(Acoustics)

Superposition & Resonance

General Physics Version

Updated 2015Apr15

Dr. Bill Pezzaglia

Physics CSUEB

1

Outline

A. Wave Superposition

B. Waveforms

C. Fourier Theory & Ohms law

2

A. Superposition

1. Galileo

2. Bernoulli

3. Example

3

1a. 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.

4

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.

5

Daniel Bernoulli1700-1782

1c. Example 6

Superposition of Waves

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0

0

20

40

60

80

10

0

12

0

14

0

16

0

18

0

20

0

22

0

24

0

26

0

28

0

30

0

32

0

34

0

36

0

Time (seconds)

Dis

pla

cem

ent

2. Interference

Two waves added together can cancel each other out if “out of phase” with each other.

7

CombinedWave

Wave 1

Wave 2

Coherent waves (in phase) add together to make bigger wave

Waves 180° out of phase will cancel each other!

3a. Beats• Two tones closer than 15 Hertz we hear as a “fused”

tone (average of frequencies) with a “beat”.

8

Demo: http://www.phys.unsw.edu.au/jw/beats.html#sounds

400401

400403

400410

400420

400440

400450

400480

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

9

4. Diffraction two sources

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

10

B. Harmonic Resonance

1. Standing Waves

2. Harmonic Series

3. Air Columns/Pipes

4. 2D Resonance (Plates & Drums)

11

1. Standing Waves • Standing wave is really the sum of two

opposing traveling waves (both at speed v)

• Makes it easy to measure wavelength

12

2. Harmonic Modes• Daniel Bernoulli (1728?) shows string can vibrate in

different modes, which are multiples of fundamental frequency (called “Harmonics” by Sauveur)

13

n=1 f1

n=2 f2=2f1

n=3 f3=3f1

n=4 f4=4f1

n=5 f5=5f1

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)

3b. Open Pipe Harmonics

• All harmonics possible (both even and odd)

15

n

Ln

2

C1

C2

G2

C3

3c. Closed Pipes

• 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!

16

3d. Closed Pipe Harmonics

• Only ODD harmonics present (n=1, 3, 5, …)

17

n

Ln

4

Open pipe Closed pipe

C1

C2

G2

C3

4a Ernst Chladni (1756—1827)

• 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.

18

4b Vibration of Rectangular Plate

Two dimensional vibration

“nodes” (places of no displacement) are now lines

19

4c. Circular Plate

“Nodes” are radial lines and circles

20

4d. Membranes (Drums) 21

Demos: http://www.falstad.com/membrane/ http://www.falstad.com/circosc/index.html

C. Timbre and Fourier Theorem

1. Wave Types and Timbre

2. Fourier Theorem

3. Ohm’s law of acoustics

22

1. Waveform Sounds

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

23

Sine Wave (flute)

Square (clarinet)

Triangular (violin)

Sawtooth (brass)

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.

24

2a. Fourier’s Theorem

Any periodic waveform can be constructed from harmonics.

25

Joseph Fourier1768-1830

2b. 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.

26

2c. FFT of a Square Wave

• Amplitude “A”

• Contains only odd harmonics “n”

• Amplitude of “n” harmonic is:

27

Ab

n

bbn

4

1

1

2d. FFT of a Sawtooth Wave

• Amplitude “A”

• Contains all harmonics “n”

• Amplitude of “n” harmonic is:

28

Ab

n

bbn

1

1

1

2e. FFT of a triangular Wave

• Amplitude “A”

• Contains ODD harmonics “n”

• Amplitude of “n” harmonic is:

29

?4

1

21

Ab

n

bbn

3a. Ohm’s Law of Acoustics 30

• 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

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0

0

20

40

60

80

10

0

12

0

14

0

16

0

18

0

20

0

22

0

24

0

26

0

28

0

30

0

32

0

34

0

36

0

Phase (Degrees)

Dis

pla

cem

ent

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 31

• 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

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0

0

20

40

60

80

10

0

12

0

14

0

16

0

18

0

20

0

22

0

24

0

26

0

28

0

30

0

32

0

34

0

36

0

Phase (Degrees)

Dis

pla

cem

ent

3c. Ohm’s Acoustic Phase Law 32

• 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

• New “physics” version april 15, 2015. May need clean up.

33

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/

34