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Lecture 3 Notes, 95.658, Spring 2012, Electromagnetic Theory II
Dr. Christopher S. Baird, UMass Lowell
1. Superposition of Waves
 Up till now we have only looked at monochromatic waves and how they behave in materials and
hinted at how a frequencydependent permittivity would lead to dispersion of a wave packet made
up of many monochromatic waves. Let us look in more detail at what this really means.
 There is no such thing as a truly monochromatic wave.
 A wave cannot always have been generated in the infinite past and continue to be generated in the
infinite future. Every electromagnetic wave therefore has a definite spatial beginning point and
spatial endpoint. A truly monochromatic wave cannot exist because it would require an infinite
extent and infinite existence.
 In this sense, every wave is actually a wave packet or pulse with finite extent.
 Many frequencies (or wavenumbers) are required in order to build up a wave packet, so every real
wave actually has a range of frequencies present.
 Wave packets that have a very large width require a narrower range of superimposed frequencies
to be mathematically constructed, and may be approximated as monochromatic waves.
 For a nearly monochromatic wave, the width of the wave's range of frequencies is known as the
“spectral linewidth”.
 The linewidth of a wave may also experience broadening due to changes in the source or due to
propagation phenomena.
 Because a dispersive material has a permittivity that depends on frequency, any wave can be
thought of as a superposition of monochromatic plane waves which each interact with the material
independently of the others, dependent on its frequency.
 For simplicity, we will investigate the behavior of one vector component of the electric
field and label it
u
(
x
,
t
) so that for instance
E
x
(
x
,
t
) =
u
(
x
,
t
). Because the components of a vector
are independent, the end solution is just the sum of each vector component after being solved
separately.
 As found previously, the principal solution of the wave equation is (showing now only one
component):
u
x ,t
=
Ae
i
k x
−
t
where
k
=±
 We will treat the wave number
k
as the independent variable so that the frequency
ω
is a function
of the wave number
k
according to:
ω
=
ω
(
k
). This is actually the more common form of the
dispersion relation.
 Assume for now that
ω
and
k
are real.
 The general solution is just the superposition of all possible solutions:
u
(
x ,t
)=
1
√
2
π
∫
−∞
∞
A
(
k
)
e
i
(
k x
−ω(
k
)
t
)
dk
General solution to the Wave Equation
 The coefficient function
A
(
k
) in the general solution is what uniquely determines the solution to a
particular problem.
 This is just a Fourier transform and we can apply Fourier Theory to find the coefficients.
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This note was uploaded on 02/13/2012 for the course PHYSICS 95.658 taught by Professor Staff during the Spring '11 term at UMass Lowell.
 Spring '11
 Staff
 Mass

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