1. Wave Superposition & Timbre1
(Acoustics)
Wave Superposition & Timbre
General Physics Version
Updated 2014Jul07
Dr. Bill Pezzaglia
Physics CSUEB

2. 2
Outline
Wave Superposition
Waveforms
Fourier Theory & Ohms law

3. A. Superposition 3
Galileo
Bernoulli
Example

4. 1. Galileo Galilei (1564 – 1642) 4
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.

5. 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

6. 3. Example 6

7. B. Waveforms Wave Types and Timbre Waveforms of Instruments Modulation7
B. Waveforms
Wave Types and Timbre
Waveforms of Instruments
Modulation

8. 1. Waveform Sounds 8
Different “shape” of wave has different “timbre” quality
Sine Wave (flute)
Square (clarinet)
Triangular (violin)
Sawtooth (brass)

9. 2. Waveforms of Instruments9
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

11. C. Fourier Theory Fourier’s Theory FFT: Frequency analyzers11
C. Fourier Theory
Fourier’s Theory
FFT: Frequency analyzers
Ohm’s law of acoustics

12. 1. Fourier’s Theorem 12
Any periodic waveform can be constructed from harmonics.
Joseph Fourier

13. 2. FFT: Fast Fourier Transform13
A device which analyzes any (periodic) waveform shape, and immediately tells what harmonics are needed to make it
Sample output: tells you its mostly 10 k Hertz, with a bit of 20k, 30k, 40k, etc.

14. 2b. FFT of a Square Wave 14 Amplitude “A”
Contains only odd harmonics “n”
Amplitude of “n” harmonic is:

15. 2c. FFT of a Sawtooth Wave 15 Amplitude “A” Contains all harmonics “n”
Amplitude of “n” harmonic is:

16. 2d. FFT of a triangular Wave16
Amplitude “A”
Contains ODD harmonics “n”
Amplitude of “n” harmonic is:

17. 3a. Ohm’s Law of Acoustics17
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”
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.

18. 3b. Ohm’s Acoustic Phase Law18
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 ( )
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.

19. 3c. Ohm’s Acoustic Phase Law19
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.

20. 20
Revision Notes
Modulation page has been cleaned up.

21. 21
D. References
Fourier Applet (waveforms)
Load Error on this page?
FFT of waveforms: