8
6
LI (dB) = 100
90
80
60
50
40
UIUC Physics 193 POM Physics of Music/Musical Instruments
Loudness Level: Phons
Phon is the perceived value of SPL (in dB) that has constant loudness for the “average”
human being:
Note that a frequency f = 1000 Hz:
Phons ≡ SPL
Present-day Sound Level Meters have 3 weighting networks:
A−weighting: 40 Phon curve of above figure. Units: dB-A SPL
B−weighting: 70 Phon curve of above figure. Units: dB-B SPL
C−weighting: flat, independent of frequency. Units: dB-C SPL
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UIUC Physics 193 POM Physics of Music/Musical Instruments
Adding (Uncorrelated) Sounds:
Two sounds with intensity levels L1 and L2 (at the same frequency)
L1 = 70 dB and L2 = 80 dB (@ f = 1000 Hz)
Now, note that L2 = 80 dB corresponds to I2 = 10 I1:
⎛I ⎞
L1 = 70 = 10 log10 ⎜⎜⎜ 1 ⎟⎟⎟
⎜⎝ I 0 ⎟⎠
⎛I ⎞
7 = log10 ⎜⎜⎜ 1 ⎟⎟⎟
⎜⎝ I 0 ⎠⎟
⎛I ⎞
107 = ⎜⎜⎜ 1 ⎟⎟⎟
⎜⎝ I ⎟⎠
0
⎛I ⎞
L2 = 80 = 10 log10 ⎜⎜⎜ 2 ⎟⎟⎟
⎜⎝ I 0 ⎟⎠
⎛I ⎞
8 = log10 ⎜⎜⎜ 2 ⎟⎟⎟
⎜⎝ I 0 ⎟⎠
⎛I ⎞
108 = ⎜⎜⎜ 2 ⎟⎟⎟
⎜⎝ I ⎟⎠
7
−12
0
8
I1 = 10 I 0 = 10 ∗10
I 2 = 10 I 0 = 10 ∗ 10−12
= 10−5 W 2
= 10−4 W 2
m
m
I 2 = 10 I1 or I1 = 1 I 2
10
7
8
Rule: For uncorrelated sounds, must add intensities, NOT loudnesses
{If sounds are not correlated → no phase coherence → no interference}
So then:
Isum = ITOTAL = I1 + I2 {If sounds are correlated, then must add
= I1 + 10 I1 = 11 I1 amplitudes → interference effect(s)}
Then:
⎛11I ⎞
Lsum = 10 log10 ⎜⎜⎜ 1 ⎟⎟⎟
⎝⎜ I 0 ⎟⎠
⎛I ⎞
= 10 log10 ⎜⎜⎜ 1 ⎟⎟⎟ + 10 log10 (11)
⎜⎝ I 0 ⎟⎠
= 70 dB + 10.4 dB
= 80.4 dB only slightly louder than 80 dB !!!
Adding Correlated Sounds:
If sounds are correlated (i.e. have a stable phase relation to each other)
then must add sounds together at amplitude level
Suppose the two {longitudinal} amplitudes are A1 and A2
“Atot = A1 + A2” but Atot depends on phase relation between A1 and A2
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UIUC Physics 193 POM Physics of Music/Musical Instruments
Use phasor diagram to calculate Atot:
A2
Atot
A1
π−δ
δ = phase angle between
A1 and A2
δ
c
Use the law of cosines:
θ
b
a
c 2 = a 2 + b 2 − 2ab cos θ
θ = (π − δ )
2
Atot
= A12 + A22 − 2 A1 A2 cos (π − δ )
Use the trig identity:
cos (π − δ ) = cos π cos δ + sin π sin δ = − cos δ
=−1
∴
2
Atot
= A12 + A22 + 2 A1 A2 cos δ
=0
~ ITot
or: Itot = I1 + I 2 + 2 I1 I 2 cos δ
Atot = A12 + A22 + 2 A1 A2 cos δ
2
But total intensity, IToT ~ Atot
{i.e. sound intensity I is proportional to square of the of the total amplitude |Atot|2.}
Relation between Loudness (Sones units) and Loudness Level (Phons units):
Straight line relation on semi-log plot (like y = mx + b):
log ( L ( sones )) = m L ( phons ) + b
slope
intercept
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UIUC Physics 193 POM Physics of Music/Musical Instruments
Critical Band:
Two sounds, slightly different in frequency are not heard as separate notes if they are
too close together in frequency.
Reason: mechanical vibrational behavior of basilar membrane
Sound of a particular frequency produces a traveling wave - which propagates along
the basilar membrane. Amplitude of this wave in the perilymph fluid is not a constant ⎯
it peaks somewhere along the basilar membrane; the position of where it peaks depends
on frequency of sound wave in the perilymph fluid (see lecture notes above).
The disturbance produced by the wave is spread out over a certain length of basilar
membrane.
The hair cells/nerve endings on the basilar membrane are excited over a narrow region
on either side of maximum amplitude of motion of basilar membrane. The range or band
of frequencies affected is known as the critical band. At center frequencies fctr ≤ 200 Hz,
the width of the critical band is Δfcrit ~ 90 Hz and increases to a width of Δfcrit ~ 900 Hz
@ fctr ~ 5000 Hz, as shown in the figure below:
900
800
Width of the
Critical Band
Δfcrit (Hz)
700
600
500
400
300
200
100
200
5000
Center Frequency fctr (Hz)
n.b. The width of the critical band is also dependent on sound intensity.
This effect is ONLY for ONE ear
⎯ it does not exist if one frequency is input to one ear, and another, different frequency
is input into the other ear. (this doesn’t happen in nature through…)
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UIUC Physics 193 POM Physics of Music/Musical Instruments
Masking:
Superposition of two (or more) sound signals can make it difficult to “decode” one of
them, e.g. listening to a friend talk to you in a crowded/noisy room – known as masking.
Masking problem increases with age and with hearing damage (exposure to loud noises).
e.g. a L = 60 dB, f = 1200 Hz masking tone.
See curve below for the JND vs. frequency for 1200 Hz masking tone vs. intensity
level of masking tone. One can see the effect of the critical band and also see effect of
2nd harmonic (difference frequency, Δf = f – fmask = 1200 Hz) – due to quadratic nonlinear response term(s) in our ear (and/or brain)! {See UIUC P498POM lecture notes on
distortion for details}
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UIUC Physics 193 POM Physics of Music/Musical Instruments
Hearing Loss/Disorders:
At birth, humans can hear over the frequency range 20 Hz – 20 KHz, but as we grow
older, we experience “natural” hearing loss, particularly at high frequencies
Very loud sounds can temporarily (and also permanently) damage sensitive hearing
nerves in the cochlea.
⎯ tinitis – ringing in the ears
⎯ ear infections can also lead to hearing loss, especially in young children, infants.
⎯ loud explosions (artillery shells – military)
⎯ exposure time * loudness level is determining factor.
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UIUC Physics 193 POM Physics of Music/Musical Instruments
Brain Processing of Sound Signals:
Signals from basilar membrane in cochlea sent to various areas of the brain – at the
frequenc(ies) of the acoustical signals incident on the ears (!!!). From recent technological
advances – development of functional Magnetic Resonance Imaging (fMRI) – researchers have
learned that a great many areas of the brain simultaneously process these sound signals –
underscoring the importance hearing (& music) to human beings. If interested in learning more
about this, the recently-published book “This is Your Brain on Music – the Science of Human
Obsession” by Daniel J. Levitin is highly recommended – see his website:
http://www.yourbrainonmusic.com/
Because many areas of the brain process sound signals, there are also many possible ways for
brain to malfunction. Again, researchers have learned a great deal on this via fMRI studies over
the past decade. If interested in this subject, the recently published book “Musicophilia – Tales of
Music and the Brain” by Oliver Sacks, MD is highly recommended – see his website:
http://musicophilia.com/
NOTE:
Table and (most) Figures are taken from the course text “The Acoustical Foundations of
Music” by John Backus, second Edition.
Questions:
1.) A SPL meter is used to measure the sound pressure level of a sound source, which is
SPL = 94.0 dB. What is the (rms) pressure amplitude corresponding to this SPL level?
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UIUC Physics 193 POM Physics of Music/Musical Instruments
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