9.2 Musical Instruments

New Ideas for today Sound and waves Pitch String and wind instruments

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 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

Pitch Western musical scale constructed around A 4, or 440 Hz. Crests repeat 440 times per second at one point in space!

IntervalsIntervals 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 Keyboard

Equal temperament Notice anything funny? G 4 / C 4 =1.498… Most Western music written around half-steps Ratio between adjacent frequencies = 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 Sound Usually involves a resonator

String as Harmonic Oscillator 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 question Why 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 Oscillation 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) 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 instrument Wine glass Wine glass II

Modes 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…) Fundamental2 nd Harmonic3 rd Harmonic4 th Harmonic Wavelength = Transverse Standing Waves Standing wave

Courtesy of PHET

Timbre Frequency (Hz) Volume Instruments have different musical fingerprint, or spectra Frequency analyzer

The 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! Guitars Speaker

Beautiful 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 System f0f0 2f 0 3f 0 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 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 Line of fire Organ pipe Singing tubes Bottle

Air 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