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 dφ = 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