(Acoustics)

Wave Superposition & Timbre

General Physics Version

Updated 2014Jul07

Dr. Bill Pezzaglia

Physics CSUEB

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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.

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Daniel Bernoulli1700-1782

3. Example 6

Superposition of Waves

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B. Waveforms

1. Wave Types and Timbre

2. Waveforms of Instruments

3. Modulation

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1. Waveform Sounds

Different “shape” of wave has different “timbre” quality

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Sine Wave (flute)

Square (clarinet)

Triangular (violin)

Sawtooth (brass)

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

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1. Fourier’s Theorem

Any periodic waveform can be constructed from harmonics.

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Joseph Fourier1768-1830

2. FFT: Fast Fourier Transform

• A device which analyzes any (periodic) waveform shape, and immediately tells what harmonics are needed to make it

• Sample output:tells you its mostly10 k Hertz, witha 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:

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Ab

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2c. FFT of a Sawtooth Wave

• Amplitude “A”

• Contains all harmonics “n”

• Amplitude of “n” harmonic is:

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Ab

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2d. FFT of a triangular Wave

• Amplitude “A”

• Contains ODD harmonics “n”

• Amplitude of “n” harmonic is:

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3a. Ohm’s Law of Acoustics 17

• 1843 Ohm's acoustic lawa 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”

Georg Simon Ohm (1789 – 1854)

Octave, in phase

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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 18

• 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.

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.

Octave, phase shifted

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3c. Ohm’s Acoustic Phase Law 19

• 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.

Revision Notes

• Modulation page has been cleaned up.

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D. References

• 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/

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