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Since last midterm:
1. the decibel scale
2. resonances
3. normal vibration modes (standing waves)
• strings
• tubes
3. human hearing
The decibel scale
(a way of measuring loudness)
Intensity (I) ~
(pressure difference)2
W/m2 = J/m2s
I
SIL  10 log
I0
where I 0  10
12
W
m2
minimum
audible sound
The only problem are the …
Logarithms
1
 0.01
10 10
1
101 
 0.1
10
100  1
102 
Powers
of 10
101  10
102  10 10  100
103  10 10 10  1000
1
10 
 0.01  log 0.01  2
10 10
1
1
10 
 0.1  log 0.1  1
10
100  1  log1  0
2
101  10 
log10  1
102  10 10  100 
log100  2
10  10 10 10  1000 
3
log1000  3
log ab = log a + log b
log 1/a = -log a
NO
CALCULATORS
DURING THE
MIDTERM
The point of using the decibel scale is the
Weber-Fetchner “Law”
A doubling of volume feels like the
same increase, regardless of how
much increase in intensity actually
occurred.
Thresholds for hearing and pain
i
c
h
Environmental
Noise
Weakest sound heard
0dB
s
Normal conversation (3-5') u
60-70dB
s
Normal piano
60-70dB
t
Telephone dial tone
80dB
a
i
City Traffic (inside car)
85dB
n
Walkman on 5/10
94dB
e
Subway train at 200‘
95dB
d
Level at which sustained exposure may result in hearing
e
9
loss
90 - 95dB
x
0
p
Power mower
107dB
o
Symphonic music peak
120-137dB
s
Pain begins
125dB
u
9
r
Jet engine at 100‘
140dB
5
e
d
Rock concert peak
150dB
B
m
Death of hearing tissue
180dB
a
Normal modes of vibration,
standing waves
If you bang on an object, it will vibrate in a
complicated way. But this complicated
motion is a superposition of NORMAL
MODES (just like a complicated sound can
be decomposed into simple sine waves).
Normal modes of strings
(standing waves)
fundamental
3rd harmonic
2nd harmonic
4th harmonic
Animation courtesy of Dr. Dan Russell, Kettering University
Standing waves are a superposition of
two counter moving waves
Animation courtesy of Dr. Dan Russell, Kettering University
Frequencies of standing modes of a string
L
1 = 2 L
2 = L
f1 = v/1 =
v/(2L)
f2 = v/2 = v/L=2 f1
…
How the velocity depends on the
string: Mersenne’s laws
v
F 1
f1 

2L
W 2L
fundamental
frequency
tension
mass per
length
length
Other objects have their normal modes too:
Square membrane:
Circular membrane:
Bottle of beer:
Standing sound waves in air tubes
This is not a
string now, it’s
the graph of
the pressure x
distance
air tubes x strings
vstring
nodes at
the ends
vsound
nodes or
antinodes at
the ends
closed end
pressure
displacement
open end
/4
Voice
Anatomy
The sound wave produces by the vocal chords
contains many frequencies that may or may
not be enhanced by the resonances (formants)
of the vocal tract
6dB/octave
Formants stay fixed as pitch changes
1. vocal chords vibrate with a given
frequency
2. formants enhance some of the
overtones (harmonics)
3. different formats = different vowels
4. consonants are formed with non-steady
changes in lips, tongue, …
Hearing
1.
Physiology
2. Place Theory
3. Psychophysics of hearing
•
•
•
Fundamental tracking
Aural harmonics
Sheppard tones and pitch perception
4. Sound localization
•
interaural level difference
• interaural time difference
•
head-transfer function
3.2 mm2
55 mm2
Uncoiled cochlea (schematic)
stiffer
http://www.howstuffworks.com/hearing1.htm
limber
Cross section of cochlea
Two frequencies f and 2f
(one octave)
3.5 mm
“same” interval corresponds
to the same frequency ratio
(fixed distance along the
cochlea)
excited hair
cells
distance along the
basilar membrane
sharpening
The amount of sharpening determined
the just noticeable difference in
frequencies
frequency up and down
by 0.001 = 0.1%
frequency up and down
by 0.005 = 0.5%
Fundamental tracking: the absence of the fundamental
does not change the perceived pitch
note D
note D minus fundamental
note D minus fundamental and 2nd
harmonic
Aural harmonics
sin(2p 50 t)
sin(2p 50 t)+ 0.2 sin(2p 100
t) +0.1 sin(2p 150 t) +…
extra frequencies
“aural harmonics”
400Hz, 400Hz+802Hz, 400Hz+1202Hz
Shepard tones
Sound localization
How do we know where the sound is coming from ?
• interaural level differences (ILD)
• interaural time differences (ITD)
• head-related transfer function (HRTF)
http://www.aip.org/pt/nov99/locsound.html
Interaural level difference:
one ear will be on the shadow cast by the head
we can
detect even
0.5 dB in
ILD
diffraction makes it
ineffective at low
frequencies
Interaural time difference: peaks and through will
arrive at ears at different times
t ~ L/v ~ (0.15 m)/(340m/s) ~ 0.0005 s
difference in
arrival time
distance
between ears
much shorter than
synaptic delays !
Phase ambiguity:
l/2=10 cm, f=340 m/s /0.2 m = 1700 Hz
distance between ears
300 Hz:
2000 Hz:
Head-related transfer function: includes the reflection,
refraction and diffraction from ears, chest, head, …