03 Wave Superposition Updated 2012Apr19 Dr. Bill Pezzaglia

Outline A.Superposition 1.Galileo 2.Bernoulli 3.Example B.Diffraction 1.Interference of Sound 2.Huygen’s Principle of wave propagation 3.Diffraction through a slit (e.g. sound through a doorway) C.Resonance 1.Mersenne’s Laws 2.Harmonic Series 3.Quality of Sound D.References 2

A. Superposition 1.Galileo 2.Bernoulli 3.Example 3

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. 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 Bernoulli

3. Example x 6

B. Diffraction & Interference 1.Huygen’s Principle 2.Interference 3.Young slit diffraction 7

1. Huygen’s Principle (1678) 8

2. Interference Two waves added together can cancel each other out if “out of phase” with each other. 9 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!

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

C. Resonance 1. Mersenne’s Laws 2. Standing Waves 3. Quality of Sound 11

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. 12

1. Mersenne’s Laws a. Frequency and String Length b. Frequency and Tension c. Frequency and String Mass 13

a. Frequency & Length (i). Consider a string under tension “plucking” the string causes it to vibrate 14

(ii). Frequency and String Length Pythagoras of Samos ( 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) is half the wavelength 15

(iii). Frequency inversely proportional to length Frequency is inversely proportional to string length More generally: Frequency is inversely proportional to wavelength 16

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: 17 Vincenzo Galilei ( )

c. Frequency and Mass (i). Mersenne (1630) states: Frequency is inversely proportional to the diameter d of the string Putting it all together: 18 ( ) 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 StringDiameterFreq E mm330 Hz B G D A E E2 is 2 octaves lower than E4 Or (1/4) the frequency Hence diameter is nearly 4x bigger! 19

(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) 20

(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 Stringgm/cmFreq E Hz B G D A E E2 is 2 octaves lower than E4 Or (1/4) the frequency Hence mass bigger by nearly a factor of 16 21

2. Standing Waves a. Wavespeed Formula b. Harmonic Modes c. Open and Closed pipes 22

a. Wavespeed Frequency “f” : oscillations per second (Hertz) Wavespeed “c”: is frequency x wavelength c = f 23

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 24

b. Harmonic Modes Daniel Bernoulli (1728?) shows string can vibrate in different modes, which are multiples of fundamental frequency (called “Harmonics” by Sauveur) 25 n=1f 1 n=2f 2 =2f 1 n=3f 3 =3f 1 n=4f 4 =4f 1 n=5f 5 =5f 1

b.2. Wavelengths of Harmonic Modes The wavelength of n-th mode is: 26

b.3. Harmonic Series The musical notes of harmonic series 27 Reference: Sound:

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) 29 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! 30

c.4. Closed Pipe Harmonics Only odd n harmonics N=1 =4L N=3 =4L/3 N=5 =4L/6 31 C4 G5 E6

3. Quality of Sound a. Waveforms b. Fourier’s Theorem c. Ohm’s Law of Acoustics 32

1. Waveform Sounds Different “shape” of wave has different “timbre” quality 33 Sine Wave (flute) Square (clarinet) Triangular (violin) Sawtooth (brass)

2. Fourier’s Theorem Any periodic waveform can be constructed from harmonics. 34 Joseph Fourier

3. Ohm’s Law of Acoustics 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: Huygen’s Animation: Hugens & Diffraction More animations