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
Waves move
f(x)
t1 > t0
t0
t2 > t1
t3 > t2
v
0
1
2
3
A wave is…
a disturbance in a medium
-propagating in space with velocity v
-transporting energy
-leaving the medium undisturbed
x
1D traveling wave
f(x)
t1 > t0
t0
t2 > t1
t3 > t2
v
0
1
2
3
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.
x
The wave equation
Let y = f (x'), where x' = x ± vt.
y
Now, use the chain rule:
So
y
x

f
x
 y
2

x
2
x
 f
So

x
and
y
x x
t
and
2
x
1
y x
2

x
y
t
 v
f
x

x
t
 v
y x
x t
 y
2

t
2
v
Combine to get the 1D differential wave equation:
 y
2
x
2
1  y
2

v
2
t
 f
2
2
works for anything that moves!
2
x
2
Harmonic waves
• periodic (smooth patterns that repeat endlessly)
• generated by undamped oscillators undergoing
harmonic motion
• characterized by sine or cosine functions
-for example, y  f ( x  vt )  A sin k  x  vt 
-complete set (linear combination of sine or cosine functions
can represent any periodic waveform)
Snapshots of harmonic waves
at a fixed time:
at a fixed point:
T
A
A
propagation constant: k = 2p/l
wave number:
k = 1/l
Note:
frequency:
n = 1/T
angular frequency: w = 2pn
n ≠v
v = velocity (m/s)
n = frequency (1/s)
v = nl
The phase of a harmonic wave
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)
dx
 v
dt
which confirms that v is the wave velocity.
Absolute phase
y = A sin[k(x ± vt) + j0]
A = amplitude
j0 =
initial phase angle (or absolute phase) at x = 0 and t =0
p
How fast is the wave traveling?
The phase
Since n = 1/T:
velocity is the wavelength/period:
v = ln
In terms of k, k = 2p/l, and
the angular frequency, w = 2p/T, this is:
v = w/k
v = l/T
Phase velocity vs. Group velocity
v phase 
w
k
v group 
dw
dk
Here, phase velocity = group velocity (the medium is non-dispersive).
In a dispersive medium, the phase velocity ≠ group velocity.
works for any periodic wave??
Complex numbers make it less complex!
P = (x,y)
x: real and y: imaginary
P =
x
= A cos(j)
where
i
1
+ iy
+ i A sin(j)
Euler’s formula
“one of the most remarkable formulas in mathematics”
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)
~ = A cos(kx – wt)
so that y = Re(y)
~ = A sin(kx – wt)
y = Im(y)
Why?
Math is easier with exponential functions than with trigonometry.
Plane waves
A plane wave’s contours of maximum field, called wavefronts, are planes.
They extend over all space.
The wavefronts of a
wave sweep along
at the speed of light.
• equally spaced
• separated by one wavelength apart
• perpendicular to direction or propagation
Usually we just draw lines;
it’s easier.
Wave vector k
represents 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
General case
k·r
= xkx = yky + zkz
kr cosq
 ks
arbitrary direction
In complex form:
Y = Aei(k·r - wt)
 Ψ
2
x
2
 Ψ
2

y
2
 Ψ
2

z
2
1  Ψ
2

v
2
t
2
Substituting the Laplacian operator:
1  Ψ
2
 Ψ 
2
v
2
t
2
Spherical waves
• harmonic waves emanating from a point source
• wavefronts travel at equal rates in all directions
 A  i  kr  w t 
Ψ   e
 r 
Cylindrical waves
 A  i  k  w t 
e
Ψ 
  


the wave nature of light
Electromagnetic waves
Derivation of the wave equation from
Maxwell’s equations
E  0
B  0
E  
B
  B  
t
E
t
http://www.youtube.com/watch?v=YLlvGh6aEIs
E and B are perpendicular
Perpendicular to the direction of propagation:
I. Gauss’ law (in vacuum, no charges):
 
 j k r  w t 
  E (r , t )    E0e
0
 
 
j k  r  ωt 
jk  E 0 e
0
 
k  E0  0
so


E  k
II. No monopoles:
 
 j k  r  w t 
  B ( r , t )    B0 e
0
 
 
j k  r  ωt 
jk  B 0 e
0
 
k  B0  0
Always!
so


B  k
Always!
E and B are perpendicular
Perpendicular to each other:
III. Faraday’s law:

E  

B
t

 j k  r  w t 
B
  E0e

t
 jk
y
E oz  jk z E oy
k y E oz  k z E oy
w
e
 
j k r  w t

B x ˆ
ˆ
i 
i
t
 B ox
so


B  E
Always!
E and B are harmonic


E  E 0 sin(


B  B 0 sin(
 
k  r  wt)
 
k  r  wt)
Also, at any specified point in time and space,
E  cB
where c is the velocity of the propagating wave,
c
1
 0 0
 2 . 998  10 m/s
8
EM wave propagation in homogeneous media
The speed of an EM wave in free space is given by: c 
1
 0 0

w
k
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)
Simple case—
plane wave with E field along x, moving along z:
General case—

2
1  E

2
 E 
c
which has the solution:
where
and
and
k 
k
x
along any direction
2
t
2
 


E ( x , y , z , t )  E 0 exp[ i k  r  w t ]
, ky , kz 

r 
k  r  kx x  ky y  kz z

E0  (E0x , E0y , E0z )
 x, y, z 

Arbitrary direction
 


E ( x , y , z , t )  E 0 sin k  r  w t



k = wave vector
k x x  k y y  k z z  const
x

r

E0

k
 
Plane of constant k  r , constant E
z
y
 
 i k  r  w t 
E (r , t )  E 0e
Electromagnetic waves transmit energy
Energy density (J/m3) in an electrostatic field:
uE 
0E
1
2
2
Energy density (J/m3) in an magnetostatic field:
uB 
1
1
u B 2
B
0
2
20
B
2
Energy density (J/m3) in an electromagnetic wave is equally divided:
u total  u E  u B   0 E 
2
1
0
B
2
Rate of energy transport
Rate of energy transport: Power (W)

k
A
P 
energy
Dt

uDV
Dt

uAc D t
Dt
P  ucA
cD t
Power per unit area (W/m2):
S  uc
S   0 c EB

 
2
S   0c E  B
2
Poynting
Pointing vector
Poynting vector oscillates rapidly
E  E 0 cos w t 
B  B 0 cos w t 
S   0 c E 0 B 0 cos
2
2
w t 
Take the time average:
S   0 c E 0 B 0 cos
2
S 
1
2
2
w t 
 0c E 0 B0
S  Ee
2
“Irradiance” (W/m2)
Light is a vector field
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 3D vector field
A light wave has both electric and magnetic 3D vector fields:
A 3D vector field assigns a 3D vector (i.e., an arrow having both
direction and length) to each point in 3D space.
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
y
x
Evolution of electric field vector
circular polarization
y
x
Exercises
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