Download The one-dimensional wave equation

We’ll start with optics
Optional optics texts:
Waves, the Wave Equation, and Phase
Velocity
What is a wave?
f(x)
f(x-2)
f(x-1)
f(x-3)
Eugene Hecht, Optics, 4th ed.
J.F. James, A Student's Guide to
Fourier Transforms
Forward [f(x-vt)] and
backward [f(x+vt)]
propagating waves
The one-dimensional wave equation
Harmonic waves
Optics played a major role in all the
physics revolutions of the 20th
century, so we’ll do some.
Wavelength, frequency, period, etc.
And waves and the Fourier
transform play major roles in all of
science, so we’ll do that, too.
Complex numbers
A wave is anything that moves.
If we let a = v t, where v is positive
and t is time, then the displacement
will increase with time.
v will be the velocity of the wave.
2
3
x
Plane waves and laser beams
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:
f(x)
f(x-2)
f(x-1)
f(x-3)
!2f
! x2
So f(x - v t) represents a rightward,
or forward, propagating wave.
Similarly, f(x + v t) represents a
leftward, or backward, propagating
wave.
1
The one-dimensional wave equation
What is a wave?
To displace any function f(x) to the
right, just change its argument from
x to x-a, where a is a positive
number.
0
Phase velocity
0
1
2
3
x
"
1 !2f
v2 ! t 2
= 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
Proof that f (x ± vt) solves the wave equation
Write f (x ± vt) as f (u), where u = x ± vt. So
The wave equation has the simple solution:
!f !f !u
=
! x !u ! x
Now, use the chain rule:
f ( x, t ) = f ( x ± vt )
So
!u
!u
= 1 and
=±v
!x
!t
!f !f
!2f !2f
=
=
!
! x !u
! x2 ! u 2
and
! f ! f !u
=
!t !u !t
!2f
!2f
!f
!f
=±v
= v2
!
2
!t
!u
!t
! u2
Substituting into the wave equation:
!2f
1 !2f
$
! x2 v2 ! t 2
where f (u) can be any twice-differentiable function.
The 1D wave equation for light waves
! 2E
! 2E
# µ" 2 = 0
2
!x
!t
where E is the
light electric field
!2f
1 " 2!2f #
$
%v
& = 0
! u 2 v2 ' ! u 2 (
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)
We’ll use cosine- and sine-wave solutions:
E ( x, t ) = B cos[k ( x ± vt )] + C sin[k ( x ± vt )]
where z = k x – ! t and y = " to obtain:
E(x,t) = A cos(kx – !t) cos(") + A sin(kx – !t) sin(")
kx ± (kv)t
or
=
E ( x, t ) = B cos(kx ± ! t ) + C sin(kx ± ! t )
which is the same result as before,
E ( x, t ) = B cos(kx " ! t ) + C sin(kx " ! t )
where:
!
k
= v =
1
µ"
The speed of light in
vacuum, usually called
“c”, is 3 x 1010 cm/s.
as long as:
A cos(") = B and
A sin(") = C
For simplicity, we’ll
just use the forwardpropagating wave.
Definitions: Amplitude and Absolute phase
E(x,t) = A cos[(k x – ! t ) – " ]
Definitions
Spatial quantities:
A = Amplitude
" = Absolute phase (or initial phase)
"
Temporal quantities:
The Phase Velocity
Human wave
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/$ :
v = #v
In terms of the k-vector, k = 2" / #, and
the angular frequency, ! = 2" / $, this is:
v =!/k
A typical human wave has a phase velocity of about 20 seats
per second.
The Phase of a Wave
Complex numbers
The phase is everything inside the cosine.
Consider a point,
P = (x,y), on a 2D
Cartesian grid.
E(x,t) = A cos(&), where & = k x – ! t – "
& = &(x,y,z,t) and is not a constant, like " !
Let the x-coordinate be the real part
and the y-coordinate the imaginary part
of a complex number.
In terms of the phase,
! = – $& /$t
So, instead of using an ordered pair, (x,y), we write:
k = $& /$x
And
– $& /$t
v = –––––––
$& /$x
This formula is useful
when the wave is
really complicated.
Euler's Formula
P = x+iy
= A cos(#) + i A sin(#)
where i = (-1)1/2
Proof of Euler's Formula exp(i&) = cos(&) + i sin(&)
Use Taylor Series:
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
f ( x) = f (0) +
x
x2
x3
f '(0) +
f ''(0) +
f '''(0) + ...
1!
2!
3!
x x 2 x3 x 4
+ + + + ...
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!
exp( x) = 1 +
If we substitute x = i&
into exp(x), then:
i! ! 2 i! 3 ! 4
"
"
+
+ ...
1! 2! 3! 4!
# !2 !4
$
#! ! 3
$
= %1 "
+
+ ...& + i % " + ...&
2!
4!
1!
3!
'
(
'
(
exp(i! ) = 1 +
= cos(! ) + i sin(! )
Complex number theorems
If exp(i! ) = cos(! ) + i sin(! )
exp(i! ) = #1
exp(i! / 2) = i
exp(-i" ) = cos(" ) # i sin(" )
1
cos(" ) = [exp(i" ) + exp(#i" ) ]
2
1
sin(" ) = [exp(i" ) # exp(#i" ) ]
2i
A1exp(i"1 ) $ A2 exp(i" 2 ) = A1 A2 exp [i ("1 + " 2 ) ]
More complex number theorems
Any complex number, z, can be written:
So
and
Re{ z } = 1/2 ( z + z* )
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
A1exp(i"1 ) / A2 exp(i" 2 ) = A1 / A2 exp [i ("1 # " 2 ) ]
We can also differentiate exp(ikx) as if
the argument were real.
z = Re{ z } + i Im{ z }
tan(#) = Im{ z } / Re{ z }
Waves using complex numbers
Recall that the electric field of a wave can be written:
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) ]
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.
We often
write these
expressions
without the
!, Re, or
+c.c.
where "+ c.c." means "plus the complex conjugate of everything
before the plus sign."
Waves using complex amplitudes
Complex numbers simplify waves!
We can let the amplitude be complex:
Adding waves of the same frequency, but different initial phase,
yields a wave of the same frequency.
E (x, t ) = A exp &$i (kx # ! t # " )'%
{
}
E (x, t ) = {A exp(#i" )} exp $&i (kx # ! t )%'
where we've separated the constant stuff from the rapidly changing stuff.
The resulting complex amplitude is:
So:
E0 = A exp("i! )
!
# (note the " ~ ")
E (x, t ) = E0 exp i (kx " ! t )
!
!
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.
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.
The 3D wave equation for the electric field
and its solution!
A wave can propagate in any
direction in space. So we must allow
the space derivative to be 3D:
or
! 2E ! 2E ! 2E
! 2E
+
+
# µ" 2 = 0
2
2
2
!x
!y
!z
!t
which has the solution:
where
and
!
"2 E
# 2 E $ µ! 2 = 0
"t
! !
E ( x, y, z , t ) = E0 exp[i (k " r # ! t )]
"
"
!
!
k ! (k x , k y , k z ) r ! (x, y, z )
! !
k ! r " kx x + k y y + kz z
k 2 ! k x2 + k y2 + k z2
! !
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 )]
!
!
w2 &
(
z
w
y
Localized wave-fronts
x
Laser beam
spot on wall