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2. Waves, the Wave Equation, and
Phase Velocity
What is a wave?
f(x)
f(x-2)
f(x-1)
f(x-3)
Forward [f(x-vt)] and
backward [f(x+vt)]
propagating waves
The one-dimensional wave equation
Harmonic waves
Wavelength, frequency, period, etc.
0
1
2
3
Phase velocity
Complex numbers
Plane waves and laser beams
x
What is a wave?
A wave is anything that moves.
To displace any function f(x) to the
right, just change its argument from
x to x-a, where a is a positive
number.
f(x)
f(x-2)
f(x-1)
f(x-3)
If we let a = v t, where v is positive
and t is time, then the displacement
will increase with time.
So f(x - v t) represents a rightward,
or forward, propagating wave.
Similarly, f(x + v t) represents a
leftward, or backward, propagating
wave.
v will be the velocity of the wave.
0
1
2
3
x
The one-dimensional wave equation
We’ll derive the wave equation from Maxwell’s equations. Here it is
in its one-dimensional form for scalar (i.e., non-vector) functions, f:
2f
2
x

1 2f
2
2
v t
 0
Light waves (actually the electric fields of light waves) will be a
solution to this equation. And v will be the velocity of light.
The solution to the one-dimensional
wave equation
The wave equation has the simple solution:
f ( x, t )  f ( x  vt )
where f (u) can be any twice-differentiable function.
Proof that f (x ± vt) solves the wave equation
Write f (x ± vt) as f (u), where u = x ± vt. So  u  1 and
x
f f u

 x u  x
Now, use the chain rule:
2
f f

f 2f


So

2
 x u
x
 u2
u
v
t
 f  f u

t u t
2
2f

f
f
f
2


v

v
and

t
u
 t2
 u2
Substituting into the wave equation:
2f
1 2f
 2
2
x
v  t2
2f
1  22f 

 2 v
 0
2
2 
u
v  u 
QED
The 1D wave equation for light waves
 2E
 2E
  2  0
2
x
t
where E is the
light electric field
We’ll use cosine- and sine-wave solutions:
E ( x, t )  B cos[k ( x  vt )]  C sin[k ( x  vt )]
kx  (kv)t
or
E ( x, t )  B cos(kx   t )  C sin(kx   t )
where:

k
 v 
1

A simpler equation for a harmonic wave:
E(x,t) = A cos[(kx – t) – q]
Use the trigonometric identity:
cos(z–y) = cos(z) cos(y) + sin(z) sin(y)
where z = k x –  t and y = q to obtain:
E(x,t) = A cos(kx – t) cos(q) + A sin(kx – t) sin(q)
which is the same result as before,
E ( x, t )  B cos(kx   t )  C sin(kx   t )
as long as:
A cos(q) = B and
A sin(q) = C
For simplicity, we’ll
just use the forwardpropagating wave.
Definitions: Amplitude and Absolute phase
E(x,t) = A cos[(k x – t ) – q]
A = Amplitude
q = Absolute phase (or initial phase)
p
Definitions
Spatial quantities:
Temporal quantities:
The Phase Velocity
How fast is the wave traveling?
Velocity is a reference distance
divided by a reference time.
The phase velocity is the wavelength / period:
v = l/t
In terms of the k-vector, k = 2p/ l, and
the angular frequency,  = 2p/ t, this is:
v = /k
The Phase of a Wave
The phase is everything inside the cosine.
E(t) = A cos(j), where j = k x – t – q
j = j(x,y,z,t)
and is not a constant, like q!
In terms of the phase,
 = – j/t
k = j/x
And
– j/t
v = –––––––
j/x
This formula is useful
when the wave is
really complicated.
Complex numbers
Consider a point,
P = (x,y), on a 2D
Cartesian grid.
Let the x-coordinate be the real part
and the y-coordinate the imaginary part
of a complex number.
So, instead of using an ordered pair, (x,y), we write:
P = x+iy
= A cos(j) + i A sin(j)
where i = (-1)1/2
Euler's Formula
exp(ij) = cos(j) + i sin(j)
so the point, P = A cos(j) + i A sin(j), can be written:
P = A exp(ij)
where
A = Amplitude
j
= Phase
Proof of Euler's Formula exp(ij) = cos(j) + i sin(j)
Use Taylor Series:
x
x2
x3
f ( x)  f (0)  f '(0) 
f ''(0) 
f '''(0)  ...
1!
2!
3!
x x 2 x3 x 4
exp( x)  1      ...
1! 2! 3! 4!
x 2 x 4 x 6 x8
cos( x)  1      ...
2! 4! 6! 8!
x x3 x5 x 7 x9
sin( x)       ...
1! 3! 5! 7! 9!
If we substitute x = ij
into exp(x), then:
ij j 2 ij 3 j 4
exp(ij )  1  


 ...
1! 2! 3! 4!
 j2 j4

j j 3

 1 

 ...  i    ...
 2! 4!

 1! 3!

 cos(j )  i sin(j )
Complex number theorems
If exp(ij )  cos(j )  i sin(j )
exp(ip )  1
exp(ip / 2)  i
exp(-ij )  cos(j )  i sin(j )
1
cos(j )   exp(ij )  exp(ij ) 
2
1
sin(j )   exp(ij )  exp(ij ) 
2i
A1exp(ij1 )  A2 exp(ij 2 )  A1 A2 exp i (j1  j 2 ) 
A1exp(ij1 ) / A2 exp(ij 2 )  A1 / A2 exp i (j1  j 2 ) 
More complex number theorems
Any complex number, z, can be written:
z = Re{ z } + i Im{ z }
So
Re{ z } = 1/2 ( z + z* )
and
Im{ z } = 1/2i ( z – z* )
where z* is the complex conjugate of z ( i  –i )
The "magnitude," | z |, of a complex number is:
| z |2 = z z* = Re{ z }2 + Im{ z }2
To convert z into polar form, A exp(ij):
A2 = Re{ z }2 + Im{ z }2
tan(j) = Im{ z } / Re{ z }
We can also differentiate exp(ikx) as if
the argument were real.
d
exp(ikx)  ik exp(ikx)
dx
Proof :
d
cos(kx)  i sin(kx)  k sin(kx)  ik cos(kx)
dx
 1

 ik   sin(kx)  cos(kx) 
 i

But  1/ i  i, so :
 ik i sin(kx)  cos(kx) 
QED
Waves using complex numbers
The electric field of a light wave can be written:
E(x,t) = A cos(kx – t – q)
Since exp(ij) = cos(j) + i sin(j), E(x,t) can also be written:
E(x,t) = Re { A exp[i(kx – t – q)] }
or
E(x,t) = 1/2 A exp[i(kx – t – q)] + c.c.
We often
write these
expressions
without the
½, Re, or
+c.c.
where "+ c.c." means "plus the complex conjugate of everything
before the plus sign."
Waves using complex amplitudes
We can let the amplitude be complex:
E  x, t   A exp i  kx   t  q  


E  x, t    A exp(iq ) exp i  kx   t  
where we've separated the constant stuff from the rapidly changing stuff.
The resulting "complex amplitude" is:
E0  A exp(iq )
So:
 (note the " ~ ")
E  x, t   E0 exp i  kx   t 
As written, this entire
field is complex!
How do you know if E0 is real or complex?
Sometimes people use the "~", but not always.
So always assume it's complex.
Complex numbers simplify optics!
Adding waves of the same frequency, but different initial phase,
yields a wave of the same frequency.
This isn't so obvious using trigonometric functions, but it's easy
with complex exponentials:
Etot ( x, t )  E1 exp i(kx   t )  E2 exp i (kx   t )  E3 exp i (kx  t )
 ( E1  E2  E3 ) exp i(kx   t )
where all initial phases are lumped into E1, E2, and E3.
The 3D wave equation for the electric field
and its solution!
A light wave can propagate in any
direction in space. So we must allow
the space derivative to be 3D:
or
 2E  2E  2E
 2E


  2  0
2
2
2
x
y
z
t
which has the solution:
where
and
2

E
2
 E   2  0
t
E( x, y, z, t )  E0 exp[i(k  r   t )]
k   kx , k y , kz 
r   x, y, z 
k  r  kx x  k y y  kz z
k 2  kx2  k y2  kz2
E0 exp[i(k  r  t )] is called a “plane wave.”
A plane wave’s contours of maximum phase, called “wave-fronts” or
“phase-fronts,” are planes. They extend over all space.
Wave-fronts
are helpful
for drawing
pictures of
interfering
waves.
A wave's wavefronts sweep
along at the
speed of light.
A plane wave's wave-fronts are equally
spaced, a wavelength apart.
They're perpendicular to the propagation
direction.
Usually, we just
draw lines; it’s
easier.
Laser beams vs. Plane waves
A plane wave has flat wave-fronts throughout
all space. It also has infinite energy.
It doesn’t exist in reality.
A laser beam is more localized. We can approximate a laser
beam as a plane wave vs. z times a Gaussian in x and y:
 x2  y 2 
E ( x, y, z, t )  E0 exp 
 exp[i(kz  t )]
2
w 

z
w
y
Localized wave-fronts
x
Laser beam
spot on wall