# what is a wave?

Urban and Civil

Nov 16, 2013 (4 years and 6 months ago)

73 views

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

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

††††††††††



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

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

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

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:

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.

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:

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