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

2 Outline Wave Superposition Waveforms Fourier Theory & Ohms law

A. Superposition 3 Galileo Bernoulli Example

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.

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 1700-1782

1c. Example 6

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!

3a. Beats 8 Two tones closer than 15 Hertz we hear as a “fused” tone (average of frequencies) with a “beat”. 400 401 400 403 400 410 400 420 400 440 400 450 400 480 Demo: http://www.phys.unsw.edu.au/jw/beats.html#sounds

9 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

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

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

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

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

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 15 All harmonics possible (both even and odd) G2 C3

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!

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

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.

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

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

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

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

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

1b. Waveforms of Instruments 24 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.

2a. Fourier’s Theorem 25 Any periodic waveform can be constructed from harmonics. Joseph Fourier 1768-1830

2b. FFT: Fast Fourier Transform 26 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.

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

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

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

3a. Ohm’s Law of Acoustics 30 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.

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.

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.

33 Revision Notes New “physics” version april 15, 2015. May need clean up.

34 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/