Chapter 1
Simple Examples of
Propagation
This chapter presents examples of wave propagation governed by hyper-
bolic equations. The ideas of propagation of singularities, group velocity,
and short wavelength asymptotics are introduced in simple situations. The
method of characteristics for problems in dimension d = 1 is presented as
well as the method of nonstationary phase. The latter is a fundamental tool
for estimating oscillatory integrals. The examples are elementary. They
could each be part of an introductory course in partial differential equa-
tions, but often are not. This material can be skipped. If needed later, the
reader may return to this
chapter.1
In sections 1.3, 1.5, and 1.6 we derive
in simple situations the three basic laws of physical geometric optics.
Wave like solutions of partial differential equations have spatially local-
ized structures whose evolution in time can be followed. The most common
are solutions with propagating singularities and solutions that are modu-
lated wave trains also called wave packets. They have the form
a(t, x)
eiφ(t,x)/
with smooth profile a, real valued smooth phase φ with = 0 on supp a.
The parameter is a wavelength and is small compared to the scale on
which a and φ vary. The classic example is light with a wavelength on the
order of 5 ×
10−5
centimeter. Singularities are often restricted to varieties
of lower codimension, hence of width equal to zero, which is infinitely small
compared to the scales of their other variations. Real world waves modeled
1Some
ideas are used which are not formally presented until later, for example the Sobolev
spaces
Hs(Rd)
and Gronwall’s lemma.
1
http://dx.doi.org/10.1090/gsm/133/01
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