Sound

• Sound waves are longitudinal mechanical waves that propagate through gases, liquids, and solids.
• Sound waves in air involve small changes in air pressure and density, associated with back-and-forth motion of the air as the wave passes.

The Speed of Sound

• For air at sea level, the speed of sound is given by:

• Where T is the air temperature in degrees Kelvin.
• Example: What is the speed of sound on a summer day when it is 25 degrees Celsius outside?

From the Preclass Survey

From the Preclass Survey

Speed of Sound

• Note that light travels almost instantaneously (300,000 km/s) and sound travels about 1/3 km/s.
• So if you count the number of seconds between seeing a flash and hearing the thunder, you can divide by 3 and get the distance to the storm in kilometres.

[image from http://www.forodecostarica.com/solaz/42155-el-jueguito-de-la-foto-13.html ]

Refraction of Sound

• Bending of waves—caused by changes in speed affected by temperature variations.

Intensity vs. Intensity Level

• The decibel (abbreviated dB) is the unit used to measure the intensity level of a sound.
• The decibel scale is a little odd because the human ear is incredibly sensitive.
• Your ears can hear everything from your fingertip brushing lightly over your skin to a loud jet engine.
• In terms of intensity, the sound of the jet engine is about 1,000,000,000,000 times more powerful than the smallest audible sound. That's a big difference!

Human Hearing and the Decibel

• For a sound with an intensity I in W/m², the intensity level β is defined as:

​

• The unit of β is called decibels, abbreviated dB.
• In this equation, I₀ is a reference intensity:

• On the decibel scale, the smallest audible sound (near total silence) is 0 dB.
• A sound 10 times more powerful is 10 dB.
• A sound 100 times more powerful than near total silence is 20 dB.
• A sound 1,000 times more powerful than near total silence is 30 dB.

Human Hearing and the Decibel

• If you are given the intensity level, β, and want to solve for intensity, you can rearrange:

From the pre-class survey

Sound Intensity Level

Note:

Every multiplicative factor of x10 in Intensity corresponds to an additive +10 decibels to the Sound Intensity Level.

• Your ears are able to detect sinusoidal sound waves with frequencies between about 20 Hz and 20 kHz.

• Low frequencies are perceived as “low pitch” bass notes, while high frequencies are heard as “high pitch” treble notes.
• Sound waves with frequencies above 20 kHz are called ultrasonic frequencies, and below 20 Hz are called infrasonic.
• Your ear is extremely good at discerning precise frequency components contained in a single sound wave.

Sound Wave Frequencies

Outer ear

Auditory canal

hammer

anvil

stirrup

Eardrum

Oval window

Cochlea

Basilar membrane

Sensory Hairs

Cochlear Nerve

to brain

Basilar membrane

Basilar membrane unwound

frequency

Amplitude

Low frequency sensitivity

High frequency sensitivity

Oval window

Sensory Hairs

Oscillations of Sensory Hairs for a Pure Tone of a specific frequency.

Basilar membrane

Basilar membrane unwound

Oscillations of Sensory Hairs for 2 Pure Tones

frequency

Amplitude

Low frequency sensitivity

High frequency sensitivity

Oval window

Fluid pressure wave

Sensory Hairs

Fourier’s Theorem:

Any sound wave can be produced mathematically as the sum of many pure tones of different amplitudes and frequencies.

• A long, narrow column of air, such as the air in a tube or pipe, can support a longitudinal standing sound wave.
• An open end of a column of air must be a pressure node (always at ambient pressure), thus the boundary conditions—nodes at the ends—are the same as for a standing wave on a string.
• A closed end forces a pressure antinode.

Standing Sound Waves

• With a wind instrument, blowing into the mouthpiece creates a standing sound wave inside a tube of air.
• The player changes the notes by using their fingers to cover holes or open valves, changing the effective length of the tube and thus its fundamental frequency:

• In both of these equations, v is the speed of sound in the air inside the tube.
• Overblowing wind instruments can sometimes produce higher harmonics such as f₂ = 2f₁ and f₃ = 3f₁.

Musical Instruments

Standing Waves in Pipes

Harmonics/Overtones On a String

Open

Open

Harmonics/Overtones in an Open-Open Wind Instrument

Pressure

Pressure

Pressure

Pressure

Closed

Open

Harmonics/Overtones in an Open-Closed Wind Instrument

Pressure

Pressure

Pressure

Pressure

Frequency Spectrum Shows Harmonics

• Both spectra below are the note C4, which has a fundamental frequency of 262 Hz.
• The flute, or open-open tube harmonics, are: 262, 524, 786, 1048, 1310, 1572 and 1834
• The clarinet, or open-closed tube harmonics are: 262, 786, 1310, and 1834
• The missing even harmonics give the clarinet its particular sound.

From https://mahamtariq.weebly.com/analysis.html

Pitch

• Here are two screen shots from Sound Spectrum Analyzer on my phone.
• They are both of me singing “LAAA”.
• The top graph was a low B. (124 Hz).
• The bottom graph was a slightly higher C-sharp (139 Hz).
• Do you see the slight increase of frequencies of the peaks?

How are frequencies on a piano determined?

Today, modern tuning standards define the musical note A4 (the 49th note from the left side of the standard 88-key full-size keyboard) is precisely 440 Hz. This is a tuning standard.

Every octave up is an increase of frequency of a factor of 2.

Every octave down is a decrease of frequency of a factor of ½.

A3 = 220 Hz

A5 = 880 Hz

A6 = 1760 Hz

…etc.

What about the other notes in between?

Equal Tempered Scale

C

D

E

F

G

A

B

C

C^#

D^#

F^#

G^#

A^#

Frequencies - Equal temperament A4=440 standard modern tuning - All 88 keys

• With a wind instrument, blowing into the mouthpiece helps set up a standing sound wave inside a tube of air.
• The player changes the notes by using their fingers to cover holes or open valves, changing the effective length of the tube and thus its fundamental frequency:

• In both of these equations, v is the speed of sound in the air inside the tube.
• f₁ is the pitch we hear. There are also harmonics in the spectrum at 2f₁, 3f₁, 4f₁, 5f₁, 6f₁, 7f₁, etc.

Wind Instruments

Waves in 2D or 3D

Doppler Effect

[image from http://www.physicsclassroom.com/class/waves/u10l3d.cfm ]

Doppler Effect

• By measuring the difference between the observed and known rest frequencies, you can determine the speed of the source.
• This is how radar guns work!

From today’s Preclass Survey

From today’s Preclass Survey

Light is also a wave

• Light from stars or galaxies moving toward or away from an observer on Earth is blue and redshifted, respectively.
• Notice that an approximate formula for the amount of shift from doppler effect is:

Gravitational Tugs in Planetary Systems

Doppler Effect Derivation: Moving Source, Stationary Observer

The Doppler effect: Moving Source

• When a wave source moves through the wave medium, the wave fronts travel outward at the speed of sound in all directions.
• But the centre of the wave fronts shifts as the source moves, making them closer together in front, and farther apart behind.
• Therefore the wavelength is reduced ahead of the source, and this makes the observed frequency higher. Also the wavelength is lengthened behind the source, making the observed frequency higher.

​

• Where f_s is the source frequency and f_obs is the observed frequency, v_w is the speed of sound, and v_s is the speed of the source.

​

source approaching.

source receding.

Doppler Effect Derivation: Stationary Sourcem, Moving Observer

The Doppler effect: Moving Observer

• When a wave source is stationary in the wave medium, a moving observer will experience a shift in frequencies.
• If the observer is moving toward the source, the effective speed of sound will be increased relative to the observer, making the observed frequency higher.
• If the observer is moving away from the source, the effective speed of sound will be decreased relative to the observer, making the observed frequency lower.

​

• Where f_s is the source frequency and f_obs is the observed frequency, v_w is the speed of sound, and v_obs is the speed of the observer.

​

observer moving toward source.

observer moving away from source.

Bow Waves

Supersonic

• Aircraft flying faster than the speed of sound.

Bow wave

• V-shape form of overlapping waves when object travels faster than wave speed.
• An increase in speed will produce a narrower V-shape of overlapping waves.

​

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Shock Waves

Shock wave

• Pattern of overlapping spheres that form a cone from objects traveling faster than the speed of sound.

​

​

​

​

​

​

�

Shock Waves

• Shock wave consists of two cones.
• a high-pressure cone generated at� the bow of the supersonic aircraft
• a low-pressure cone that follows �toward (or at) the tail of the aircraft
• It is not required that a moving source be noisy.

Sonic boom

• Sharp cracking sound generated by a supersonic aircraft
• Intensity due to overpressure and underpressure of atmospheric pressure between the two cones of the shock waves
• Continually produced by any object traveling faster than the speed of sound – an observer on the ground hears a single “boom” after the object passes overhead.

�Examples:

• supersonic bullet
• crack of circus whip

Shock Waves

• On Feb. 15, 2013, a meteor struck the Earth’s atmosphere over Russia
• It was traveling at supersonic speeds, so produced a sonic boom

[still from video at https://youtu.be/fBLjB5qavxY

• Many people were injured by the shaking and broken glass from this sonic boom
• The meteor exploded and thousands of small fragments fell in the countryside.

The Doppler effect: Moving Source

• When a wave source moves through the wave medium, the wavelength is reduced ahead of the source, and this makes the observed frequency higher.
• Also, the wavelength is lengthened behind the source, making the observed frequency lower:

​

• Where f_s is the source frequency and f_obs is the observed frequency, v_w is the speed of sound, and v_s is the speed of the source.

​

source approaching.

source receding.

The Doppler effect: Moving Observer (Review)

• If the observer is moving toward the source, the effective speed of sound will be increased relative to the observer, making the observed frequency higher.
• If the observer is moving away from the source, the effective speed of sound will be decreased relative to the observer, making the observed frequency lower.

​

• Where f_s is the source frequency and f_obs is the observed frequency, v_w is the speed of sound, and v_obs is the speed of the observer.

​

observer moving toward source.

observer moving away from source.

Doppler Effect

• If both the source and the observer are moving relative to the air, two of these four equations can be used in sequence.
• First, find the observed frequency as if the observer was stationary.
• Next, use this as the new source frequency as if the source was stationary, and find the observed frequency for the moving observer.

Source approaching:

Source receding:

Observer moving toward source:

Observer moving away from source:

• People were asked to judge the relative loudness between two pure-tone sounds of different frequencies.
• For each curve shown, there was a particular reference tone with a freq. of 1000 Hz, and a particular sound intensity level.
• Observers compared this reference tone to tones of different frequencies, and the experimenters varied the sound intensity level of those tones until the observers claimed that they sounded equally as loud as the reference at 1000 Hz.

From today’s Preclass Survey

Standing Waves

• For a string which is fixed at both ends, the boundary conditions at each end require that any standing wave mode have nodes at the ends.
• The animation below shows the first four allowed mode shapes for a fixed-fixed string.
• The number of "humps" (antinodes) corresponds to the value of n, a whole number called “the harmonic number”

Standing Waves