Download 1843 Ohm`s acoustic law

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(Acoustics)
Wave Superposition & Timbre
General Physics Version
Updated 2014Jul07
Dr. Bill Pezzaglia
Physics CSUEB
Outline
A. Wave Superposition
B. Waveforms
C. Fourier Theory & Ohms law
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A. Superposition
1. Galileo
2. Bernoulli
3. Example
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1. Galileo Galilei (1564 – 1642)
• If a body is subjected to two
separate influences, each
producing a characteristic type of
motion, it responds to each without
modifying its response to the
other.
• In projectile motion, for example,
the horizontal motion is
independent of the vertical motion.
• Linear Superposition of Velocities:
The total motion is the vector sum
of horizontal and vertical motions.
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2 Bernoulli’s Superposition principle 1753
• The motion of a string
is a superposition of its
characteristic
frequencies.
• When 2 or more waves
pass through the same
medium at the same
time, the net
disturbance of any
point in the medium is
the sum of the
disturbances that
would be caused by
each wave if alone in
the medium at that
point.
Daniel Bernoulli
1700-1782
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3. Example
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Superposition of Waves
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-6.0
Time (seconds)
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Displacement
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B. Waveforms
1. Wave Types and Timbre
2. Waveforms of Instruments
3. Modulation
1. Waveform Sounds
Different “shape” of wave has different “timbre” quality
Sine Wave (flute)
Square (clarinet)
Triangular (violin)
Sawtooth (brass)
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2. Waveforms of Instruments
•
Helmholtz resonators (e.g. blowing on a bottle)
make a sine wave
•
As the reed of a Clarinet vibrates it
open/closes the air pathway, so its either “on”
or “off”, a square wave (aka “digital”).
•
Bowing a violin makes a kink in the string, i.e.
a triangular shape.
•
Brass instruments have a “sawtooth” shape.
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3. Modulation
• AM: Amplitude
Modulation, aka
“tremolo”. The
loudness is varied (e.g.
a beat frequency).
• FM: Frequency
Modulation aka
“vibrato”. The pitch is
wiggled
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C. Fourier Theory
1. Fourier’s Theory
2. FFT: Frequency analyzers
3. Ohm’s law of acoustics
1. Fourier’s Theorem
Any periodic
waveform
can be
constructed
from
harmonics.
Joseph Fourier
1768-1830
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2. FFT: Fast Fourier Transform
•
A device which analyzes any (periodic)
waveform shape, and immediately tells
what harmonics are needed to make it
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Sample output:
tells you its mostly
10 k Hertz, with
a bit of 20k, 30k, 40k,
etc.
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2b. FFT of a Square Wave
• Amplitude “A”
• Contains only odd harmonics “n”
• Amplitude of “n” harmonic is:
b1
bn 
n
4
b1  A

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2c. FFT of a Sawtooth Wave
• Amplitude “A”
• Contains all harmonics “n”
• Amplitude of “n” harmonic is:
b1
bn 
n
1
b1   A

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2d. FFT of a triangular Wave
• Amplitude “A”
• Contains ODD harmonics “n”
• Amplitude of “n” harmonic is:
b1
bn  2
n
4
b1  A ?

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3a. Ohm’s Law of Acoustics
1843 Ohm's acoustic law
a musical sound is perceived by the ear as a set of a
number of constituent pure harmonic tones, i.e. acts as
a “Fourier Analyzer”
Octave, in phase
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-4.0
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Phase (Degrees)
Georg Simon Ohm
(1789 – 1854)
For example:, the ear does not really “hear” the combined
waveform (purple above), it “hears” both notes of the
octave, the low and the high individually.
3b. Ohm’s Acoustic Phase Law
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Hermann von Helmholtz elaborated the law (1863?)
into what is often today known as Ohm's acoustic law,
by adding that the quality of a tone depends solely on
the number and relative strength of its partial simple
tones, and not on their relative phases.
Octave, phase shifted
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Displacement
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Phase (Degrees)
Hermann von Helmholtz
(1821-1894)
The combined waveform here looks completely different,
but the ear hears it as the same, because the only
difference is that the higher note was shifted in phase.
3c. Ohm’s Acoustic Phase Law
• Hence Ohm’s acoustic law favors the “place”
theory of hearing over the “telephone” theory.
• Review:
– The “telephone theory” of hearing (Rutherford,
1886) would suggest that the ear is merely a
microphone which transmits the total waveform to
the brain where it is decoded.
– The “place theory” of hearing (Helmholtz 1863,
Georg von Békésy’s Nobel Prize): different pitches
stimulate different hairs on the basilar membrane of
the cochlea.
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Revision Notes
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Modulation page has been cleaned up.
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D. References
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Fourier Applet (waveforms) http://www.falstad.com/fourier/
http://www.music.sc.edu/fs/bain/atmi02/hs/index-audio.html
Load Error on this page?
http://www.music.sc.edu/fs/bain/atmi02/wt/index.html
FFT of waveforms: http://beausievers.com/synth/synthbasics/