• So far we’ve essentially treated light in terms of rays– But in the 19th century is was realized that light really

exhibited wave-like behavior• To understand this part of the nature of light, we need to

explore some of the characteristics of waves• The basics: what is a wave?

– It’s a transfer of energy from one place to another withouttransferring matter

• Common examples are:– waves in water (some of the energy from a pebble thrown

into a pond is transmitted to the shore)– sound waves (energy from my voice is transmitted to your

ears)• In both cases, matter wiggles as the wave passes by, but

then returns to its normal position

Lecture 19: Waves

Types of Waves• The way that matter wiggles is different for waves on a

string and sound waves– for waves on a string, the molecules in the string wiggle in a

direction perpendicular to the direction of the wave• any wave that behaves this way is called a transverse wave

– for sound waves, the air molecules wiggle back and forth inthe same direction the wave is traveling

• any wave that behaves this way is called a longitudinal wave

Sinusoidal Waves• Waves typically oscillate back and forth periodically• We can represent this behavior with a sine wave

– the distance by which the wiggling matter is displaced fromits equilibrium position is a function of both position andtime:

y x,t( ) = Asin kx !"t( )

Amplitude

Wave number

Angular frequency

Wavelength, Frequency, and Period• The wave is periodic in both space and time• First, consider a wave “frozen” at an instant in time (t = 0):

• Wave repeats whenWavelength (λ)

sin k x +  = sin kx( ) = sin kx + 2"( )

k! = 2"

k =2"

!

• We can similarly consider a single position (x = 0)• As the wave goes by, the matter at that position wiggles up

and down and back to where it started in one period (T):

• The frequency is the inverse of the period:

sin ! t +T[ ]( ) = sin !t( ) = sin !t + 2"( )

!T = 2"

! =2"

T

f =1

T=

!

2"

Speed and Direction• Since we defined a wave as something that’s carrying

energy from place to place, the wave must have a speedand direction

• For sinusoidal waves, we can find this by considering themotion of a point at the peak of the wave:

• Note also that:

Want y = Asin kx !"t( ) = A

That means: kx !"t = #

x =# +"t

k

v =dx

dt="

k

Note that wave is movingin the +x direction

v =!

k=2" f

2"

#

= f#

Superposition and Interference• What happens if two waves are traveling in the same

medium?– i.e., two waves on the same string, or two sources of sound

in the same room• The medium responds to the sum of all the waves:

• This is the principle of superposition• Consider two waves with the same amplitude and

frequency, but different values at x = t =0– this value is known as the phase of the wave

y x,t( ) = y1 x,t( ) + y2 x,t( ) +…+ y

Nx,t( )

• The nature of the resulting wave depends on φ:

• Special cases:1. φ = 0:

2. φ = 180o:

y1x,t( ) = Asin kx !"t( )

y2x,t( ) = Asin kx !"t +#( )

y x,t( ) = A sin kx !"t( ) + sin kx !"t +#( )$% &'

= A sin kx !"t( ) + sin kx !"t( )cos# + cos kx !"t( )sin#$% &'

= A sin kx !"t( ) 1+ cos#( ) + cos kx !"t( )sin#$% &'

y x,t( ) = A sin kx !"t( ) 1+1( ) + cos kx !"t( ) 0( )#$ %&

= 2Asin kx !"t( )

y x,t( ) = A sin kx !"t( ) 1!1( ) + cos kx !"t( ) 0( )#$ %&

= 0

Twice the amplitude of the original wave

No wave at all!

Interference in Two Dimensions• If waves are traveling in more than one dimension (across

the surface of water, for example) even more interestingpatterns of interference can result

• Assume we have two identical wave sources– lines below represent wave crests When two crests

(or troughs)overlap, get alarge amplitude(constructiveinterference)

When a crestoverlaps atrough, getsmallamplitude(destructiveinterference)

• We can tell whether a given point in space will have amaximum or minimum of the amplitude by finding thedistance from that point to each source (d1 and d2)

• If the distances differ by an integer number ofwavelengths:

then crests and troughs will “line up” to produce largeamplitudes

• On the other hand, if:

then the crest of one wave will coincide with the trough ofthe other, and the two waves will cancel

d1! d

2= n",n = 0,1,2...

d1! d

2= n +

1

2

"#$

%&'(,n = 0,1,2...

Non-sinusiodal waves• So far we’ve only considered waves that can be

represented with a sine function– but there are many examples of waves with different shapes– why do we care so much about sine waves then?

• Due to Fourier’s Theorem, which tells us that any periodicfunction can be be formed from the sum of sine waves:

• The fundamental frequency ω1is the frequencycorresponding to the period of the function, T:

y t( ) = An sin!nt + Bn cos!nt( )n

"

2!

"1

= f1=1

T

• The other frequencies are multiples of the fundamentalfrequency:

• The coefficients An and Bn represent the amplitude of eachsine wave in the sum– you’ll learn how to calculate them in a more advanced

course!

!n= n!

1