Chapter 2: Wave Motion

Lecture 4

3D wave equation Spherical, cylindrical waves

Chapter 3: Electromagnetic theory Review of basic laws of electromagnetism Maxwell’s equations

Plane waves: Cartesian coordinates

trkiAetr , zkykxkrk zyx

tzyxkiAetzyx ,,, , , - direction cosines of k

tzkykxki zyxAetzyx ,,,

Wave eq-ns in Cartesian coordinates:

222zyx kkkk

1222

Importance of plane waves:• easy to generate using any harmonic generator• any 3D wave can be expressed as superposition of plane waves

Three dimensional differential wave equationTaking second derivatives for tzyxkiAetzyx ,,,can derive the following:

222

2

kx

222

2

ky

222

2

kz

+

+

22

2

2

2

2

2

kzyx

22

2

t

2

2

21

t

combine and use: vk

3-D differential wave equation

2

2

22

2

2

2

2

2 1tzyx

v

Three dimensional differential wave equation

2

2

22

2

2

2

2

2 1tzyx

vAlternative expression

Use Laplacian operator:

2

2

2

2

2

22

zyx

2

2

22 1

t

v

Using =kv, we can rewrite tzyxkiAetzyx ,,, tzyxikAetzyx v ,,,function of tzyx v

tzyxftzyx v ,,,It can be shown, that:

tzyxgtzyx v ,,,

f, g are plane-wave solutions of the diff. eq-n, provided that are twice differentiable.Not necessarily harmonic!

In more general form, the combination is also a solution: tkkrgCtkkrfCtzyx vv //,,, 21

ExampleGiven expression ,where a>0, b>0: btaxtx 2,

Does it correspond to a traveling wave? What is its speed?

Solution:1. Function must be twice differentiable

32 x

x

42

2

6 ax

x

bt

02

2

t

2

2

22

2

2

2

2

2 1tzyx

v

2. Wave equation:

06 4 ax Is not solution of wave equation!This is not a wave traveling at constant speed!

Spherical waves

2-D concentric water wavesSpherical waves originate from a point source and propagate at constant speed in all directions: waveforms are concentric spheres. Isotropic source - generates waves in all directions.

spherical waveSymmetry: introduce spherical coordinates

cossinsincossin

rzryrx

Symmetry: the phase of wave should only depend on r, not on angles:

rrr ,,

Spherical waves

cossinsincossin

rzryrx

2

2

22

2

22

2

sin1

sinsin1

1

r

r

rr

rr

2

2

22 1

t

v

r

rrr

22

2 1

Since depends only on r:

rrr 2

22 1

evaluates to the same

2

2

22

2 11t

rrr

v

Wave equation:

×r

rt

rr 2

2

22

2 1

v

Spherical waves

rt

rr 2

2

22

2 1

v

This is just 1-D wave equationIn analogy, the solution is:

trftrr v,

r

trftr v, - propagates outwards (diverging)

+ propagates inward (converging)

Note: solution blows up at r=0

In general, superposition works too:

r

trgCr

trfCtr vv

21,

Harmonic Spherical waves

r

trftr v,

Harmonic spherical wave

trkr

tr vcos, A

In analogy with 1D wave:

triker

tr vA, - source strengthA

Constant phase at any given time: kr=constAmplitude decreases with r A

Single propagatingpulse

Spherical harmonic waves

trkr

tr vcos, A

Decreasing amplitude makes sense:

Waves can transport energy (even though matter does not move)

The area over which the energy is distributed as the wave moves outwards increases

Amplitude of the wave must drop!

Note: spherical waves far from source approach plane waves:

Cylindrical wavesWavefronts form concentric cylinders of infinite length

zzryrx

sincos

2

2

2

2

2

2

1

1

zr

rr

rr

Symmetry: work in cylindrical coordinates rzrr ,,

2

2

211

trr

rr

v

It is similar to Bessel’s eq-n.At larger r the solution can be approximated:

Harmonic cylindrical wave

trkr

tr vcos, A

triker

tr vA,

Cylindrical waves

Harmonic cylindrical wave

trkr

tr vcos, A

triker

tr vA,

Can create a long wave source by cutting a slit and directing plane waves at it:emerging waves would be cylindrical.

Chapter 3

Electromagnetic theory, Photons.and Light

Review of basic laws of electromagnetism Maxwell’s equations

Basic laws of electromagnetic theory

Q1 Q2

F FBlack box

rrQE

QEF

ˆ4

122

0

1

Interaction occurs via electric fieldElectric field can exist even when charge disappears (annihilation in black box)

Electric field

electric permittivity of free space

221

041

rQQ

FF

Coulomb force law:

Basic laws of electromagnetic theoryGauss’s Law: electric

Karl Friedrich Gauss (1777-1855)Electric field flux from an enclosed volume is proportional to the amount of charge inside

qE0

1

qSdES

0

1

If there are no charges (no sources of E field), the flux is zero: 0S SdE

More general form:

VS

dVSdE 0

1

Charge density

Basic laws of electromagnetic theory

Magnetic field Moving charges create magnetic field

20 ˆ

4 rrvqB

The Biot-Savart law for

moving charge

Magnetic field interacts with moving charges: BvqFmagnetic

Charges interact with both fields:

BvqEqF

(Lorentz force)

permeability of free space

Basic laws of electromagnetic theoryGauss’s Law: magnetic

Magnetic field flux from an enclosed volume is zero (no magnetic monopoles)

0M

0S SdB

Basic laws of electromagnetic theoryFaraday’s Induction Law1822: Michael Faraday Changing magnetic field can result in variable electric field

dtdemf M

AC

SdBdtdldE

Formalversion

dAnSd ˆ

dAnBSdBd M ˆ

normal to area

cosBdAd M

Changing current in the solenoid produces changing magnetic field B. Changing magnetic field flux creates electric field in the outer wire.

area

angle between B and normal to the area dA

Basic laws of electromagnetic theoryAmpère’s Circuital Law

1826: (Memoir on the Mathematical Theory of Electrodynamic Phenomena, Uniquely Deduced from Experience)

All the currents in the universe contribute to Bbut only ones inside the path result in nonzero path integral

A wire with current creates magnetic field around it

Ampere’s law

pathinsideCIldB _0

AC

SdJldB

0

Current density

Incomplete!

Basic laws of electromagnetic theoryAmpère’s-Maxwell’s Law

Maxwell considered all known laws and noticed asymmetry:

AC

SdBdtdldE

0S SdB

qSdES

0

1

AC

SdJldB

0

Gauss’s

Gauss’s

Faraday’s

Ampère’s

Changing magnetic field leads to changing electric field

No similar term here

Hypothesis: changing electric field leads to variable magnetic field

Basic laws of electromagnetic theoryAmpère’s-Maxwell’s Law

AC

SdJldB

0

Ampère’s law

iSdJldBAC 00

1

02

0 ACSdJldB

The B will depend on area:

Workaround: Include term that takes into account changing electric field flux in area A2:

AC

SdtEJldB

00 Ampère’s-Maxwell’s Law:

displacement current density

Maxwell equations

AC

SdBdtdldE

0S SdB

qSdES

0

1

Gauss’s

Gauss’s

Faraday’s

Ampère-Maxwell’s

AC

SdtEJldB

00

+Lorentz force: BvqEqF

fields are defined through interaction with charges

Inside the media electric and magnetic fields are scaled. To account for that the free space permittivity 0 and 0 are replaced by and :

0 EKdielectric constant, KE>1

0 MKrelative permeability

In vacuum(free space)

Maxwell equations

AC

SdBdtdldE

0S SdB

qSdES

1

Gauss’s

Gauss’s

Faraday’s

Ampère-Maxwell’s

AC

SdtEJldB

Lorentz force: BvqEqF

+

fields are defined through interaction with charges

In matter

Maxwell equations: free space, no chargesCurrent J and charge are zero

AC

SddtBdldE

0S SdB

0S SdE

AC

SdtEldB

00

There is remarkable symmetry between electric and magnetic fields!

Integral form of Maxwell equations in free space:

no magnetic ‘charges’

no electric charges

changing magnetic field creates changing electric fieldchanging electric field creates changing magnetic field