double the double the distancedistance

reduce intensity reduce intensity by a factor of 4by a factor of 4

BUT

Half of intensity does not “feel” like half Half of intensity does not “feel” like half as loud !as loud !

Before going into other properties Before going into other properties like interference, diffraction, … we like interference, diffraction, … we will make a mathematical excursion will make a mathematical excursion (basically covered in Chapter 1 of (basically covered in Chapter 1 of

Berg&Stork)Berg&Stork)

How to read graphsHow to read graphs

• figure what’s in the horizontal axis (w/ units)figure what’s in the horizontal axis (w/ units)

• figure what’s in the vertical axis (w/ units)figure what’s in the vertical axis (w/ units)

• find the value of “savings” at any particular timefind the value of “savings” at any particular time

22P x x

visualvisual analyticalanalytical

WATCH !!!WATCH !!!

same informationsame information

Let us watch now the graph of sound Let us watch now the graph of sound pressure variation as a function of time for pressure variation as a function of time for

some real soundssome real sounds

• Which sounds are periodic ?

• What distinguishes noise from “musical” sounds ?

• What makes a sound louder ?

• What makes a sound lower or higher in pitch ?

WavetoolsWavetools

http://www.physics.mcgill.ca/~grant/225B/http://www.physics.mcgill.ca/~grant/225B/Wavetools/Computerstuff.htmlWavetools/Computerstuff.html

A very “pure” (but annoying) sound:A very “pure” (but annoying) sound:

““Musical” sounds (“tones”) are periodicMusical” sounds (“tones”) are periodic

period (T)period (T)

amplitude (A)

frequency (f) = 1/T, T=1/ffrequency (f) = 1/T, T=1/f

The mathematics of periodic wavesThe mathematics of periodic waves

wavelength: distance between two crests v

vTf

period: time between two crests

Tv

frequency: how many crests per second1 v

fT

• Period (T) = time for one cycle (measured in s, …)

• Frequency (f) = number of cycles per second (measured in 1/s = Hz)

Example1 1 1

2 0.5 0.52

T s f HzT s s

1 1400 0.0025 2.5

400f Hz T s ms

f Hz

amplitude = loudnessamplitude = loudness

frequency = pitchfrequency = pitch

shape = timbreshape = timbre

With a few qualifications …

It is a good time now to read Berg & Stork, It is a good time now to read Berg & Stork,

Chapter 1Chapter 1

““Pure” tones are sine wavesPure” tones are sine waves

Harmonic oscillatorHarmonic oscillator

[Using a wave generator, find the lowest and [Using a wave generator, find the lowest and highest audible frequencies and calculate highest audible frequencies and calculate

their period and wavelength]their period and wavelength]

sin(2 )P A f t

amplitude frequency

phasephase

How sounds combine: adding two wavesHow sounds combine: adding two waves

300 Hz

450 Hz

300 Hz+

450 Hz

BeatsBeats

Tartini tones

Back to sound (wave) properties …Back to sound (wave) properties …

• reflection

• refraction

• 1/r2 law

• diffraction

• interference

• Doppler effect

Waves interfere:Waves interfere:

• constructively (amplitudes have the same sign and enhance the wave)

• destructively (amplitudes have opposite sign and partially or completely cancel)

Interference movieInterference movie

http://www.colorado.edu/physics/2000/schroedinger/big_interference.html

Interference appletInterference applet

http://www.falstad.com/ripple/ex-2source.html

Doppler effectDoppler effect

Doppler effect appletDoppler effect applet

http://www.lon-capa.org/~mmp/applist/doppler/http://www.lon-capa.org/~mmp/applist/doppler/d.htmd.htm

[go to the blackboard and do some real physics][go to the blackboard and do some real physics]

Applications:Applications:

• Doppler radarDoppler radar

• Doppler sonogramDoppler sonogram

• Expansion of the UniverseExpansion of the Universe

mostly mostly constructive constructive interferenceinterference

mostly mostly destructive destructive interferenceinterference

•Larger wavelength (~ obstacles) : more diffractionLarger wavelength (~ obstacles) : more diffraction

•Shorter wavelength (<< obstacles) : less diffractionShorter wavelength (<< obstacles) : less diffraction

DiffractionDiffraction

Ultrasound and sonogramsUltrasound and sonograms

Typical frequencies: 300 MHz

1 1340 0.000001

1300.000.000

mvT v m

f ss

It is a good time now to read Berg & It is a good time now to read Berg & Stork, Stork,

Chapter 2.4, 2.5, 2.6, 2.7, 2.8, 2.9Chapter 2.4, 2.5, 2.6, 2.7, 2.8, 2.9