1.5. A cautionary example in geometric optics 27

The sin t|ξ|/|ξ| factor is ≤ |t|. For small, the distance of e1/ to the support

of χ is ≥ C/. Therefore,

χ(ξ)

sin t|ξ|

|ξ|

1

ξ − e1/ s

L∞(Rd)

ξ

≤ Cs |t|

s

, 0 ≤ 1 .

It follows that

χ(ξ)

sin t|ξ|

|ξ|

1

ξ − e1/ s

ks(ξ − e1/)

L2(Rd)

≤ Cs |t|

s

ks

L2(Rd)

,

with s arbitrarily large. The small frequency contribution is negligible in

the limit → 0. It is removed with a cutoff as above, and then the analysis

away from ξ = 0 proceeds by decomposition into plane wave as in the case

with ut(0) = 0. It yields left and right moving waves with the same phases

as before.

Exercise 1.4.3. Solve the Cauchy problem for the anisotropic wave equa-

tion, utt = uxx + 4uyy with initial data given by

u (0,x) = γ(x)

eixξ/

, ut(0,x) = 0 , γ ∈

s

Hs(Rd)

.

Find the leading term in the approximate solution to u+. In particular, find

the velocity of propagation as a function of ξ. Discussion. The velocity is

equal to the group velocity from §1.3.

1.5. A cautionary example in geometric optics

A typical science text discussion of a mathematics problem involves simplify-

ing the underlying equations. The usual criterion applied is to ignore terms

which are small compared to other terms in the equation. It is striking that

in many of the problems treated under the rubric of geometric optics, such

an approach can lead to completely inaccurate results. It is an example of

an area where more careful mathematical consideration is not only useful

but necessary.

Consider the initial value problems

∂tu + ∂xu + u = 0 , u

t=0

= a(x) cos(x/)

in the limit → 0. The function a is assumed to be smooth and to vanish

rapidly as |x| → ∞, so the initial value has the form of a wave packet.

The initial value problem is uniquely solvable, and the solution depends

continuously on the data. The exact solution of the general problem

∂tu + ∂xu + u = 0 , u

t=0

= f(x) ,

is u(t, x) =

e−t

f(x − t), so the exact solution u is

u (t, x) =

e−t

a(x − t) cos((x − t)/) .