Physics 1251 The Science and Technology of Musical Sound Unit 1 Session 9 Transients and Resonances Unit 1 Session 9 Transients and Resonances

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 9Session 9

Transients and Transients and ResonancesResonances

Unit 1Unit 1

Session 9Session 9

Transients and Transients and ResonancesResonances

Physics 1251 Physics 1251 Unit 1 Session 9 Unit 1 Session 9 Transients and ResonancesTransients and Resonances

Foolscap Quiz:Foolscap Quiz:What are the following quantities for the waveform What are the following quantities for the waveform

below? Period, Frequency, Amplitude and below? Period, Frequency, Amplitude and Phase.Phase.

P =5.0 ms =0.0050 sec; P =5.0 ms =0.0050 sec; f f = 1/P = 200. = 1/P = 200. Hz;Hz; A = 2.0 mPa; Phase = +25% (90 A = 2.0 mPa; Phase = +25% (90⁰)⁰)


80/2080/20PeriodPeriod is the time from one point on a is the time from one point on a wave to a corresponding point on the next.wave to a corresponding point on the next.

80/2080/20FrequencyFrequency (in Hz)= 1/Period (in seconds). (in Hz)= 1/Period (in seconds).

80/2080/20AmplitudeAmplitude is the maximum excursion from is the maximum excursion from equilibrium for a wave.equilibrium for a wave.

80/2080/20PhasePhase is the shift (measured in fraction of is the shift (measured in fraction of a cycle) for a wave relative to a time a cycle) for a wave relative to a time standard.standard.


11′ Lecture: ′ Lecture: • Sound carries mechanical Sound carries mechanical energyenergy..

• Frequency spectra can change with time Frequency spectra can change with time producing producing transientstransients as energy is as energy is dissipated by radiation and frictional forces.dissipated by radiation and frictional forces.

• ResonanceResonance is the strongly driven oscillation, is the strongly driven oscillation, occurring at natural frequencies of the occurring at natural frequencies of the natural modes of oscillation.natural modes of oscillation.


80/2080/20WorkWork=Force x distance moved=Force x distance movedW =FW =F‧d ‧d ..

80/2080/20EnergyEnergy is the work performed or is the work performed or stored. stored.

E = W; [W]= Joule = NE = W; [W]= Joule = N∙m∙m80/2080/20Energy is Energy is conservedconserved; it changes ; it changes

form but is never really lost .form but is never really lost .

80/2080/20HeatHeat is a form of energy. is a form of energy.


TheThe Simple Harmonic Oscillator of DoomSimple Harmonic Oscillator of Doom


Why doesn’t the bowling Why doesn’t the bowling ball return to the same ball return to the same height?height?(1)(1)Energy is conversed, i.e. total energy Energy is conversed, i.e. total energy

is constant.is constant.

(2)(2) The bowling ball does work (a little) The bowling ball does work (a little) moving the air out of the way. moving the air out of the way. { Recall: W = F{ Recall: W = F‧d }‧d }

(3)(3) Therefore, the bowling ball has less Therefore, the bowling ball has less energy to make it up the “hill.”energy to make it up the “hill.”


80/2080/20Sound waves carry energy.Sound waves carry energy.

80/2080/20PowerPower is the rate of energy is the rate of energy dissipation or radiation.dissipation or radiation.

ℙ ℙ = W/t; = W/t; [ℙ] [ℙ]= Watt = = Watt = Joule/secJoule/sec80/2080/20IntensityIntensity is the power/square is the power/square meter.meter.

I = I = ℙ/A ; [ℙ/A ; [I I ]] = =Watt/m Watt/m 22 . .


Energy Dissipation:Energy Dissipation:

If a constant If a constant fractionfraction of the energy is of the energy is dissipated every second, the amplitude dissipated every second, the amplitude of an oscillator “decays” exponentially.of an oscillator “decays” exponentially.


““Damped”Damped”Simple Harmonic OscillatorSimple Harmonic Oscillator


Transients are due to work done by the Transients are due to work done by the

SHO.SHO.

Energy of Energy of SHOSHO

HeatHeat


Where does the “lost Where does the “lost energy” go?energy” go?(1)(1)Energy is conversed, i.e. total energy Energy is conversed, i.e. total energy

is constant.is constant.

(2)(2) The SHO does work (a little each The SHO does work (a little each cycle) moving the air or water out of cycle) moving the air or water out of the way. the way. { Recall: W = F{ Recall: W = F‧d }‧d }

(3)(3) Therefore, the SHO slows down a Therefore, the SHO slows down a little each cycle. It has less energy.little each cycle. It has less energy.

(4)(4) Ultimately, the energy goes into Ultimately, the energy goes into heat (random motion of air heat (random motion of air molecules).molecules).


““Dr. M’s Energy Conservation Dr. M’s Energy Conservation Furnace”Furnace”

1.1. Rub your hands together briskly. Rub your hands together briskly.

2.2. What do you feel?What do you feel?

3.3. Why?Why?


80/2080/20Higher frequencies tend to “damp” Higher frequencies tend to “damp” more rapidly than do lower frequenciesmore rapidly than do lower frequencies..

Why?Why?

ΔΔE = Energy dissipated in each cycle E = Energy dissipated in each cycle

= (damping) force x distance.= (damping) force x distance.

Energy dissipated per second = Energy Energy dissipated per second = Energy dissipated per cycle x cycles per second dissipated per cycle x cycles per second = = ΔΔE E ∙ ∙ f.f.


Time Dependent Fourier Spectrum:Time Dependent Fourier Spectrum:

Observe TransientsObserve Transients


Time Dependent Fourier Spectrum:Time Dependent Fourier Spectrum:

Observe TransientsObserve Transients

Piano Transient Frequency Piano Transient Frequency SpectrumSpectrum

FrequencyFrequency

Am

plitu

de o

f A

mp

litu

de o

f C

om

pon

en

tC

om

pon

en

t


Resonantly DrivenResonantly DrivenSimple Harmonic OscillatorSimple Harmonic Oscillator


Resonantly DrivenResonantly DrivenSimple Harmonic OscillatorSimple Harmonic Oscillator

PushPush

Wait 1 Wait 1 periodperiod


Resonance:Resonance:

An oscillator will absorb energy and grow in An oscillator will absorb energy and grow in amplitude when the energy is supplied at a amplitude when the energy is supplied at a frequency near that of a natural mode.frequency near that of a natural mode.

Helmholtz Resonator Video DemonstrationHelmholtz Resonator Video Demonstration


““Resonantly Driven”Resonantly Driven”Simple Harmonic OscillatorSimple Harmonic Oscillator


Summary:Summary:• Transients occur because radiation and Transients occur because radiation and

friction damp the various harmonics friction damp the various harmonics differently.differently.

• Resonance is the phenomenon of Resonance is the phenomenon of sympathetic vibration at a natural sympathetic vibration at a natural frequency with energy supplied by an frequency with energy supplied by an outside source.outside source.

• Sound transmits energy since pressure Sound transmits energy since pressure = force/area and a displacement occurs = force/area and a displacement occurs as a sound wave passes.as a sound wave passes.

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Physics 1251 The Science and Technology of Musical Sound Unit 1 Session 9 Transients and Resonances Unit 1 Session 9 Transients and Resonances