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
t
0
t
1
> t
0
t
2
> t
1
t
3
> t
2
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
t
0
t
1
> t
0
t
2
> t
1
t
3
> t
2
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
††††††††††
慮搠a†††††††††††
Combine
to get the 1D differential wave equation
:
2
2
2
2
2
1
t
y
v
x
y
1
x
x
v
t
x
x
x
x
y
x
y
t
x
x
y
t
y
x
f
x
y
2
2
2
2
x
f
x
y
x
f
v
t
y
2
2
2
2
2
x
f
v
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)
vt
x
k
A
vt
x
f
y
sin
)
(
Snapshots of harmonic waves
T
A
at a fixed time:
at a fixed point:
A
frequency:
n
=
1/
T
angular frequency:
w
=
2
pn
propagation constant:
k
=
2
p
/
l
wave number:
k
=
1
/
l
n
≠
v
v
= velocity (m/s)
n
㴠牥煵敮捹
ㄯ猩
v
=
nl
Note:
The
phase,
j
Ⱐ
is
everything inside the
sine or cosine (the argument).
y
=
A
sin[
k(x
±
vt
)]
j
=
k(x
±
vt
)
For constant phase,
d
j
=
0 =
k(dx
±
vdt
)
which confirms that
v
is the wave velocity.
v
dt
dx
The phase of a harmonic wave
A
=
amplitude
j
0
=
initial phase angle (or absolute phase) at
x
= 0 and
t
=0
p
Absolute phase
y
=
A
sin[
k(x
±
vt
) +
j
0
]
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
= 2
p
/
l
Ⱐ,
and
the angular frequency
,
w
= 2
p
/
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.
k
v
phase
w
dk
d
v
group
w
works for any periodic wave
any
??
Complex numbers make it less complex!
x
: real
and
y
: imaginary
P
=
x
+
i
y
= A
cos
(
j
⤠
⬠+
i
䄠獩渨
j
)
where
P
= (
x,y
)
1
i
“one of the most remarkable formulas in mathematics”
Euler’s formula
Links the trigonometric functions and the complex exponential function
e
i
j
=
cos
j
+
i
sin
j
so the point,
P
=
A
cos
(
j
) +
i
A
sin(
j
)
, can also be written:
P
=
A
exp(
i
j
) =
A
e
iφ
where
A
= Amplitude
j
= Phase
Harmonic waves as complex functions
Using Euler’s formula, we can describe the wave as
y = A
e
i
(
kx

w
t
)
so
that
y =
Re(
y
)
= A
cos
(
kx
–
w
t
)
y =
Im(
y
) =
A
sin(
kx
–
w
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 k
represents direction of propagation
Y
=
A
sin(
kx
–
w
t
)
Consider a snapshot in time, say
t
= 0:
Y
=
A
sin(
kx
)
Y
=
A
sin(
kr
cos
q
)
If we turn the propagation constant (
k
= 2
p
/
l
into a vector,
kr
cos
q
=
k ∙ r
Y
=
A
sin(
k ∙ r
–
w
t
)
•
wave disturbance defined by
r
•
propagation along the
x
axis
arbitrary direction
General case
k ∙ r
=
xk
x
=
yk
y
+
zk
z
kr
cos
q
ks
In complex form:
Y
=
Ae
i
(
k
∙
r

w
t
)
2
2
2
2
2
2
2
2
2
1
t
Ψ
v
z
Ψ
y
Ψ
x
Ψ
Substituting the Laplacian operator:
2
2
2
2
1
t
Ψ
v
Ψ
Spherical waves
•
harmonic waves emanating from a point source
•
wavefronts
travel at equal rates in all directions
t
kr
i
e
r
A
Ψ
w
Cylindrical waves
t
k
i
e
A
Ψ
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
B
E E
t
E
B B
t
I. Gauss’
law (in vacuum
, no charges):
so
II. No
monopoles
:
so
Always!
Always!
E
and
B
are perpendicular
0
)
,
(
0
t
r
k
j
e
E
t
r
w
E
0
0
ωt
r
k
j
e
E
k
j
0
0
E
k
0
)
,
(
0
t
r
k
j
e
B
t
r
w
B
0
0
ωt
r
k
j
e
B
k
j
0
0
B
k
k
E
k
B
Perpendicular to the direction of propagation:
III. Faraday’s
law
:
so
Always!
t
B
E
t
e
E
t
r
k
j
B
w
0
i
t
B
i
e
E
jk
E
jk
x
t
r
k
j
oy
z
oz
y
ˆ
ˆ
w
ox
oy
z
oz
y
B
E
k
E
k
w
E
B
Perpendicular to each other:
E
and
B
are perpendicular
E
and
B
are harmonic
)
sin(
)
sin(
0
0
t
t
w
w
r
k
B
B
r
k
E
E
Also, at any specified point in time and space,
cB
E
where c is the velocity of the propagating wave,
m/s
10
998
.
2
1
8
0
0
c
The speed of an EM wave in free space is given by:
0
= permittivity of free space,
0
㴠=慧湥瑩挠a敲浥慢楬楴i潦o敥獰s捥
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.
㴠
0
(1+
⤠⬠
i
w
㴠
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
)
k
c
w
0
0
1
EM wave propagation in homogeneous media
Simple case
—
plane
wave with
E
field along
x
, moving along
z
:
which has the solution:
where
and
and
x y z
k r k x k y k z
,,
x y z
k k k k
,,
r x y z
General case
—
along any direction
2
2
2
2
1
t
c
E
E
]
exp[
)
,
,
,
(
0
t
i
t
z
y
x
w
r
k
E
E
)
,
,
(
0
0
0
0
z
y
x
E
E
E
E
z
y
x
Arbitrary direction
const
z
k
y
k
x
k
z
y
x
k
= wave vector
k
0
E
r
t
i
e
t
w
r
k
E
r
E
0
)
,
(
k
Plane of constant , constant
E
r
k
r
k
t
t
z
y
x
w
r
k
E
E
sin
)
,
,
,
(
0
Energy density (
J/m
3
) in an electrostatic field:
Energy density (
J/m
3
) in an
magnetostatic
field:
2
0
2
1
B
u
B
Energy
density (
J/m
3
)
in an electromagnetic wave is equally divided:
2
1
2
0
0
B
E
u
u
u
B
E
total
Electromagnetic waves transmit energy
2
0
2
1
E
u
E
2
0
2
1
B
u
B
Rate of energy transport: Power (
W
)
c
D
t
Power per unit area (
W/m
2
):
Pointing
vector
Poynting
Rate of energy transport
uc
S
EB
c
S
2
0
B
E
S
2
0
c
t
t
uAc
t
V
u
t
energy
P
D
D
D
D
D
ucA
P
k
Take the time average:
“Irradiance” (
W/m
2
)
Poynting vector oscillates rapidly
t
E
E
w
cos
0
t
B
B
w
cos
0
t
B
E
c
S
w
2
0
0
2
0
cos
t
B
E
c
S
w
2
0
0
2
0
cos
0
0
2
0
2
1
B
E
c
S
e
E
S
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
x
y
Evolution of electric field vector
circular
polarization
x
y
You are encouraged to solve
all problems in the textbook
(Pedrotti
3
).
The following may be
covered in the werkcollege
on 14 September 2011:
Chapter
4
:
3
,
5
,
7
,
13
,
14
,
17
,
18
,
24
Exercises
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