03 Wave Superposition

Updated 2012Apr19

Dr. Bill Pezzaglia

OutlineA. Superposition

1. Galileo2. Bernoulli3. Example

B. Diffraction1. Interference of Sound2. Huygen’s Principle of wave propagation3. Diffraction through a slit (e.g. sound through a doorway)

C. Resonance1. Mersenne’s Laws2. Harmonic Series3. Quality of Sound

D. References

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A. Superposition

1. Galileo

2. Bernoulli

3. Example

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1.a 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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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.

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Daniel Bernoulli1700-1782

3. Example

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Superposition of Waves

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B. Diffraction & Interference

1. Huygen’s Principle

2. Interference

3. Young slit diffraction

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1. Huygen’s Principle (1678) 8

2. Interference

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

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CombinedWave

Wave 1

Wave 2

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

Waves 180° out of phase will cancel each other!

3. Diffraction Patterns

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

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C. Resonance

1. Mersenne’s Laws

2. Standing Waves

3. Quality of Sound

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Angers Bridge

Angers Bridge was a suspension bridge over the Maine River in Angers, France. The bridge is famous for having collapsed on April 15, 1850, when 478 French soldiers marched across it in lockstep. Since the soldiers were marching together, they caused the bridge to vibrate and twist from side to side, dislodging an anchoring cable from its concrete mooring. 226 soldiers died in the river below the bridge.

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1. Mersenne’s Laws

a. Frequency and String Length

b. Frequency and Tension

c. Frequency and String Mass

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a. Frequency & Length

(i). Consider a string under tension• “plucking” the string causes it to vibrate

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(ii). Frequency and String Length

• Pythagoras of Samos (569-475 BC) found that if you put your finger midway on the string, the string would sing an octave higher (i.e. double the frequency).

• Fundamental Frequency

• One octave higher(double frequency) ishalf the wavelength

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(iii). Frequency inversely proportional to length

• Frequency is inversely proportional to string length

• More generally:Frequency is inverselyproportional to wavelength

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Lf

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1

f

b. Frequency and String Tension

• Vincenzo Galilei, the father of Galileo Galilei, was an Italian lutenist, composer, and music theorist.

• Vincenzo determined: To double the frequency of a violin string, one must quadruple the tension!

• Hence:

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Vincenzo Galilei(1520-1591)

L

Ff

c. Frequency and Mass

(i). Mersenne (1630) states:

• Frequency is inversely proportional to the diameter d of the string

• Putting it all together:

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Ld

Ff

(1588-1648)Marin Mersenne

“The Father of Acoustics”

(ii). Guitar String Diameters For guitar, all strings same length, and want tensions the same, so to get different frequencies, must vary diameters of strings

String Diameter FreqE4 0.010 mm 330 HzB3 0.013 247G3 0.017 196D3 0.026 147A2 0.036 111E2 0.046 82.5

•E2 is 2 octaves lower than E4•Or (1/4) the frequency•Hence diameter is nearly 4x bigger!

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Ld

Ff

(iii). Frequency and Mass

Mersenne (1630) further states:

Frequency is inversely proportional:

• to the root of the mass

• Or to the root of the mass density

Putting it together,

(=Mass per length)

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F

Lf

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(iv). Guitar String Masses For guitar, all strings same length, and want tensions the same, so to get different frequencies, the masses of strings must be different

String gm/cm FreqE4 0.0057 330 HzB3 0.0101 247G3 0.0209 196D3 0.0375 147A2 0.0656 111E2 0.1017 82.5

•E2 is 2 octaves lower than E4•Or (1/4) the frequency

•Hence mass bigger by nearly a factor of 16

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F

Lf

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2. Standing Waves

a. Wavespeed Formula

b. Harmonic Modes

c. Open and Closed pipes

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a. Wavespeed

• Frequency “f” : oscillations per second (Hertz)• Wavespeed “c”: is frequency x wavelength

c = f

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a.2 Standing Waves • Standing wave is really the sum of two

opposing traveling waves (both at speed v)

• Makes it easy to measure wavelength

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

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

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n=1 f1

n=2 f2=2f1

n=3 f3=3f1

n=4 f4=4f1

n=5 f5=5f1

b.2. Wavelengths of Harmonic Modes

• The wavelength of n-th mode is:

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n

Ln

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L2

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L

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L

b.3. Harmonic Series

The musical notes of harmonic series

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Reference: http://www.music.sc.edu/fs/bain/atmi02/hs/index-audio.htmlSound: http://www.music.sc.edu/fs/bain/atmi02/hs/playback/partials/hs1-12-c.mov

c.1. Open Pipes 28

• 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)

c.2. Open Pipe Harmonics

• All harmonics possible (both even and odd)

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n

Ln

2

C1

C2

G2

C3

c.3. 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!

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c.4. Closed Pipe Harmonics

• Only odd n harmonics

• N=1 =4L

• N=3 =4L/3

• N=5 =4L/6

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n

Ln

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C4

G5

E6

3. Quality of Sound

a. Waveforms

b. Fourier’s Theorem

c. Ohm’s Law of Acoustics

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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. Fourier’s Theorem

Any periodic waveform can be constructed from harmonics.

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Joseph Fourier1768-1830

3. Ohm’s Law of Acoustics35

• 1843 Ohm's acoustic law, (acoustic phase 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”

• 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. Georg Simon Ohm

(1789 – 1854)

References• Wave Animations: http://www.sciencejoywagon.com/physicszone/09waves/

• Huygen’s Animation: http://www.sciencejoywagon.com/physicszone/otherpub/wfendt/huygens.htm

• Hugens & Diffraction http://www.launc.tased.edu.au/online/sciences/Physics/diffrac.html

• More animations http://www.launc.tased.edu.au/online/sciences/Physics/tutes1.htmlh

• http://id.mind.net/~zona/mstm/physics/waves/propagation/huygens1.html

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