Musical Instruments A glass pane exposed to a loud, short sound A. - PDF document

Musical Instruments 1 Musical Instruments 2 Introductory Question Sound can break glass. Which is most likely to � break: Musical Instruments A glass pane exposed to a loud, short sound A. A glass pane exposed to a certain loud tone B. A crystal glass exposed to a loud, short sound C. A crystal glass exposed to a certain loud tone D. Musical Instruments 3 Observations about Musical Instruments 4 4 Questions about Musical Instruments Musical Instruments � They can produce different notes � Why do strings produce specific notes? � They must be tuned to produce the right notes � Why does a vibrating string sound like a string? � They sound different, even on the same note � Why do stringed instruments need surfaces? � They require power to create sound � What is vibrating in a wind instrument? Musical Instruments 5 Musical Instruments 6 Question 1 Oscillations of a Taut String � Why do strings produce specific notes? � A taut string has � a mass that provides it with inertia � a tension that provides restoring forces � a stable equilibrium shape (straight line) � restoring forces proportional to displacement � A taut string is a harmonic oscillator � It oscillates about its equilibrium shape � Its pitch is independent of its amplitude (volume)! 1

Musical Instruments 7 Musical Instruments 8 A Taut String’s Pitch Fundamental Vibration � Stiffness of a string’s restoring forces are set by � A string has a fundamental vibrational mode � the string’s tension � in which it vibrates as a single arc, up and down, � the string’s curvature (or, equivalently, length) � with a velocity antinode at its center � and velocity nodes at its two ends � The inertial characteristics of a string are set by � Its fundamental pitch (frequency of vibration) is � the string’s mass per length � proportional to its tension, � inversely proportional to its length, � and inversely proportional to its mass per length Musical Instruments 9 Musical Instruments 10 Question 2 Overtone Vibrations � Why does a vibrating string sound like a string? � A string can also vibrate as � two half-strings (one extra antinode) � three third-strings (two extra antinodes) � etc. � These higher-order vibrational modes � have higher pitches than the fundamental mode � and are called “overtones” Musical Instruments 11 Musical Instruments 12 A String’s Harmonics (Part 1) A String’s Harmonics (Part 2) � A string’s overtones are special: harmonics � Integer overtones are called “harmonics” � First overtone involves two half-strings � Bowing or plucking a string excites a mixture of � Twice the fundamental pitch: 2 nd harmonic fundamental and harmonic vibrations, giving the string its characteristic sound � One octave above the fundamental frequency � Second overtone involves three third-strings � Three times the fundamental pitch: 3 rd harmonic � An octave and a fifth above the fundamental � Etc. 2

Musical Instruments 13 Musical Instruments 14 Question 3 Projecting Sound � In air, sound consists of density fluctuations � Why do stringed instruments need surfaces? � Air has a stable equilibrium: uniform density � Disturbances from uniform density make air vibrate � Vibrating strings barely project sound because � air flows around thin vibrating objects � and is only slightly compressed or rarefied � Surfaces project sound much better because � air can’t flow around surfaces easily � and is substantially compressed or rarefied Musical Instruments 15 Musical Instruments 16 Plucking and Bowing Introductory Question (revisited) � Plucking a string transfers energy instantly Sound can break glass. Which is most likely to � break: � Bowing a string transfers energy gradually � Bow does a little work on the string every cycle � Excess energy builds up gradually in the string A glass pane exposed to a loud, short sound A. � This gradual buildup is resonant energy transfer A glass pane exposed to a certain loud tone B. � The string will vibrate sympathetically when A crystal glass exposed to a loud, short sound C. � another object vibrates at its resonant frequency A crystal glass exposed to a certain loud tone D. � and it gradually obtains energy from that object Musical Instruments 17 Musical Instruments 18 Question 4 Oscillations of Air in a Tube � What is vibrating in a wind instrument? � Air in a tube has � a mass that provides it with inertia � a pressure distribution that provides restoring forces � a stable equilibrium structure (uniform density) � restoring forces proportional to displacement � Air in a tube is a harmonic oscillator � It oscillates about its equilibrium density distribution � Its pitch is independent of its amplitude (volume)! 3

Musical Instruments 19 Musical Instruments 20 Fundamental Vibration Air in a Tube’s Pitch Open-Open Column � Stiffness of the air’s restoring forces are set by � Air column vibrates as a single object � the air’s pressure � Pressure antinode occurs at column center � the air’s pressure gradient (or, equivalently, length) � Pressure nodes occur at column ends � The inertial characteristics of the air are set by � Pitch (frequency of vibration) is � the air’s mass per length � proportional to air pressure � inversely proportional to column length � inversely proportional to air density Musical Instruments 21 Musical Instruments 22 Fundamental Vibration Air Harmonics (Part 1) Open-Closed Column � Air column vibrates as a single object � In open-open pipe, the overtones are at � Pressure antinode occurs at closed end � twice fundamental (two pressure antinodes) � Pressure node occurs at open end � three times fundamental (three antinodes) � Air column in open-closed pipe vibrates � etc. (all integer multiples or “harmonics”) � In open-closed pipe, the overtones are at � as half the column in an open-open pipe � at half the frequency of an open-open pipe � three times fundamental (two antinodes) � five times fundamental (three antinodes) � etc. (all odd integer multiples or “harmonics”) Musical Instruments 23 Musical Instruments 24 Air Harmonics (Part 2) Surface Instruments � Blowing across the column tends to excite a � Most 1-dimensional instruments mixture of fundamental and harmonic vibrations � can vibrate at half, third, quarter length, etc. � Examples � harmonic oscillators with harmonic overtones � Most 2- or 3- dimensional instruments � Organ pipes � Recorders � have complicated higher-order vibrations � Flutes � harmonic oscillators with non-harmonic overtones � Whistles � Examples: drums, cymbals, bells � Reeds and horns also use a vibrating air column 4

Musical Instruments 25 Musical Instruments 26 Drumhead Vibrations Summary of Musical Instrument � use strings, air, etc. as harmonic oscillators � pitches independent of amplitude/volume � tuned by tension/pressure, length, density � often have harmonic overtones � project vibrations into the air as sound 5