Chapter 4: Wave equations

Chapter 4: Wave equations

What is a wave?

what is a wave?

anything that moves

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

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 f(x -vt) represents a rightward, or forward, propagating wave.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 Þ and Þ

Combine to get the 1D differential wave equation:

2

2

22

2 1ty

vxy

1

xx v

tx

xx

xy

xy

tx

xy

ty

xf

xy

2

2

2

2

xf

xy

xfv

ty

2

22

2

2

xfv

ty

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: n = 1/T angular frequency: w = 2pn

propagation constant: k = 2p/lwave number: k = 1/l

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

v = nl

Note:

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

y = A sin[k(x ± vt)] j = k(x ± vt)

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

which confirms that v is the wave velocity.

vdtdx

The phase of a harmonic wave

A = amplitude

j0 = initial phase angle (or absolute phase) at x = 0 and t =0

p

Absolute phase

y = A sin[k(x ± vt) + j0]

How fast is the wave traveling?

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

Since n = 1/T:

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

v = ln

v = w/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

w

dkdvgroupw

works for any periodic waveany ??

Complex numbers make it less complex!

x: real and y: imaginary

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

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

eij = cosj + i sinj

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

P = A exp(ij) = A eiφ

where

A = Amplitude

j = Phase

Harmonic waves as complex functions

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

y = Aei(kx-wt)

so that y = Re(y) = A cos(kx – wt)y = Im(y) = A sin(kx – wt)

~

~~

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

Y = A sin(kx – wt)

Consider a snapshot in time, say t = 0:

Y = A sin(kx)Y = A sin(kr cosq)

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

Y = A sin(k · r – wt)

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

arbitrary direction

General case

k · r = xkx = yky + zkz

kr cosq ks

In complex form:

Y = Aei(k·r - wt)

2

2

22

2

2

2

2

2 1tΨ

vzΨ

yΨ

xΨ

Substituting the Laplacian operator:

2

2

22 1

tΨ

vΨ

Spherical waves

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

tkrierAΨ w

Cylindrical waves

tkieAΨ w

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 EtEB Bt

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

soII. No monopoles:

so

Always!

Always!

E and B are perpendicular

0),( 0 trkjeEtr wE

00 ωtrkjeEkj

00 Ek

0),( 0 trkjeBtr wB

00 ωtrkjeBkj

00 Bk

k

E

k

B

Perpendicular to the direction of propagation:

III. Faraday’s law:

so

Always!

t

BE

t

eE trkj

B

w

0

it

BieEjkEjk xtrkjoyzozy

ˆˆ

w

oxoyzozy B

EkEk

w

EB

Perpendicular to each other:

E and B are perpendicular

E and B are harmonic

)sin(

)sin(

0

0

t

t

w

w

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, 0 = 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 w = 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 w

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

and

x 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

EE

]exp[),,,( 0 titzyx w rkEE

),,( 0000 zyx EEEE

z

y

x

Arbitrary direction

constzkykxk zyx

k

= wave vectork

0E

r

tiet w rkErE

0),(

k Plane of constant , constant E rk

rk

ttzyx w rkEE

sin),,,( 0

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

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

2

021 BuB

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

2120 0

BEuuu BEtotal

Electromagnetic waves transmit energy

202

1 EuE

2

021 BuB

Rate of energy transport: Power (W)

cDt

A

Power per unit area (W/m2):

Pointing vectorPoynting

Rate of energy transport

ucS

EBcS 20

BES

20c

ttuAc

tVu

tenergyP

DD

DD

D

ucAP

k

Take the time average:

“Irradiance” (W/m2)

Poynting vector oscillates rapidly

tEE wcos0 tBB wcos0

tBEcS w 200

20 cos

tBEcS w 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 vectorlinear polarization

x

y

Evolution of electric field vectorcircular polarization

x

y

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

The following may be covered in the werkcollege on 14 September 2011:

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

Exercises