Waves, the Wave Equation, and Phase WeÕll start with ...
Transcript of Waves, the Wave Equation, and Phase WeÕll start with ...
We’ll start with optics
Optional optics texts:
Eugene Hecht, Optics, 4th ed.
J.F. James, A Student's Guide to
Fourier Transforms
Optics played a major role in all the
physics revolutions of the 20th
century, so we’ll do some.
And waves and the Fourier
transform play major roles in all of
science, so we’ll do that, too.
Waves, the Wave Equation, and Phase
Velocity
What is a wave?
Forward [f(x-vt)] andbackward [f(x+vt)]propagating waves
The one-dimensional wave equation
Harmonic waves
Wavelength, frequency, period, etc.
Phase velocity
Complex numbers Plane waves and laser beams
f(x)
f(x-3)
f(x-2)
f(x-1)
x0 1 2 3
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.
If we let a = v t, where v is positive
and t is time, then the displacement
will increase with time.
So represents a rightward,
or forward, propagating wave.
Similarly, represents a
leftward, or backward, propagating
wave.
v will be the velocity of the wave.
f(x)
f(x-3)
f(x-2)
f(x-1)
x0 1 2 3
f(x - v t)
f(x + v t)
The one-dimensional wave equation
2 2
2 2 2
10
v
f f
x t
! !
! !" =
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:
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
where f (u) can be any twice-differentiable function.
( , ) ( v )f x t f x t= ±
The wave equation has the simple solution:
Proof that f (x ± vt) solves the wave equation
Write f (x ± vt) as f (u), where u = x ± vt. So and
Now, use the chain rule:
So ! and !
Substituting into the wave equation:
1u
x
!
!= v
u
t
!
!= ±
f f u
x u x
! ! !
! ! !=
f f u
t u t
! ! !
! ! !=
f f
x u
! !
! != v
f f
t u
! !
! != ±
2 2
2
2 2v
f f
t u
! !
! !=
2 2
2 2
f f
x u
! !
! !=
2 2 2 2
2
2 2 2 2 2 2
1 1v 0
v v
f f f f
x t u u
! ! ! !
! ! ! !
" #$ = $ =% &
' (
The 1D wave equation for light waves
We’ll use cosine- and sine-wave solutions:
or
where:
2 2
2 20
E E
x t
! !µ"
! !# =
( , ) cos[ ( v )] sin[ ( v )]E x t B k x t C k x t= ± + ±
( , ) cos( ) sin( )E x t B kx t C kx t! != ± + ±
1v
k
!
µ"= =
( v)kx k t±
where E is the
light electric field
The speed of light in
vacuum, usually called
“c”, is 3 x 1010 cm/s.
A simpler equation for a harmonic wave:
E(x,t) = A cos[(kx – !t) – "]
Use the trigonometric identity:
cos(z–y) = cos(z) cos(y) + sin(z) sin(y)
where z = k x – ! t and y = " to obtain:
E(x,t) = A cos(kx – !t) cos(") + A sin(kx – !t) sin(")
which is the same result as before,
as long as:
A cos(") = B and A sin(") = C
( , ) cos( ) sin( )E x t B kx t C kx t! != " + "
For simplicity, we’ll
just use the forward-
propagating wave.
Definitions: Amplitude and Absolute phase
E(x,t) = A cos[(k x – ! t ) – " ]
A = Amplitude
" = Absolute phase (or initial phase)
"
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 = # / $
Since % = 1/$ :
In terms of the k-vector, k = 2" / #, and
the angular frequency, ! = 2" / $, this is:
v = # v
v = ! / k
Human wave
A typical human wave has a phase velocity of about 20 seats
per second.
The phase is everything inside the cosine.
E(x,t) = A cos(&), where & = k x – ! t – "
& = &(x,y,z,t) and is not a constant, like " !
In terms of the phase,
! = – $& /$t
k = $& /$x
And – $& /$t
v = ––––––– $& /$x
The Phase of a Wave
This formula is useful
when the wave is
really complicated.
Complex numbers
So, instead of using an ordered pair, (x,y), we write:
P = x + i y
= A cos(#) + i A sin(#)
where i = (-1)1/2
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.
Euler's Formula
exp(i&) = cos(&) + i sin(&)
so the point, P = A cos(#) + i A sin(#), can be written:
P = A exp(i&)
where
A = Amplitude
& = Phase
Proof of Euler's Formula
Use Taylor Series:
2 3 4
2 4 3
exp( ) 1 ...1! 2! 3! 4!
1 ... ...2! 4! 1! 3!
cos( ) sin( )
i ii
i
i
! ! ! !!
! ! ! !
! !
= + " " + +
# $ # $= " + + + " +% & % &' ( ' (
= +
2 3
( ) (0) '(0) ''(0) '''(0) ...1! 2! 3!
x x xf x f f f f= + + + +
exp(i&) = cos(&) + i sin(&)
2 3 4
2 4 6 8
3 5 7 9
exp( ) 1 ...1! 2! 3! 4!
cos( ) 1 ...2! 4! 6! 8!
sin( ) ...1! 3! 5! 7! 9!
x x x xx
x x x xx
x x x x xx
= + + + + +
= ! + ! + +
= ! + ! + +
If we substitute x = i&
into exp(x), then:
Complex number theorems
[ ]
[ ]
[ ]
[ ]
1 1 2 2 1 2 1 2
1 1 2 2 1 2 1 2
exp( ) 1
exp( / 2)
exp(- ) cos( ) sin( )
1 cos( ) exp( ) exp( )
2
1 sin( ) exp( ) exp( )
2
exp( ) exp( ) exp ( )
exp( ) / exp( ) / exp ( )
i
i i
i i
i i
i ii
A i A i A A i
A i A i A A i
!
!
" " "
" " "
" " "
" " " "
" " " "
= #
=
= #
= + #
= # #
$ = +
= #
exp( ) cos( ) sin( )i i! ! != +If
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(i&):
A2 = Re{ z }2 + Im{ z }2
tan(#) = Im{ z } / Re{ z }
We can also differentiate exp(ikx) as if
the argument were real.
[ ]
[ ]
exp( ) exp( )
cos( ) sin( ) sin( ) cos( )
1 sin( ) cos( )
1/ sin( ) cos( )
dikx ik ikx
dx
dkx i kx k kx ik kx
dx
ik kx kxi
i i ik i kx kx
=
+ = ! +
" #= ! +$ %& '
! = = +
Proof :
But , so :
Waves using complex numbers
Recall that the electric field of a wave can be written:
E(x,t) = A cos(kx – !t – ")
Since exp(i&) = cos(&) + i sin(&), E(x,t) can also be written:
E(x,t) = Re { A exp[i(kx – !t – ")] }
or
E(x,t) = 1/2 A exp[i(kx – !t – ")] + c.c.
where "+ c.c." means "plus the complex conjugate of everythingbefore the plus sign."
We often
write these
expressions
without the
!, Re, or
+c.c.
Waves using complex amplitudes
We can let the amplitude be complex:
where we've separated the constant stuff from the rapidly changing stuff.
The resulting complex amplitude is:
So:
( ) ( )
( ) { } ( ){ }
, exp
ex, ex( pp )
E x t A i kx t
E x t i k ti xA
! "
!"
= # #$ %& '
= ## $ %& '
0 exp( ) E A i!= " #!
(note the " ~ ")
( ) ( )0, expE x t E i kx t!= "! !
How do you know if E0 is real or complex?
Sometimes people use the "~", but not always.
So always assume it's complex.
As written, this entire
field is complex!
Complex numbers simplify waves!
This isn't so obvious using trigonometric functions, but it's easy
with complex exponentials:
1 2 3
1 2 3
( , ) exp ( ) exp ( ) exp ( )
( ) exp ( )
totE x t E i kx t E i kx t E i kx t
E E E i kx t
! ! !
!
= " + " + "
= + + "! ! ! !
! ! !
where all initial phases are lumped into E1, E2, and E3.
Adding waves of the same frequency, but different initial phase,
yields a wave of the same frequency.
The 3D wave equation for the electric field
and its solution!
or
which has the solution:
where
and
2 2 2 2
2 2 2 20
E E E E
x y z t
! ! ! !µ"
! ! ! !+ + # =
0( , , , ) exp[ ( )]E x y z t E i k r t!= " #
! !
" "
x y zk r k x k y k z! " + +
! !
2 2 2 2
x y zk k k k! + +
2
2
20
EE
tµ!
"# $ =
"
!A wave can propagate in any
direction in space. So we must allow
the space derivative to be 3D:
( ), ,x y zk k k k!
!
( ), ,r x y z!!
is called a plane wave.
A plane wave's wave-fronts are equally
spaced, a wavelength apart.
They're perpendicular to the propagation
direction.
Wave-fronts
are helpful
for drawing
pictures of
interfering
waves.
A wave's wave-
fronts sweep
along at the
speed of light.
A plane wave’s contours of maximum phase, called wave-fronts or
phase-fronts, are planes. They extend over all space.
0 exp[ ( )]E i k r t!" #
! !
"
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:
2 2
20( , , , ) exp[ ( )exp ]x y
E x y z t E i kz tw
!=" #+$%
&$'
(! !
Laser beam
spot on wall
w
x
y
Localized wave-fronts
z