Violins and Pipe Organs 1 Violins and Pipe Organs 2

Question:

Violins and Pipe Organs Sound can break glass. Which is easiest to break:

• A glass pane exposed to a loud, short sound • A glass pane exposed to a certain loud tone • A crystal glass exposed to a loud, short sound • A crystal glass exposed to a certain loud tone

Violins and Pipe Organs 3 Violins and Pipe Organs 4 Observations About Strings as Violins and Pipe Organs Harmonic Oscillators

• They can produce different pitches • A string is a harmonic oscillator • They must be tuned – Its mass gives it inertia • They sound different, even on same pitch – Its tension gives it a restoring force • Sound character is adjustable – It has a stable equilibrium – Restoring forces proportional to displacement • Both require power to create sound • Pitch independent of amplitude (volume)! • Can produce blended or dissonant sounds

Violins and Pipe Organs 5 Violins and Pipe Organs 6 String’s Inertia and Fundamental Restoring Forces Vibration

• String’s restoring force stiffness set by • String vibrates as single arc, up and down – string’s tension – velocity antinode occurs at center of string – string’s curvature (or, equivalently, length) – velocity nodes occur at ends of string • String’s inertial characteristics set by • This is the fundamental vibrational mode – string’s mass per length • Pitch (frequency of vibration) is – proportional to tension – inversely proportional to string length – inversely proportional to mass per length

•1 Violins and Pipe Organs 7 Violins and Pipe Organs 8 Overtone String Harmonics Vibrations Part 1

• String can also vibrate as • In a string, the overtone pitches are at – two half-strings (one extra antinode) – twice the fundamental frequency – three third-strings (two extra antinodes) • One octave above the fundamental frequency –etc. • Produced by two half-string vibrational mode • These are higher-order vibrational modes – three times the fundamental frequency • An octave and a fifth above the fundamental • They have higher pitches • Produced by three half-string vibrational mode • They are called “overtones” –etc.

Violins and Pipe Organs 9 Violins and Pipe Organs 10 String Harmonics Producing Part 2 Sound

• Integer overtones are called “harmonics” • Thin objects don’t project sound well • Bowing or plucking a string tends to excite – Air flows around objects a mixture of fundamental and harmonic – Compression and rarefaction is minimal vibrations, giving character to the sound • Surfaces project sound much better – Air can’t flow around surfaces easily – Compression and rarefaction is substantial • Many instruments use surfaces for sound

Violins and Pipe Organs 11 Violins and Pipe Organs 12 Plucking Question: and Bowing

• Plucking a string transfers energy instantly Sound can break glass. Which is easiest to break: • Bowing a string transfers energy gradually – Rhythmic excitation at the right frequency • A glass pane exposed to a loud, short sound causes sympathetic vibration • A glass pane exposed to a certain loud tone – Bowing always excites string at the right • A crystal glass exposed to a loud, short sound frequency – The longer the string’s resonance lasts, the • A crystal glass exposed to a certain loud tone more effective the gradual energy transfer

•2 Violins and Pipe Organs 13 Violins and Pipe Organs 14 Air as a Air’s Inertia and Harmonic Oscillator Restoring Forces

• A column of air is a harmonic oscillator • Air’s restoring force stiffness set by – Its mass gives it inertia – pressure – Pressure gives it a restoring force – pressure gradient (or, equivalently, length) – It has a stable equilibrium • Air’s inertial characteristics set by – Restoring forces proportional to displacement – air’s mass per length (essentially density) • Pitch independent of amplitude (volume)!

Violins and Pipe Organs 15 Violins and Pipe Organs 16 Fundamental Vibration Fundamental Vibration Open-Open Column Open-Closed Column

• Air column vibrates as a single object • Air column vibrates as a single object – Pressure antinode occurs at column center – Pressure antinode occurs at closed end – Pressure nodes occur at column ends – Pressure node occurs at open end • Pitch (frequency of vibration) is • Air column in open-closed pipe vibrates – proportional to air pressure – as half the column in an open-open pipe – inversely proportional to column length – at half the frequency of an open-open pipe – inversely proportional to air density

Violins and Pipe Organs 17 Violins and Pipe Organs 18 Air Harmonics Air Harmonics Part 1 Part 2

• In open-open pipe, the overtones are at • Blowing across column tends to excite a – twice fundamental (two pressure antinodes) mixture of fundamental and harmonic – three times fundamental (three antinodes) vibrations – etc. (all integer multiples or “harmonics”) • In open-closed pipe, the overtones are at – three times fundamental (two antinodes) – five times fundamental (three antinodes) – etc. (all odd integer multiples or “harmonics”)

•3 Violins and Pipe Organs 19 Violins and Pipe Organs 20 Summary of Other Instruments Violins and Pipe Organs

• Most 1-dimensional instruments • use strings and air as harmonic oscillators – can vibrate at half, third, quarter length, etc. • pitches independent of amplitude/volume – harmonic oscillators with harmonic overtones • tuned by tension/pressure, length, density • Most 2- or 3- dimensional instruments • have harmonic overtones – have complicated higher-order vibrations • project vibrations into the air as sound – harmonic osc. with non-harmonic overtones • Examples: drums, cymbals, glass balls

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