Chapter 4: Wave equations

Features of today’s lecture:55 slides10 meters of rope1 song

(condensed version)

This idea has guided my research: for matter, just as much as for radiation, in particular light, we must introduce at one and the same time the corpuscle concept and wave concept.

Louis de Broglie

0 1 2 3

Waves move

t0 t1 > t0 t2 > t1 t3 > t2

x

f(x)

v

A wave is…a disturbance in a medium

-propagating in space with velocity v-transporting energy -leaving the medium undisturbed

let’s make some waves

To displace f(x) to the right: x x-a, where a >0.

Let a = vt, where v is velocity and t is time-displacement increases with time, and-pulse maintains its shape.

So x' = f(x -vt) represents a rightward, or forward, propagating wave.x' = f(x+vt) represents a leftward, or backward, propagating

wave.

0 1 2 3

t0 t1 > t0 t2 > t1 t3 > t2

x

f(x)

v

1D traveling wave

The wave equation

works for anything that moves!

Let y = f (x'), where x' = x ± vt. So and

 

Now, use the chain rule:

 

So and

  

Combine to get the 1D differential wave equation:

2

2

22

2 1

t

y

vx

y

1

x

xv

t

x

x

x

x

f

x

y

t

x

x

f

t

y

x

f

x

y

2

2

2

2

x

f

x

y

x

fv

t

y

2

22

2

2

x

fv

t

y

Harmonic waves

• periodic (smooth patterns that repeat endlessly)•generated by undamped oscillators undergoing harmonic motion

• characterized by sine or cosine functions -for example,

-complete set (linear combination of sine or cosine functions can represent any periodic waveform)

vtxkAvtxfy sin)(

Snapshots of harmonic waves

T

A

at a fixed time: at a fixed point:

A

frequency: = 1/T angular frequency: = 2

propagation constant: k = 2/wave number: = 1/

≠ v v = velocity (m/s) = frequency (1/s)

v =

Note:

The phase, , is everything inside the sine or cosine (the argument).

y = A sin[k(x ± vt)]

= k(x ± vt)

For constant phase, d = 0 = k(dx ± vdt)

which confirms that v is the wave velocity.

vdt

dx

The phase of a harmonic wave

A = amplitude

0 = initial phase angle (or absolute phase) at x = 0 and t =0

Absolute phase

y = A sin[k(x ± vt) + 0]

How fast is the wave traveling?

The phase velocity is the wavelength/period: v = /T

Since = 1/T:

In terms of k, k = 2/, and the angular frequency, = 2/T, this is:

v = v =

v = /k v = /k

Phase velocity vs. Group velocity

Here, phase velocity = group velocity (the medium is non-dispersive).

In a dispersive medium, the phase velocity ≠ group velocity.

kvphase

dk

dvgroup

works for any periodic waveany ??

Complex numbers make it less complex!

x: real and y: imaginary

P = x + i y = A cos() + i A sin()

where

P = (x,y)

1i

“one of the most remarkable formulas in mathematics”Euler’s formula

Links the trigonometric functions and the complex exponential function

ei = cos + i sin

so the point, P = A cos() + i A sin(), can also be written:

P = A exp(i) = A eiφ

where

A = Amplitude

= Phase

Harmonic waves as complex functions

Using Euler’s formula, we can describe the wave as

y = Aei(kx-t)

so that y = Re(y) = A cos(kx – t)

y = Im(y) = A sin(kx – t)

~

~

~

Why? Math is easier with exponential functions than with trigonometry.

Plane waves

• equally spaced• separated by one wavelength apart• perpendicular to direction or propagation

The wavefronts of a wave sweep along

at the speed of light.

A plane wave’s contours of maximum field, called wavefronts, are planes. They extend over all space.

Usually we just draw lines; it’s easier.

Wave vector krepresents direction of propagation

= A sin(kx – t)

Consider a snapshot in time, say t = 0:

= A sin(kx)

= A sin(kr cos)

If we turn the propagation constant (k = 2/into a vector, kr cos = k · r

= A sin(k · r – t)

• wave disturbance defined by r• propagation along the x axis

arbitrary direction

General case

k · r = xkx = yky + zkz

kr cosks

In complex form:

= Aei(k·r - t)

2

2

22

2

2

2

2

2 1

t

Ψ

vz

Ψ

y

Ψ

x

Ψ

Substituting the Laplacian operator:

2

2

22 1

t

Ψ

vΨ

Spherical waves

• harmonic waves emanating from a point source• wavefronts travel at equal rates in all directions

tkrier

AΨ

Cylindrical waves

tkieA

Ψ

the wave nature of light

Electromagnetic waves

http://www.youtube.com/watch?v=YLlvGh6aEIs

Derivation of the wave equation from Maxwell’s equations

0

0

BE E

t

EB B

t

Derivation:

I. Gauss’ law (in vacuum, no charges):

so

II. No monopoles:

so

Always!

Always!

E and B are perpendicular

0),( 0 trkjeEtr E

00 ωtrkjeEkj

00 Ek

0),( 0 trkjeBtr B

00 ωtrkjeBkj

00 Bk

k

E

k

B

Perpendicular to the direction of propagation:

III. Faraday’s law:

so

Always!

t

B

E

t

eE trkj

B

0

it

BieEjkEjk xtrkj

oyzozyˆˆ

oxoyzozy B

EkEk

EB

Perpendicular to each other:

E and B are perpendicular

E and B are harmonic

)sin(

)sin(

0

0

t

t

rkBB

rkEE

Also, at any specified point in time and space,

cBE where c is the velocity of the propagating wave,

m/s 10998.21 8

00

c

The speed of an EM wave in free space is given by:

0= permittivity of free space, = magnetic permeability of free space

To describe EM wave propagation in other media, two properties of the medium are important, its electric permittivity ε and magnetic permeability μ. These are also complex parameters.

= 0(1+ ) + i = complex permittivity

= electric conductivity = electric susceptibility (to polarization under the influence of an external field)

Note that ε and μ also depend on frequency (ω)

kc

00

1

EM wave propagation in homogeneous media

x

z

y

Simple case— plane wave with E field along x, moving along z:

which has the solution:

 

where

and

andx y zk r k x k y k z

, ,x y zk k k k

, ,r x y z

General case— along any direction

2

2

22 1

tc

E

E

]exp[),,,( 0 titzyx rkEE

),,( 0000 zyx EEEE

z

y

x

Arbitrary direction

constzkykxk zyx

k

= wave vectork

0E

r

tiet rkErE

0),(

k Plane of constant , constant E rk

rk

ttzyx rkEE

sin),,,( 0

Energy density (J/m3) in an electrostatic field:

Energy density (J/m3) in an magnetostatic field:

2

02

1BuB

Energy density (J/m3) in an electromagnetic wave is equally divided:

2120 0

BEuuu BEtotal

Electromagnetic waves transmit energy

202

1 EuE

2

02

1BuB

Rate of energy transport: Power (W)

ct

A

Power per unit area (W/m2):

Pointing vector

Poynting

Rate of energy transport

ucS

EBcS 20

BES

20c

t

tuAc

t

Vu

t

energyP

ucAP

k

Take the time average:

“Irradiance” (W/m2)

Poynting vector oscillates rapidly

tEE cos0 tBB cos0

tBEcS 200

20 cos

tBEcS 200

20 cos

002

021 BEcS

eES

A 2D vector field assigns a 2D vector (i.e., an arrow of unit length having a unique direction) to each point in the 2D map.

Light is a vector field

Light is a 3D vector field

A 3D vector field assigns a 3D vector (i.e., an arrow having both direction and length) to each point in 3D space.

A light wave has both electric and magnetic 3D vector fields:

Polarization

• corresponds to direction of the electric field• determines of force exerted by EM wave on

charged particles (Lorentz force)• linear,circular, eliptical

Evolution of electric field vector

linear polarization

Evolution of electric field vector

circular polarization

You are encouraged to solve all problems in the textbook (Pedrotti3).

The following may be covered in the werkcollege on 8 September 2010:

Chapter 4:3, 5, 7, 13, 14, 17, 18, 24

Exercises