physics 1251 the science and technology of musical sound unit 1 session 7 good vibrations unit 1...

Physics 1251Physics 1251The Science and The Science and

Technology of Musical Technology of Musical SoundSound

Physics 1251Physics 1251The Science and The Science and

Technology of Musical Technology of Musical SoundSound

Unit 1Unit 1

Session 7Session 7

Good VibrationsGood Vibrations

Unit 1Unit 1

Session 7Session 7

Good VibrationsGood Vibrations

Physics 1251 Physics 1251 Unit 1 Session 7 Unit 1 Session 7 Good VibrationsGood Vibrations

Foolscap Quiz:Foolscap Quiz:Two tuning forks that Two tuning forks that

each, separately, each, separately, oscillate at 128 Hz oscillate at 128 Hz are joined together are joined together by one tine. How by one tine. How many modes of many modes of oscillation are there oscillation are there with them joined? with them joined? Are the frequencies Are the frequencies the same as before?the same as before?


80/2080/20The number of normal or natural The number of normal or natural modes of vibration of an oscillator modes of vibration of an oscillator is equal to the number of simple is equal to the number of simple harmonic oscillators that harmonic oscillators that comprise the complex oscillator.comprise the complex oscillator.


Example:Example:Two coupled masses on springs with same k Two coupled masses on springs with same k

and m.and m.Separately: Separately:

ff11 = (1/2 = (1/2π)π)√k /m ,√k /m ,

ff22 = (1/2 = (1/2π)π)√k /m .√k /m .Coupled Normal Modes:Coupled Normal Modes:

ffii = (1/2 = (1/2π)π)√k /2m :[m√k /2m :[m1 1 && mm2 2 move as one],move as one],

ffiiii = (1/2 = (1/2π)π)√2k /m :[√2k /m :[mm1 1 && mm2 2 move in move in opposition].opposition].


Spring Spring ——————→→

Mass Mass ————→————→

Two Masses on Two Coupled Two Masses on Two Coupled SpringsSprings

Spring Spring ————→————→

Mass Mass ————→————→

Mode 1Mode 1 Mode 2Mode 2


The “Fingered” BellThe “Fingered” Bell

• Bell is constructed Bell is constructed of many “tuning forks.”of many “tuning forks.”

• New normal modesNew normal modesappear when coupledappear when coupled


11′ Lecture:′ Lecture:• Any waveform can be broken up Any waveform can be broken up

into simple sinusoidal components.into simple sinusoidal components.

• Any waveform can be built up from Any waveform can be built up from single frequency components in single frequency components in the correct combination.the correct combination.


Doctor M’s Stupendous, Colossal, Remarkable Doctor M’s Stupendous, Colossal, Remarkable

BBCBBC

Beer Bottle ChoirBeer Bottle Choir


Why don’t we see “Jug Bands” and Why don’t we see “Jug Bands” and

““Beer Bottle Choirs” except asBeer Bottle Choirs” except as

curiosities?curiosities?

• How would you describe the How would you describe the “timbre” (quality of the sound) of a “timbre” (quality of the sound) of a Helmholtz resonator?Helmholtz resonator?

• What gives a “jug” its timbre?What gives a “jug” its timbre?


Normal ModesNormal Modes

Of Hand BellOf Hand Bell• Bell is shaped to Bell is shaped to produce many normal produce many normal modes of oscillation.modes of oscillation.

• Timbre is determined Timbre is determined by combination of by combination of normal modes.normal modes.


Peregrination in Existential Peregrination in Existential PhysicsPhysics::

• Move you hand in a circle.Move you hand in a circle.• Notice its height.Notice its height.• Now move your other hand up and Now move your other hand up and

down in a straight line at the same down in a straight line at the same height.height.

• This is sinusoidal (like a sine) motion.This is sinusoidal (like a sine) motion.



Sine of the angleSine of the angle



• Move you hand in a circle while Move you hand in a circle while waving it up and down.waving it up and down.

• Notice its height.Notice its height.• This is a complex motion that can be This is a complex motion that can be

described as the sum of simple described as the sum of simple sinusoidal motions.sinusoidal motions.


80/2080/20A A microphonemicrophone converts pressure converts pressure fluctuations into electrical signals.fluctuations into electrical signals.

TransducerTransducer

ElectronicElectronicss


80/2080/20An An oscilloscopeoscilloscope displays electrical displays electrical signals versus time. Repeating signals versus time. Repeating oscillations can be viewed on the oscillations can be viewed on the screen.screen.


80/2080/20The term The term ““sine wavesine wave”” describes describes anything that varies with time or anything that varies with time or position like the trigonometric position like the trigonometric function called “sine” (abbreviated function called “sine” (abbreviated sin).sin).

Sine (A) Sine (A) Argument Argument AA


80/2080/20A simple harmonic oscillator moves A simple harmonic oscillator moves like a sine wave. Its motion is called like a sine wave. Its motion is called ““sinusoidalsinusoidal [=like a sine].” [=like a sine].”

In one place, the pressure produced In one place, the pressure produced by the sound from a simple by the sound from a simple harmonic oscillator varies with harmonic oscillator varies with time as a sine wave.time as a sine wave.

Physics 1251 Physics 1251 Unit 1 Session 5 Unit 1 Session 5 Simple Harmonic OscillatorsSimple Harmonic Oscillators

CompressionCompression

RarefactioRarefactionn

At an instant of time, the pressure At an instant of time, the pressure produced by the sound from a produced by the sound from a simple harmonic oscillator varies simple harmonic oscillator varies with position as a sine wave.with position as a sine wave.

Physics 1251 Physics 1251 Unit 1 Session 5 Unit 1 Session 5 Simple Harmonic OscillatorsSimple Harmonic Oscillators

CompressionCompression

RarefactioRarefactionn


80/2080/20Complex sounds can be Complex sounds can be produced from simple sounds.produced from simple sounds.

Example:Example:Play CPlay C44, C, C44+G+G44, C, C44+G+G44+E+E44

Observe oscilloscope.Observe oscilloscope.


Fourier Analysis and SynthesisFourier Analysis and Synthesis

• Baron Jean BaptisteBaron Jean BaptisteJoseph FourierJoseph Fourier(1768-1837)(1768-1837)

• MathematicianMathematician

• Fourier AnalysisFourier Analysis

• Fourier SynthesisFourier Synthesis

[Pronounced “Four-yeah”][Pronounced “Four-yeah”]


80/2080/20Fourier AnalysisFourier Analysis is the is the decomposition of a wave into the decomposition of a wave into the sine wave components from which sine wave components from which it can be built up.it can be built up.


80/2080/20The The lowest frequencylowest frequency component component of a of a wave is equal to 1/Pwave is equal to 1/Prepeatrepeat the the reciprocal of the repeat period Preciprocal of the repeat period Prepeatrepeat

Time Time →→

Repeat Period PRepeat Period Prepeatrepeat fflowestlowest = = 1/P1/Prepeatrepeat


80/20 80/20 A A Fourier AnalysisFourier Analysis is a is a representation of all the representation of all the components that comprise a components that comprise a waveform, waveform, amplitudeamplitude versus versus frequency and frequency and phasephase versus versus frequency.frequency.


Example: Synthesis of a Square Example: Synthesis of a Square wavewaveffoo

3f3f

oo5f5f

oo7f7f

oo

Square WaveSquare Wave


Example: Synthesis of a Square wave-Example: Synthesis of a Square wave-

The Fourier SpectrumThe Fourier Spectrum

FrequencyFrequencyffoo 3f3f

oo

5f5f

oo

7f7f

oo

11stst Overtone – 3 Overtone – 3rdrd HarmonicHarmonic

22ndnd Overtone – 5 Overtone – 5thth HarmonicHarmonic

Fundamental – 1Fundamental – 1stst HarmonicHarmonic

Am

plitu

de

Am

plitu

de


Summary:Summary:

• Waveforms can be broken down into Waveforms can be broken down into simple sine waves, called simple sine waves, called Fourier Fourier Analysis.Analysis.

• Waveforms can be built up from simple Waveforms can be built up from simple sine waves, called sine waves, called Fourier Synthesis.Fourier Synthesis.

• The lowest frequency needed is 1/PThe lowest frequency needed is 1/Prepeatrepeat..

• A A Fourier SpectrumFourier Spectrum is the recipe for is the recipe for building the waveform.building the waveform.

physics 1251 the science and technology of musical sound unit 1 session 7 good vibrations unit 1...

Documents

good vibrations unit

good vibrations peregrination

existential physics

good vibrations example

good vibrations spring

good vibrations foolscap

angle slide

combination of normal