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Waves, the Wave Equation, and Phase Velocity

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This formula is useful when the wave is really complicated. ... This isn't so obvious using trigonometric functions, but it's easy. with complex exponentials: ... – PowerPoint PPT presentation

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Title: Waves, the Wave Equation, and Phase Velocity


1
Waves, the Wave Equation, and Phase Velocity
  • What is a wave?
  • Forward f(x-vt) and backward f(xvt)
    propagating waves
  • The one-dimensional wave equation
  • Harmonic waves
  • Wavelength, frequency, period, etc.
  • Phase velocity
  • Complex numbers Plane waves
    and laser beams

2
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 - v t)
f(x v t)
3
The one-dimensional wave equation
Well derive the wave equation from Maxwells
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.
4
The solution to the one-dimensional wave equation
The wave equation has the simple solution
  • where f (u) can be any twice-differentiable
    function.

5
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 Þ and
    Þ
  •   
  • Substituting into the wave equation

6
The 1D wave equation for light waves
where E is the light electric field
  • Well use cosine- and sine-wave solutions
  •  
  • or
  • where

The speed of light in vacuum, usually called c,
is 3 x 1010 cm/s.
7
A simpler equation for a harmonic wave
  • E(x,t) A cos(kx wt) q
  • Use the trigonometric identity
  • cos(zy) cos(z) cos(y)
    sin(z) sin(y)
  • where z k x w t and y q to obtain
  • E(x,t) A cos(kx wt) cos(q) A
    sin(kx wt) sin(q)
  • which is the same result as before,
  • as long as
  • A cos(q) B and
    A sin(q) C

For simplicity, well just use the
forward-propagating wave.
8
Definitions Amplitude and Absolute phase
  • E(x,t) A cos(k x w t ) q
  • A Amplitude
  • q Absolute phase (or initial phase)

9
Definitions
  • Spatial quantities

Temporal quantities
10
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,
w 2p / t, this is v w / k
11
Human wave
A typical human wave has a phase velocity of
about 20 seats per second.
12
The Phase of a Wave
  • The phase is everything inside the cosine.
  • E(t) A cos(j), where j k
    x w t q
  • j j(x,y,z,t) and is not a
    constant, like q !
  • In terms of the phase,
  • w j /t
  • k j /x
  • And
  • j /t
  • v
  • j /x

This formula is useful when the wave is really
complicated.
13
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 i y
  • A cos(j) i A sin(j)
  • where i (-1)1/2

14
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

15
Proof of Euler's Formula
exp(ij) cos(j) i sin(j)
  • Use Taylor Series

If we substitute x ij into exp(x), then
16
Complex number theorems
17
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

18
We can also differentiate exp(ikx) as if the
argument were real.
19
Waves using complex numbers
  • The electric field of a light wave can be
    written
  • E(x,t) A cos(kx wt q)
  • Since exp(ij) cos(j) i sin(j), E(x,t) can
    also be written
  • E(x,t) Re A expi(kx wt q)
  • or
  • E(x,t) 1/2 A expi(kx wt q) c.c.
  • where " c.c." means "plus the complex conjugate
    of everything before the plus sign."

We often write these expressions without the ½,
Re, or c.c.
20
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

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.
21
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
where all initial phases are lumped into E1, E2,
and E3.
22
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
  • which has the solution
  •  
  • where
  • and

23
is called a plane wave.
A plane waves contours of maximum phase, called
wave-fronts or phase-fronts, are planes.
They extend over all space.
A wave's wave-fronts sweep along at the speed of
light.
Wave-fronts are helpful for drawing pictures of
interfering waves.
A plane wave's wave-fronts are equally spaced, a
wavelength apart. They're perpendicular to the
propagation direction.
24
Laser beams vs. Plane waves
A plane wave has flat wave-fronts throughout all
space. It also has infinite energy.It doesnt
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
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