9.2 Musical Instruments9.2 Musical Instruments9.2 Musical Instruments9.2 Musical Instruments

New Ideas for todayNew Ideas for today

•Sound and waves•Pitch•String and wind instruments

Sound is a wave!Sound is a wave!Sound is a wave!Sound is a wave!

• Sound is a longitudinal pressure wave in a medium (gas, liquid or solid)

• Anything that vibrates a medium produces sound

• Air is the most common medium for carrying sound

• Sound is a longitudinal pressure wave in a medium (gas, liquid or solid)

• Anything that vibrates a medium produces sound

• Air is the most common medium for carrying sound

Bell in vacuumBell in vacuum

•Waves have a wavelength: distance to next crest or trough

•Waves have an amplitude: peak change in pressure for sound in air

•Any mechanical wave “represents the natural motion of an extended object around stable equilibrium shape or situation”

Watch one place – the period is the time for the next crest or trough to appear

Watch one crest – moves at the speed of sound (330 m/s in air)

The frequency, or pitch, of sound is the number of times per second that the wave repeats itself (or 1/period)

wave speed = frequency × wavelength

http://phet.colorado.edu/simulations/sound/sound.jnlp

PitchPitch

Western musical scale constructed around A4, or 440 Hz. Crests repeat 440 times per second at one point in space!

IntervalsIntervalsIntervalsIntervals

Playing two frequencies together sounds nice when they are in small integer ratios

• Octave 2:1• Fifth 3:2• Fourth 4:3• Third 5:4

Playing two frequencies together sounds nice when they are in small integer ratios

• Octave 2:1• Fifth 3:2• Fourth 4:3• Third 5:4

KeyboardKeyboard

Equal temperamentEqual temperament

•Notice anything funny?

G4 / C4=1.498…

•Most Western music written around half-steps

Ratio between adjacent frequencies = 12 2 1.05946...

•Makes every key sound similar (every key equally mistuned)

•Orchestras will sometimes tune specially for particular pieces

An object’s natural vibration or resonant frequency is determined by its: Mass Size and shape Elastic nature (stiffness) Composition

Musical resonators Stretched strings (violin string) Hollow tubes (flute) Stretched membrane (drum)

An object’s natural vibration or resonant frequency is determined by its: Mass Size and shape Elastic nature (stiffness) Composition

Musical resonators Stretched strings (violin string) Hollow tubes (flute) Stretched membrane (drum)

Tacoma Narrows Bridge – a wind instrument!

Producing Musical SoundProducing Musical SoundUsually involves a resonator

String as Harmonic String as Harmonic OscillatorOscillatorString as Harmonic String as Harmonic OscillatorOscillator Its mass gives it inertia Its tension and curvature give it a restoring force It has a stable equilibrium Its restoring force is proportional to displacement

Its mass gives it inertia Its tension and curvature give it a restoring force It has a stable equilibrium Its restoring force is proportional to displacement

Clicker questionClicker questionWhy does the low E string have copper wrapped around it?

A. To increase the tension of the string.B. To increase the mass of the string.C. To change the length of the string.D. To increase the thermal conductivity of the string.

Modes of OscillationModes of OscillationModes of OscillationModes of OscillationFundamental Vibration (First Harmonic)

Center of string vibrates up and down Frequency of vibration (pitch) is

proportional to tension inversely proportional to length inversely proportional to density (mass/length)

Fundamental Vibration (First Harmonic) Center of string vibrates up and down Frequency of vibration (pitch) is

proportional to tension inversely proportional to length inversely proportional to density (mass/length)

Stringed instrumentStringed instrument

Wine glassWine glass

Wine glass IIWine glass II

Modes of OscillationModes of OscillationModes of OscillationModes of Oscillation Higher-Order Vibrations (Overtones or Harmonics)

Second harmonic is like two half-strings Third harmonic is like three third-strings, …

Each higher mode has one more node in its oscillation Harmonics come in integer multiples (1,2,3,4…)

Higher-Order Vibrations (Overtones or Harmonics) Second harmonic is like two half-strings Third harmonic is like three third-strings, …

Each higher mode has one more node in its oscillation Harmonics come in integer multiples (1,2,3,4…)

Fundamental

2nd Harmonic 3rd Harmonic 4th Harmonic

Wavelength =

Transverse Standing Waves

Standing waveStanding wave

Courtesy of PHET

TimbreTimbre

Frequency (Hz)

Volu

me

Instruments have different musical fingerprint, or spectra

Frequency analyzerFrequency analyzer

The Sound BoxThe Sound BoxThe Sound BoxThe Sound Box Strings don’t project sound well

Air flows around objects Surfaces project sound much better

Air can’t flow around surfaces easily Movement of air is substantial!

Strings don’t project sound well Air flows around objects

Surfaces project sound much better Air can’t flow around surfaces easily Movement of air is substantial!

GuitarsGuitars

SpeakerSpeaker

Beautiful MusicBeautiful MusicBeautiful MusicBeautiful Music Transfer of vibration to a “sound box” is important in

instrument design helps to project the sound effectively helps to “color” the sound, making the instrument sound

unique The method of exciting the string also affects the sound.

Plucking a string transfers energy quickly and excites many vibrational modes

Bowing a string transfers energy slowly excites the string at its fundamental frequency each stroke adds to the string’s vibrational energy

Transfer of vibration to a “sound box” is important in instrument design helps to project the sound effectively helps to “color” the sound, making the instrument sound

unique The method of exciting the string also affects the sound.

Plucking a string transfers energy quickly and excites many vibrational modes

Bowing a string transfers energy slowly excites the string at its fundamental frequency each stroke adds to the string’s vibrational energy

Bowing is a neat way to sustain a note using the properties of friction!

Air Column as Resonant Air Column as Resonant SystemSystemAir Column as Resonant Air Column as Resonant SystemSystem

f0

2f0

3f0

A column of air is a harmonic oscillator Its mass gives it inertia Pressure gives it a restoring force It has a stable equilibrium Restoring forces are proportional

to displacement Stiffness of restoring forces

determined by pressure pressure gradient

A column of air is a harmonic oscillator Its mass gives it inertia Pressure gives it a restoring force It has a stable equilibrium Restoring forces are proportional

to displacement Stiffness of restoring forces

determined by pressure pressure gradient

Air Column PropertiesAir Column PropertiesAir Column PropertiesAir Column Properties An air column vibrates as a single object

Pressure antinode occurs at center of open column Velocity antinode occurs at ends of open column

Pitch (frequency) is inversely proportional to column length inversely proportional to air density

An air column vibrates as a single object Pressure antinode occurs at center of open column Velocity antinode occurs at ends of open column

Pitch (frequency) is inversely proportional to column length inversely proportional to air density

Length

Length

Line of fireLine of fire

Organ pipeOrgan pipe

Singing tubesSinging tubes

BottleBottle

Air Column PropertiesAir Column PropertiesAir Column PropertiesAir Column Properties Just like a string, an air column can vibrate at

many different frequencies. A closed pipe vibrates as half an open pipe

pressure antinode occurs at sealed end frequency is half that of an open pipe

Just like a string, an air column can vibrate at many different frequencies.

A closed pipe vibrates as half an open pipe pressure antinode occurs at sealed end frequency is half that of an open pipe

Open Column Closed Column

See you next class!

For next class: Read Sections 10.1 and 10.2