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

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Musical Instruments 1

Musical Instruments

Musical Instruments 2

Introductory Question

• Sound can break glass. Which is most likely to

break:

A.

A glass pane exposed to a loud, short sound

B.

A glass pane exposed to a certain loud tone

C.

A crystal glass exposed to a loud, short sound

D.

A crystal glass exposed to a certain loud tone

Musical Instruments 3 Observations about

Musical Instruments

They can produce different notes They must be tuned to produce the right notes They sound different, even on the same note They require power to create sound

Musical Instruments 4

4 Questions about Musical Instruments

Why do strings produce specific notes? Why does a vibrating string sound like a string? Why do stringed instruments need surfaces? What is vibrating in a wind instrument?

Musical Instruments 5

Question 1

Why do strings produce specific notes?

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Oscillations of a Taut String

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)!

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A Taut String’s Pitch

Stiffness of a string’s restoring forces are set by

the string’s tension the string’s curvature (or, equivalently, length)

The inertial characteristics of a string are set by

the string’s mass per length

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Fundamental Vibration

A string has a fundamental vibrational mode

in which it vibrates as a single arc, up and down, with a velocity antinode at its center and velocity nodes at its two ends

Its fundamental pitch (frequency of vibration) is

proportional to its tension, inversely proportional to its length, and inversely proportional to its mass per length

Musical Instruments 9

Question 2

Why does a vibrating string sound like a string?

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Overtone Vibrations

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”

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A String’s Harmonics (Part 1)

A string’s overtones are special: harmonics First overtone involves two half-strings

Twice the fundamental pitch: 2nd harmonic One octave above the fundamental frequency

Second overtone involves three third-strings

Three times the fundamental pitch: 3rd harmonic An octave and a fifth above the fundamental

Etc.

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A String’s Harmonics (Part 2)

Integer overtones are called “harmonics” Bowing or plucking a string excites a mixture of

fundamental and harmonic vibrations, giving the string its characteristic sound

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Question 3

Why do stringed instruments need surfaces?

Musical Instruments 14

Projecting Sound

In air, sound consists of density fluctuations

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

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Plucking and Bowing

Plucking a string transfers energy instantly Bowing a string transfers energy gradually

Bow does a little work on the string every cycle Excess energy builds up gradually in the string This gradual buildup is resonant energy transfer

The string will vibrate sympathetically when

another object vibrates at its resonant frequency and it gradually obtains energy from that object

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Introductory Question (revisited)

• Sound can break glass. Which is most likely to

break:

A.

A glass pane exposed to a loud, short sound

B.

A glass pane exposed to a certain loud tone

C.

A crystal glass exposed to a loud, short sound

D.

A crystal glass exposed to a certain loud tone

Musical Instruments 17

Question 4

What is vibrating in a wind instrument?

Musical Instruments 18

Oscillations of Air in a Tube

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)!

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Musical Instruments 19

Air in a Tube’s Pitch

Stiffness of the air’s restoring forces are set by

the air’s pressure the air’s pressure gradient (or, equivalently, length)

The inertial characteristics of the air are set by

the air’s mass per length

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Fundamental Vibration Open-Open Column

Air column vibrates as a single object

Pressure antinode occurs at column center Pressure nodes occur at column ends

Pitch (frequency of vibration) is

proportional to air pressure inversely proportional to column length inversely proportional to air density

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Fundamental Vibration Open-Closed Column

Air column vibrates as a single object

Pressure antinode occurs at closed end Pressure node occurs at open end

Air column in open-closed pipe vibrates

as half the column in an open-open pipe at half the frequency of an open-open pipe

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Air Harmonics (Part 1)

In open-open pipe, the overtones are at

twice fundamental (two pressure antinodes) three times fundamental (three antinodes)

• 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”)

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Air Harmonics (Part 2)

Blowing across the column tends to excite a

mixture of fundamental and harmonic vibrations

Examples

Organ pipes Recorders Flutes Whistles

Reeds and horns also use a vibrating air column

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Surface Instruments

Most 1-dimensional instruments

can vibrate at half, third, quarter length, etc. harmonic oscillators with harmonic overtones

Most 2- or 3- dimensional instruments

have complicated higher-order vibrations harmonic oscillators with non-harmonic overtones

Examples: drums, cymbals, bells

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Drumhead Vibrations

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Summary of Musical Instrument

use strings, air, etc. as harmonic oscillators pitches independent of amplitude/volume tuned by tension/pressure, length, density

• ften have harmonic overtones

project vibrations into the air as sound