chapter 2 wave motion - purdue university
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Chapter 2Wave Motion
Lecture 4
Plane waves 3D Differential wave equation Spherical waves Cylindrical waves
3-D waves: plane waves(simplest 3-D waves)
All the surfaces of constant phase of disturbance form parallel planes that are perpendicular to the propagation direction
3-D waves: plane waves(simplest 3-D waves)
All the surfaces of constant phase of disturbance form parallel planes that are perpendicular to the propagation direction
Unit vectors
An equation of plane that is perpendicular to kji ˆˆˆ
zyx kkkk
aconstrk
All possible coordinates of vector r are on a plane k
Can construct a set of planes over which varies in space harmonically:
rkAr sin
rkAr cosor
rkiAer or
Plane waves
rkr sin The spatially repetitive nature
can be expressed as:
kkrr
In exponential form:
kirkikkrkirki eAeAeAer /
For that to be true: 12 ie
2k
2
k
Vector k is called propagation vector
Plane waves: equation
rkiAer This is snap-shot in time, no time dependence
To make it move need to add time dependence the same way as for one-dimensional wave:
trkiAetr , Plane wave equation
Plane wave: propagation velocity
Can simplify to 1-D case assuming that wave propagates along x:
trkiAetr ,
i||r tkxiAetr ,
We have shown that for 1-D wave phase velocity is:
k
vThat is true for any direction of k+ propagate with k- propagate opposite to k
More general case: see page 26
Example: two plane wavesSame wavelength: k1= k2=k=2/,Write equations for both waves.Solution:
Same speed v:1=2==kv
trkiAe
Dot product:zkykxkrk zyx
Wave 1: zkrk 11
tzkieA 111
directionWave 2: zkykrk cossin 222
tzykieA cossin22
2
tkzA cos11 tzykA cossincos22
Note: in overlapping region = 1 + 2
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 2, cbtaxtx
Does it correspond to a traveling wave? What is its speed?
Solution:1. Function must be twice differentiable
acbtaxx
2
22
2
2ax
bcbtaxt
2
22
2
2bt
2
2
22
2
2
2
2
2 1tzyx
v
2. Speed:
22
2 212 bav
ab
v
Direction: negative x direction
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 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 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
Lecture 5
Basic laws of electromagnetic theory Maxwell’s equations Electromagnetic waves Polarization of EM waves Energy and momentum
Basic laws of electromagnetic theory
221
041
rQQ
FF
Q1 Q2
F F
Coulomb force law:
Black 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
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: 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 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
Maxwell equations: differential form(free space)
0 E
0 B
tBE
tEB
00
Notation: kz
jy
ix
ˆˆˆ
2
2
2
2
2
22
zyx
Laplacian:
0)(
z
Ey
Ex
EEdivE zyx
ky
Ex
Ej
xE
zEi
zE
yEE xyzxyz ˆˆˆ
tB
zE
yE xyz
tB
xE
zE yzx
tB
yE
xE zxy
)(EcurlE
Electromagnetic waves
(free space)
tBE
tEB
00
Changing E field creates B fieldChanging B field creates E field
Is it possible to create self-sustaining EM field?
Can manipulate mathematically into:
2
2
002
tEE
2
2
002
tBB
Electromagnetic waves
2
2
002
tEE
2
2
002
tBB
2
2
2
2
2
22
ˆˆˆ
zyx
kx
jx
ix
2
2
002
2
2
2
2
2
2
2
002
2
2
2
2
2
2
2
002
2
2
2
2
2
tE
zE
yE
xE
tE
zE
yE
xE
tE
zE
yE
xE
zzzz
yyyy
xxxx
2
2
002
2
2
2
2
2
2
2
002
2
2
2
2
2
2
2
002
2
2
2
2
2
tB
zB
yB
xB
tB
zB
yB
xB
tB
zB
yB
xB
zzzz
yyyy
xxxx
Resembles wave equation: 2
2
22
2
2
2
2
22 1
tzyx
v
Each component of the EM field obeys the scalar wave equation, provided that
00
1
v
Light - electromagnetic wave?
00
1
vMaxwell in ~1865 found that EM wave must move at speed
At that time permittivity 0 and permeability 0 were known from electric/magnetic force measurements and Maxwell calculated
km/s 740,3101
00
v
Speed of light was also measured by Fizeau in 1949: 315,300 km/s
Maxwell wrote: This velocity is so nearly that of light, that it seems we have strong reason to conclude that light itself (including radiant heat, and other radiations if any) is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws.
Exact value of speed of light: c = 2.997 924 58 × 108 m/sceler (lat. - fast)
Electromagnetic waveAssume: reference frame is chosen so that E=(Ex,0,0)
longitudinal wave, propagates along x
0 E
0 B
tBE
tEB
00
0
xEx Ex does not vary with x
This cannot be a wave!
Conclusion: it must be transverse wave, i.e. Ex=0. Similarly Bx=0.
Since E is perpendicular to x, we must specify its direction as a function of time
Direction of vector E in EM wave is called polarization
Simple case: polarization is fixed, i.e. direction of E does not change
0
zE
yE
xE zyx
Polarized electromagnetic waveWe are free to chose y-axis so that E field propagating along x is polarized along y: (0, Ey ,0).
0 E
0 B
tBE
tEB
00
tB
zE
yE xyz
tB
xE
zE yzx
tB
yE
xE zxy
tB
xE zy
Also: Bx=By=const (=0)
E-field of wave has only y componentB-field of wave has only z component(for polarized wave propagating along x)
In free space, the plane EM wave is transverse
Harmonic polarized electromagnetic wave
cxtEtxE yy /cos, 0
Harmonic functions are solution for wave equation:
polarized along y axis propagates along x axis
tB
xE zy
dtx
EB y
z
Find B:
cxtEc
txB yz /cos1, 0
zy cBE
This is true for any wave:- amplitude ratio is c- E and B are in-phase
Harmonic polarized electromagnetic wave
* direction of propagation is in the direction of cross-product:
BE
* EM field does not ‘move’ in space, only disturbance does.Changing E field creates changing B field and vice versa
Electromagnetic waves
Energy of EM wave
It was shown (in Phys 272) that field energy densities are:
20
2EuE
2
021 BuB
Since E=cB and c=(00)-1/2:
BE uu
- the energy in EM wave is shared equally between electric and magnetic fields
Total energy: 2
0
20
1 BEuuu BE (W/m2)
The Poynting vector
EM field contains energy that propagates through space at speed cEnergy transported through area A in time t: uAct
EBEBcBEcEcuctAtuAcS
00
000
20
11
Energy S transported by a wave through unit area in unit time:
E c2
The Poynting vector:
BES
0
1
power flow per unit area for a wave, direction of propagation is direction of S.
(units: W/m2)
John Henry Poynting (1852-1914)
The Poynting vector: polarized harmonic wave
BES
0
1
Polarized EM wave:
trkEE
cos0
trkBB
cos0
Poynting vector:
trkBES
200
0
cos1
This is instantaneous value: S is oscillating
Light field oscillates at ~10 15 Hz -most detectors will see average value of S.
Irradiance
trkBES
200
0
cos1Average value for periodic function: need to average one period only.
It can be shown that average of cos2 is: 21cos2 T
t20
000
0 221 EcBES
T
And average power flow per unit time:
Irradiance:20
0
2EcSI
T
Alternative eq-ns:
TTBcEcI 2
0
20
Usually mostly E-field component interacts with matter, and we will refer to E as optical field and use energy eq-ns with E
Irradiance is proportional to the square of the amplitude of the E field
For linear isotropic dielectric:
TEI 2v
Optical power radiant flux total power falling on some area (Watts)
Spherical wave: inverse square lawSpherical waves are produced by point sources. As you move away from the source light intensity drops
trkr
tr vcos, A
Spherical wave eq-n:
trkr
EE
cos0 trk
rBB
cos0
trkr
Br
ES
200
0
cos1
202
0 12
Er
cSIT
Inverse square law: the irradiance from a point source drops as 1/r2
Classical EM waves versus photonsThe energy of a single light photon is E=h
The Planck’s constant h = 6.626×10-34 JsVisible light wavelength is ~ 0.5 m J 104 19
1
chhE
Example: laser pointer output power is ~ 1 mWnumber of photons emitted every second:
photons/s 105.2J/photon 104J/s10 15
19
3
1
EP
Conclusion: in many every day situations the quantum nature of light is not pronounced and light could be treated as a classical EM wave