Sound Waves

Chapter 11

Longitudinal wave
Requires elastic medium for propagation
Series of compressions and rarefactions in the medium (what the sound is passing through)
Elastic forces of the medium transfer wave energy from particle to particle

Frequency range over which sound waves can be transmitted
Upper limit determined by particle spacing of medium
For air and other gases upper limit is about 1 billion Hz; higher in liquids & solids
No clearly defined lower limit

Sound is produced by vibration in an elastic medium
Energy of vibration is passed to particles of medium
As object pushes against medium it creates compressed area; as it moves back, it creates rarefied area
Change in amplitude of vibration changes amount of compression, but not frequency
Vibrating source can be string, reed, air column, vocal chord, etc.

At 0° C, sound travels 331.5 m/s in air
Speed increases about 0.6 m/s for each degree above zero
Speed of sound is about 4 times faster in water, much faster in elastic solids like steel

Intensity: time rate of flow of sound energy through a unit area
I = P / A where P is power in watts and A is area in square meters.
Intensity (and power) is proportional to square of wave amplitude
Sound wave spreads in spherical pattern from source so intensity depends inversely on square of distance from source
Loudness is physiological response to intensity and depends on sensitivity of listener
Human hearing more sensitive to some frequencies than others

The faintest audible sound is called the threshold of hearing, 10^-12 W/m² (I₀)
Intensity is measured with respect to this minimum value, called relative intrensity
Relative intensities placed on logarithmic scale called the decibel scale
1 decibel = 10 bels, the original unit of sound intensity
β = 10 log(I/I0) where β is relative intensity in decibels (dB)
Relative intensity of 120 dB called threshold of pain

A change in frequency due to relative motion between wave source and observer
Applies to all wave types
With sound waves, effect is change of pitch heard when either source or listener are moving (or both)
Apparent frequency and pitch higher than actual frequency when source and listener move closer
Apparent frequency and pitch lower when source and listener move apart
Used to measure speed of moving objects: planets, stars, cars when applied to light or radar waves.
Good Doppler effect demos can be viewed at these sites: Doppler demo, http://www.kettering.edu/~drussell/Demos/doppler/doppler.html, http://webphysics.davidson.edu/Applets/Examples_From_Others/doppler.html, http://www.colorado.edu/physics/2000/applets/doppler.html

Listener moving : f´ = f (v ± v_D)/v where v_D is speed of listener (detector), f is actual frequency of source and f´ is frequency heard (detected)
Source moving: f´ = f v / (v ± v_S) where v_S is speed of source

The lowest frequency produced by a vibrating object is the fundamental
Produced by strings vibrating as a whole
Strings and other vibrating objects can vibrate in segments producing harmonics: frequencies that are whole number multiples of the fundamental
Fundamental is called first harmonic 2 x fundamental frequency is second harmonic, etc.
Music term for harmonic is overtone; 2nd harmonic is 1st overtone, etc.
Most sounds are combinations of fundamental and harmonics

Harmonic content determines sound quality or tone.
Presence of many high harmonics produces bright tone (treble)
Different instruments producing same fundamental frequency sound different because of different harmonic content

Frequency is inversely proportional to string length: f / f´ = l´ /l
Frequency is inversely proportional to string diameter: f / f´ = d´ /d
Frequency is directly proportional to square root of tension: f / f´ = (T/T´ )^½
Frequency is inversely proportional to string density: f / f´ = (D´ /D)^½
Forced Vibrations cause a second object to vibrate by contact with another vibrating object
Vibrating strings & reeds produce little sound but transmit vibration to larger object to increase sound: body of instrument, sounding board, vibrating air column

All objects have certain natural frequencies of vibration
When forced vibration occurs at natural frequency of object, resonance occurs and vibration amplitude is greatly increased
Can be desirable or not

Basis of organ pipes, flute, clarinet, trumpet, etc.
Air column resonates with applied frequency depending on its length
Wave is reflected at opposite end of tube creating standing wave at resonant frequency
Can be open ended or closed ended tube

Standing wave in closed tube has node at closed end, anti-node at open end
Fundamental frequency has approx. 1/4 of wave inside tube (node to anti-node)
Wavelength about 4 x length of tube
Resonance will occur at all harmonics that have same node/anti-node arrangement
Closed end tubes thus resonate at odd quarter wavelengths and produce only odd numbered harmonics

Standing wave in open ended tubes have antinode at each end
Wavelength of fundamental is approx. 2 x length of tube since about half the wave is contained in tube
Harmonics with antinode at each end will also be produced, thus open ended tubes produce all harmonics

Two waves of nearly the same frequency sounded simultaneously will interfere with each other
Alternating constructive and destructive interference causes amplitude pulsation
Sound heard is average frequency with beat frequency equal to difference in two sound frequencies
Used to tune instruments

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