chapter 4: wave equations. what is a wave?
TRANSCRIPT
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
t0 t1 > t0 t2 > t1 t3 > t2
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
t0 t1 > t0 t2 > t1 t3 > t2
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 Þ and Þ
Combine to get the 1D differential wave equation:
2
2
22
2 1
t
y
vx
y
1
x
xv
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
fv
t
y
2
22
2
2
x
fv
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)
vtxkAvtxfy sin)(
Snapshots of harmonic waves
T
A
at a fixed time: at a fixed point:
A
frequency: n = 1/T angular frequency: w = 2pn
propagation constant: k = 2p/lwave number: k = 1/l
n ≠ v v = velocity (m/s)n = frequency (1/s)
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, dj = 0 = k(dx ± vdt)
which confirms that v is the wave velocity.
vdt
dx
The phase of a harmonic wave
A = amplitude
j0 = initial phase angle (or absolute phase) at x = 0 and t =0
p
Absolute phase
y = A sin[k(x ± vt) + j0]
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 = 2p/l, and the angular frequency, w = 2p/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.
kvphase
dk
dvgroup
works for any periodic waveany ??
Complex numbers make it less complex!
x: real and y: imaginary
P = x + i y = A cos(j) + i A sin(j)
where
P = (x,y)
1i
“one of the most remarkable formulas in mathematics”Euler’s formula
Links the trigonometric functions and the complex exponential function
eij = cosj + i sinj
so the point, P = A cos(j) + i A sin(j), can also be written:
P = A exp(ij) = A eiφ
where
A = Amplitude
j = Phase
Harmonic waves as complex functions
Using Euler’s formula, we can describe the wave as
y = Aei(kx-wt)
so that y = Re(y) = A cos(kx – wt)
y = Im(y) = A sin(kx – wt)
~
~
~
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 krepresents direction of propagation
Y = A sin(kx – wt)
Consider a snapshot in time, say t = 0:
Y = A sin(kx)
Y = A sin(kr cosq)
If we turn the propagation constant (k = 2p/l) into a vector, kr cosq = k · r
Y = A sin(k · r – wt)
• wave disturbance defined by r• propagation along the x axis
arbitrary direction
General case
k · r = xkx = yky + zkz
= kr cosq ks
In complex form:
Y = Aei(k·r - wt)
2
2
22
2
2
2
2
2 1
t
Ψ
vz
Ψ
y
Ψ
x
Ψ
Substituting the Laplacian operator:
2
2
22 1
t
Ψ
vΨ
Spherical waves
• harmonic waves emanating from a point source• wavefronts travel at equal rates in all directions
tkrier
AΨ
Cylindrical waves
tkieA
Ψ
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
BE E
t
EB B
t
I. Gauss’ law (in vacuum, no charges):
so
II. No monopoles:
so
Always!
Always!
E and B are perpendicular
0),( 0 trkjeEtr E
00 ωtrkjeEkj
00 Ek
0),( 0 trkjeBtr B
00 ωtrkjeBkj
00 Bk
k
E
k
B
Perpendicular to the direction of propagation:
III. Faraday’s law:
so
Always!
t
B
E
t
eE trkj
B
0
it
BieEjkEjk xtrkj
oyzozyˆˆ
oxoyzozy B
EkEk
EB
Perpendicular to each other:
E and B are perpendicular
E and B are harmonic
)sin(
)sin(
0
0
t
t
rkBB
rkEE
Also, at any specified point in time and space,
cBE where c is the velocity of the propagating wave,
m/s 10998.21 8
00
c
The speed of an EM wave in free space is given by:
0 = permittivity of free space, 0 = magnetic permeability of free space
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 = complex permittivity
s = electric conductivity c = electric susceptibility (to polarization under the influence of an external field)
Note that ε and μ also depend on frequency (ω)
kc
00
1
EM wave propagation in homogeneous media
x
z
y
Simple case— plane wave with E field along x, moving along z:
which has the solution:
where
and
and
x y zk r k x k y k z
, ,x y zk k k k
, ,r x y z
General case— along any direction
2
2
22 1
tc
E
E
]exp[),,,( 0 titzyx rkEE
),,( 0000 zyx EEEE
z
y
x
Arbitrary direction
constzkykxk zyx
k
= wave vectork
0E
r
tiet rkErE
0),(
k Plane of constant , constant E rk
rk
ttzyx rkEE
sin),,,( 0
Energy density (J/m3) in an electrostatic field:
Energy density (J/m3) in an magnetostatic field:
2
02
1BuB
Energy density (J/m3) in an electromagnetic wave is equally divided:
2120 0
BEuuu BEtotal
Electromagnetic waves transmit energy
202
1 EuE
2
02
1BuB
Rate of energy transport: Power (W)
cDt
A
Power per unit area (W/m2):
Pointing vector
Poynting
Rate of energy transport
ucS
EBcS 20
BES
20c
t
tuAc
t
Vu
t
energyP
ucAP
k
Take the time average:
“Irradiance” (W/m2)
Poynting vector oscillates rapidly
tEE cos0 tBB cos0
tBEcS 200
20 cos
tBEcS 200
20 cos
002
021 BEcS
eES
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 (Pedrotti3).
The following may be covered in the werkcollege on 14 September 2011:
Chapter 4:3, 5, 7, 13, 14, 17, 18, 24
Exercises