1. Superposition & Resonance1
(Acoustics)
Superposition & Resonance
General Physics Version
Updated 2015Apr15
Dr. Bill Pezzaglia
Physics CSUEB

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

3. A. Superposition 3
Galileo
Bernoulli
Example

4. 1a. 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. 1b 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. 1c. Example 6

7. 2. Interference 7
Two waves added together can cancel each other out if “out of phase” with each other.
Combined Wave
Wave 1
Wave 2
Coherent waves (in phase) add together to make bigger wave
Waves 180° out of phase will cancel each other!

8. 3a. Beats 8
Two tones closer than 15 Hertz we hear as a “fused” tone (average of frequencies) with a “beat”.
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Demo:

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3b. Modulation
AM: Amplitude Modulation, aka “tremolo”. The loudness is varied (e.g. a beat frequency).
FM: Frequency Modulation aka “vibrato”. The pitch is wiggled

10. 4. Diffraction two sources10
Two wave sources close together (such as two speakers) will create “diffraction patterns”. At certain angles the waves cancel!

11. B. Harmonic Resonance Standing Waves Harmonic Series Air Columns/Pipes11
B. Harmonic Resonance
Standing Waves
Harmonic Series
Air Columns/Pipes
2D Resonance (Plates & Drums)

12. 1. Standing Waves 12
Standing wave is really the sum of two opposing traveling waves (both at speed v)
Makes it easy to measure wavelength

13. 2. Harmonic Modes 13
Daniel Bernoulli (1728?) shows string can vibrate in different modes, which are multiples of fundamental frequency (called “Harmonics” by Sauveur)
n=1 f1 n=2 f2=2f1
n=3 f3=3f1
n=4 f4=4f1
n=5 f5=5f1

14. 3a. Open Pipes 14 Pressure node at both ends
Displacement antinode at both ends
Fundamental wavelength is 2x Length
A two foot pipe approximately hits “middle C” (C4)
All harmonics are present (but higher harmonics are excited only when the air flow is big)

15. 3b. Open Pipe Harmonics 15 All harmonics possible (both even and odd)G2 C3

16. 3c. Closed Pipes 16 Pressure antinode at closed end, node at mouth
Displacement node at closed end, antinode at mouth
Fundamental wavelength is 4x Length
A one foot pipe approximately hits “middle C” (C4)
Only odd harmonics present!

17. 3d. Closed Pipe Harmonics17
3d. Closed Pipe Harmonics
Only ODD harmonics present (n=1, 3, 5, …)
Open pipe
Closed pipe C1 C2 G2 C3

18. 4a Ernst Chladni (1756—1827) 18
First measurement of speed of sound in solids (up to 40x faster than in air!)
Measures speed of sound in different gases (slower in heavier gases)
1787 “Chladni Plate” shows vibration of sound using sand on a plate.

19. 4b Vibration of Rectangular Plate19
Two dimensional vibration
“nodes” (places of no displacement) are now lines

20. 4c. Circular Plate 20
“Nodes” are radial lines and circles

21. 4d. Membranes (Drums) Demos: http://www.falstad.com/membrane/21
Demos:

22. C. Timbre and Fourier Theorem22
C. Timbre and Fourier Theorem
Wave Types and Timbre
Fourier Theorem
Ohm’s law of acoustics

23. 1. Waveform Sounds 23
Different “shape” of wave has different “timbre” quality
Sine Wave (flute)
Square (clarinet)
Triangular (violin)
Sawtooth (brass)

24. 1b. Waveforms of Instruments24
1b. 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.

25. 2a. Fourier’s Theorem 25
Any periodic waveform can be constructed from harmonics.
Joseph Fourier

26. 2b. FFT: Fast Fourier Transform26
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.

27. 2c. FFT of a Square Wave 27 Amplitude “A”
Contains only odd harmonics “n”
Amplitude of “n” harmonic is:

28. 2d. FFT of a Sawtooth Wave 28 Amplitude “A” Contains all harmonics “n”
Amplitude of “n” harmonic is:

29. 2e. FFT of a triangular Wave29
Amplitude “A”
Contains ODD harmonics “n”
Amplitude of “n” harmonic is:

30. 3a. Ohm’s Law of Acoustics30
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.

31. 3b. Ohm’s Acoustic Phase Law31
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.

32. 3c. Ohm’s Acoustic Phase Law32
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
New “physics” version april 15, May need clean up.

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