Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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Vortex Filament Methods in 3D

Vortex simulations in 3D differ from those in 2D in

that the stretching term in eqn [4] needs to be

incorporated. The vortex filament method approx-

imates the fluid vorticity by a finite number of

filaments whose circulation remains constant in

time. Each filament is marked by computational

mesh points which move with the regularized

induced velocity. The regularization is necessary to

prevent the infinite self-induced velocities of curved

vortex filaments. As in 2D, this method automati-

cally conserves circulation. Vorticity stretching is

accounted for by the stretching between computa-

tional mesh points. As the filament length increases,

more meshpoints are typically introduced to keep it

resolved. Also, the number of filaments can be

increased throughout the simulation to maintain

resolution.

Viscous Vortex Methods

While inviscid models are expected to approximate

small viscosity fluids well far from boundaries, near

boundaries, where vortex shedding is an inherently

viscous mechanism, it is important to incorporate

the effects of viscosity. The first methods to do so

used operator splitting in which inviscid and viscous

terms of the Navier–Stokes equations were solved in

a sequential manner. In each time step, the compu-

tational elements would first be convected, and then

they would be diffused by a random-walk scheme.

The particle strength exchange method, introduced

more recently, does not rely on operator splitting

and has better accuracy. The particle position and

vorticity evolve simultaneously, and viscous

diffusion is accounted for in a consistent manner.

Vortex dynamics continues to be a source of

interesting problems of theoretical and practical

importance. In particular, much remains to be

learned to better understand turbulence and the

transition to turbulence, a process dominated by

deterministic vortex dynamics.

Further Remarks

Finally, some remarks on relevant literature on this

subject are in order. Lugt (1983) and Tritton (1988)

are recommended as elementary introduction to

vortex flows. van Dyke (1982) presents beautiful and

instructive flow visualizations. Comprehensive treat-

ments of incompressible fluid dynamics are given in

Batchelor (1967), Chorin and Marsden (1992), Lamb

(1932),andSaffman (1992), and compressible flow is

treated in Anderson (1990). Cottet and Koumoutsakos

(2000) give an overview of numerical vortex methods.

Special topics have also been addressed; atmosphere

(Andrews et al. 1987), point vortex motion and chaos

(Aref 1983, Newton 2001, Ottino 1989), superfluids

and superconductors (Blatter et al. 1994, Donnelly

1991), turbulence theory using statistical mechanics (

Chorin 1994), vortex reconnection (Kida and

Takaoka 1994), theory for Euler and Navier–Stokes

equations (Majda and Bertozzi 2002), contour

dynamics (Pullin 1992), vortex rings (Shariff and

Leonard 1992), and aircraft trailing vortices (Spalart

1998). Green (1995) includes survey articles on

various topics.

Nomenclature

a vortex ring core size

g gravitational field

H Hamiltonian

singular velocity kernel

2D, 

regularized velocity kernel

(x, t) fluid density

R vortex ring radius

T(x, t) temperature

U translation velocity

u(x, t) = u(x, t)iþ

v(x, t)j þ w(x, t)k

fluid velocity

w(k) dispersion relation

w = ru vorticity

! = v

 u

scalar vorticity

 ring circulation

See also: Abelian Higgs Vortices; Incompressible Euler

Equations: Mathematical Theory; Integrable Systems:

Overview; Interfaces and Multicomponent Fluids;

Intermittency in Turbulence; Newtonian Fluids and

Thermohydraulics; Point-Vortex Dynamics; Stochastic

Hydrodynamics; Superfluids; Topological Knot Theory

and Macroscopic Physics; Turbulence Theories.

Further Reading

Anderson JD (1990) Modern Compressible Flow with Historical

Perspective, 2nd edn. New York: McGraw-Hill.

Andrews DG, Holton JR, and Leovy CB (1987) Middle Atmo-

sphere Dynamics. Orlando: Academic Press.

Aref H (1983) Integrable, chaotic, and turbulent vortex motion in

two-dimensional flows. Annual Review of Fluid Mechanics

15: 345–389.

Batchelor GK (1967) An Introduction to Fluid Dynamics.

Cambridge: Cambridge University Press.

Blatter G, Feigel’man MV, Geshkenbein VB, Larkin AI, and

Vinokur VM (1994) Vortices in high-temperature

superconductors. Reviews of Modern Physics 66(4):

1125–1388.

Chorin AJ (1994) Vorticity and Turbulence. New York: Springer.

Chorin AJ and Marsden JE (1992) A Mathematical Introduction

to Fluid Mechanics, 3rd edn. New York: Springer.

Cottet G-H and Koumoutsakos PD (2000) Vortex Methods:

Theory and Practice. Cambridge: Cambridge University Press.

398 Vortex Dynamics

Donnelly RJ (1991) Quantized Vortices in Helium II. Cambridge:

Cambridge University Press.

Green SI (ed.) (1995) Fluid Vortices. Dordrecht: Kluwer Academic.

Kida S and Takaoka M (1994) Vortex reconnection Annual

Review of Fluid Mechanics 26: 169–189.

Lamb H (1932) Hydrodynamics, 6th edn. New York: Dover.

Lugt HJ (1983) Vortex Flow in Nature and Technology. New

York: Wiley.

Majda AJ and Bertozzi AL (2002) Vorticity and Incompressible

Flow. Cambridge: Cambridge University Press.

Newton PK (2001) The N-Vortex Problem: Analytical Techni-

ques. New York: Springer.

Pullin DI (1992) Contour dynamics methods. Annual Review of

Fluid Mechanics 24: 89–115.

Ottino JM (1989) The Kinematics of Mixing: Stretching, Chaos,

and Transport. Cambridge: Cambridge University Press.

Saffman PG (1992) Vortex Dynamics. Cambridge: Cambridge

University Press.

Shariff K and Leonard A (1992) Vortex rings. Annual Review of

Fluid Mechanics 24: 235–279.

Spalart PR (1998) Airplane trailing vortices. Annual Review of

Fluid Mechanics 30: 107–138.

Tritton DJ (1988) Physical Fluid Dynamics, 2nd edn. Oxford:

Clarendon Press.

van Dyke M (1982) Album of Fluid Motion. Stanford: The

Parabolic Press.

Vortices see Abelian Higgs Vortices; Point-Vortex Dynamics

Vortex Dynamics 399

Wave Equations and Diffraction

M E Taylor, University of North Carolina, Chapel Hill,

NC, USA

Introduction

The most basic wave equation is

 u ¼ 0 ½1

for u = u(t , x), where  is the Laplace operator,

given by u = @

u=@x

þþ@

u=@x

on n-dimen-

sional Euclidean space R

. More generally, u might

be defined on R  M, where R is the t-axis and M is

a Riemannian manifold, with a metric tensor given

in local coordinates by (g

). Then the Laplace–

Beltrami operator is given, in local coordinates, by

u ¼ g

1=2

j;k

1=2



½2

where (g

) is the matrix inverse to (g

)and

g = det (g

). Even if one concentrates on wave

propagation in Euclidean space, one frequently

wants to use curvilinear coordinates, and [2] is

useful. Equation [1] is supplemented by initial

conditions of the form

uð0; xÞ¼f ðxÞ;@

uð0; xÞ¼gðxÞ½3

called Cauchy data. If the spatial domain M has a

boundary @M (e.g., if M is a bounded region in R

then boundary conditions are imposed. The most

common are the Dirichlet boundary condition

uðt; xÞ¼0 for x 2 @M ½4

and the Neumann boundary condition



uðt; xÞ¼0 for x 2 @M ½5

where @



u denotes the normal derivative of u at the

boundary. More generally, one might have a driving

force, and replace 0 on the right-hand side of [1] by

a function F(t, x). Similarly, one can consider

nonzero boundary data in [4] and [5].

The wave equation [1] models a number of

physical phenomena, at least in the linear approxi-

mation. The vibration of a drum head is modeled by

[1], with M a planar domain, and with the Diri chlet

boundary condition [4]. The motion of sound waves

in a room with hard walls is modeled by [1], with M

a region in R

, and with the Neumann bounda ry

condition [5]. The pro pagation of electromagnetic

waves is given by Maxwell’s equations:

 curl B ¼J

þ curl E ¼ 0

div E ¼ 

div B ¼ 0

½6

where  is the electric charge density and J the

current. These equations yield [1] (with the right-

hand side replaced by some function F(t, x)ifJ and 

are not zero) for the components of the electric field

E and the magnetic field B. If the propagation is in a

region M in R

bounded by a perfect cond uctor,

then the boundary conditions are that E is normal to

@M and B is tangential to @M.If@M is flat, these

equations can be decomposed into Dirichlet pro-

blems for some components and Neumann problems

for the rest, but if @M is curved such a decomposi-

tion is not possible.

Other models of vibrating objects produce var-

iants of [1]. Examples include vibrating elastic

solids, yielding an equation like [1] with u

replaced by u þ (  þ )grad div u, for linear

elasticity. Here  acts componentwise on u, and 

and  are constants, called Lame´ constants. Other

examples model vibrations of crystals and propaga-

tion of electromagnetic waves in crystal s. Further

interesting phenomena arise in these various cases,

such as Rayleigh waves in linear elastici ty and

conical refraction in crystal optics.

Here we discuss the propagation of waves and their

reflection and diffraction at boundaries. In the interest

of providing reasonable coverage in a brief space, we

restrict attention to the wave equation [1].

Basic Propagation Phenomena

The simplest examples of waves propagating accord-

ing to [1] are plane waves, of the form

uðt; xÞ¼’ðx  !  tÞ½7

with (t, x) 2 R  R

, ! a unit vector in R

,and’ a

function on R.If’ has two continuous derivatives,

[7] defines a classical solution of [1]. More

generally, one can allow ’ to be less regular. For

example, it could be piecewise smooth with a jum p

discontinuity at some point a 2 R. In such a case, u

will be piecewise smooth with a jump across the

n-dimensional surface x  !  t = a in R  R

, which

will solve [1] in a weak, or distributional, sense. For

fixed t, u(t,  ) has a jump across the (n  1)-

dimensional surface 

= {x 2 R

: x  ! = t þ a}. As

t varies, 

moves in the direction ! with unit speed.

There are also spherical wave solutions to [1] on

R  R

, such as

uðt; xÞ¼

sgn t

2

ðt

jxj

1=2

½8

for n = 2, and

uðt; xÞ¼

4t

ðjxjjtjÞ ½9

for n = 3. Here s

= s for s > 0, s

= 0 for s  0, and

(s) is the Dirac delta function. In fact, [8] and [9]

are ‘‘fundamental solutions’’ (more on which in the

section on ha rmonic a nalysis) to the wave equation

on R  R

, for n = 2 and 3, respectively. In such

cases, the singularity in u(t,  ) for each fixed t lies in



= {x 2 R

: jxj= jtj}, a family of surfaces in R

that moves, in the direction of the normal to 

,at

unit speed.

The examples mentioned above illustrate two

general phenomena about the behavior of solutions

to [1]. The first is finite propagation speed. Its

general formulation is that, given a closed set

K  M,

supp f ; g  K ) sup p uðt; Þ

fx 2 M : distð x; KÞjtjg ½10

In fact, given that [8]–[9] are fundamental solutions,

[10] is a consequence of these formulas when

M = R

or R

. The result [10] is true in great

generality, with well-known demonstrations invol-

ving energy estimates.

The second phenomenon involves propagation of

singularities. Typically, if the Cauchy data f and g in

[3] are smooth on the complement of an (n  1)-

dimensional surface 

, perhaps with a jump across



, or such a singularity as in [8] or [9], the solution

u(t, x) will be a sum of two terms, with singularities

of a similar nature on the surfaces 



, moving at

unit speed in the direction of their normals, 

flowing from 

in one direction and 



in the

other. This also holds for the manifold case [2]. That

happens at least until such surfaces develop singula-

rities, when matters become more elaborate.

An alternative way to describe how the set of

singularities evolves is the following. Let S

M denote

the space of unit vectors tangent to M; this is

a submanifold of the tangent bundle of M, TM.

There is a natural projection  : S

M ! M. Asso-

ciated to a smooth surface 

of dimension n  1in

M (of dimension n) are two preimages 

and 



M, consisting of unit vectors lying ov er points of



and normal to 

. The geodesic flow is a flow on

M, and it takes 



to smooth (n  1)-dimensional

surfaces 



in S

M. The sets 



are the images of





under . The geodesics starting out at points in





and sweeping out 



are the rays along which

the singularities of the solution u propagate.

This latter description works for all t if M has no

boundary and is complete, that is, all geodesics are

defined for all t, although singularities develop in

the images (



) =



, at points p 2 



, where 



meets T

M nontransversally. The behavior of u near

such singular points of 



, known as caustics, is

more complicated than that near regular points, but

it can be captured in terms of integrals. Methods of

establishing this propagation of singularities are

discussed in the section on geometrical optics.

Such a description needs further elaboration if M

has a bounda ry. One of the principal problems of

diffraction theory is to explain how singularities of

solutions to [1], with a boundary condition such as

[4] or [5], propagate and reflect off the bounda ry.

Considering the case where M is a half-space

in R

M ¼ R

¼fx 2 R

: x

 0g½11

provides a guide to the simplest reflection phenom-

ena. In such a case, one can solve the Dirichlet or

Neumann boundary problem for the wave equation

[1] by the method of images. One extends f and g

from R

to R

. For the Dirichlet problem [4], one

takes odd extensions, f (x

, x

) = f (x

, x

), and

similarly for g . For the Neumann problem [5], one

takes even extensions, f (x

, x

) = f (x

, x

), etc.

One then solves the wave equation [1] on R  R

with the extended Cauchy data, and the restriction

to R  R

solves the respective boundary proble m.

Suppose 

is a smooth (n  1)-dimensional surface

that does not meet @R

, and that f and g have

singularities on 

, as above. (Suppose for simplicity

that f and g vanish near @R

.) Those rays issuing

402 Wave Equations and Diffraction

from normals to 

have mirror images, which are

rays in R



. If such a ray hits @R

, its mirror image

does so also, and continues into R

, as the reflected

ray. The singularities of u propagate along such

reflected rays.

Such a description extends to a general complete

Riemannian manifold with boundary M, in the case

of rays that hit the boundary transversally. Such a

ray is reflected by retaining the tangential compo-

nent of its velocity vector at the point of intersection

@M and rever sing the sign of the normal component.

One says that the ray is reflected according to the

laws of geometrical optics. Singularities of u carried

by such rays that hit @M are correspondingly

reflected. Methods to establish such transversal

reflection of singularities are natural extensions of

those developed to treat the propagation away from

@M, mentioned above.

Matters become more delicate when there are rays

that are tangent to @M. A model example is given by

M ¼ R

n B; B ¼fx 2 R

: jxj < 1g½12

which one takes when studying the scattering of

waves in R

by the obstacle B. Consider a solution

to [1] with boundary condition given by [4] or [5]

that has a simple singularity on 

= {x 2 R

: x

= t}

for t < 1. The associated rays are of the form



(t) = (x

, t), for t < 1, with x

2 R

n1

.Ifjx

j > 1,

these rays continue on in R

nB, for all t 1. If

j < 1, these rays hit @M = @B transversally, and

their reflection is as described above. If jx

j= 1,

these rays hit @B tangentia lly, at t = 0; they are

sometimes called grazing rays. One also continues

them past t = 0. One defines in this fashion 

for

t 1. The region

S¼fx ¼ðx

; x

Þ2R

nB : jx

j < 1; x

> 0g½13

is called the ‘‘shadow region.’’ It is disjoint from 

for all t. The solution u is smoot h in S for all t,

although it is not identically zero. The set

¼fx ¼ðx

; x

Þ2R

nB : jx

j¼1; x

 0g½14

is the ‘‘shadow boundary.’’

One can replace B in [12] by a more general

smooth, convex obstacle K, with positive Gauss

curvature everywhere, and the same considerations

of transversal and grazing rays and shadow regions

apply. These notions also extend to a more general

class of Riemannian manifolds with boundary,

called manifolds with diffractive boundary. In the

case K = B, one can use separation of variables to

reduce the problem of analyzing solutions to [1] and

showing that singularities propagate along such rays

to a problem in harmonic analysis on the sphere

n1

. For more general convex obstacles K or

manifolds with diffractive boundary, other techni-

ques are required, to show that waves reflect off the

boundary in a fashion similar to the case [12] .

Another situation arises if instead of [12] one

takes M = B, or more generally M = K, a convex

region as described above. A ray starting off from

a point in @M, almost tangent to @M but with a

small component in the direction of the normal

pointing into M, will undergo many reflections in

a short time. Upon shrinking the normal c ompo-

nent of the initial velocity to zero, one obtains in

thelimitageodesicin@M, known as a g liding ray.

In such a case, singularities of solutions to [1],

with such a boundary condition as [4] or [5],

propagate along both transversally reflected and

gliding rays.

For the generic smooth obstacle K in R

, the

second fundamental form can have a variety of

signatures at various boundary points. Various types

of ‘‘generalized rays’’ occur – generally speaking

limits of sequences of transversally reflected rays.

This situation also holds for general complete

Riemannian manifolds with smooth boundary. The

main result about propagation of singularities in

such a case is that it is always along such generalized

rays. This was established by Melrose and Sjo¨ strand

(1978).

Further diffraction effects arise when @M has

singularities, such as edges and corners. The simplest

example is

M ¼fx 2 R

: a    b; r  0g½15

where (r, ) are the polar coordinates of x 2 R

, and

we assume 0  a < b  2 . Here one is studying the

diffraction of waves by a wedge. In the limiting case

a = 0, b = 2, the wedge becomes a half-line, that is,

M ¼ R

nfðx

; 0Þ : x

> 0g½16

Singularities of solutions to [1] on R  M with

such a boundary condition as [4] or [5] propagate in

the interior of M and reflect off the regular points of

@M as before. If a family of continuous, piecewise

smooth curves 

carrying the singularity of u hit the

corner x = 0att = a, this reflection creates a tear in



for t > a.Inaddition,adiffractedwavespreads

out from the corner at unit speed. This diffracted

wave carries a singularity that is weaker than that of

the incident wave. For example, if one has a solution

like [8], but shifted to have support in a disk of radius

jtj about a point p 6¼ 0inR

, for small jtj,thenthe

diffracted wave will have a jump discontinuity.

The space M in [15] is a special case of a cone.

More generally, if N is a complete Riemannian

Wave Equations and Diffraction 403

manifold (possibly with boundary), then the cone

C(N) with base N is the set

CðNÞ¼½0; 1Þ  N ½17

with all points (0, x), x 2 N, identified, with the

metric tensor

¼ dr

þ r

g ½18

where g is the metric tensor on N, and points on

C(N) are denoted (r, x), r 2 [0, 1), x 2 N. The space

in [14] has the form M = C(N) with N = [a, b], an

interval. A cone in Euclidean space R

is of the form

C(N) with N a domain in the unit sphere S

n1

The propagation of singularities for solutions to [1]

on C(N), when N has smooth boundary, has a

description similar to that above for the case [15].

Again, there is a diffracted wave set off from the conic

point {r = 0} when a singularity of a wave hits it. The

diffracted wave is typically (n  1)=2 units smoother

than the singular wave producing it, where

n = dim C(N). For example, the fundamental solution

to the wave equation on C(N) produces a diffracted

wave which is the sum of a jump discontinuity and (in

general) a logarithmic singularity.

In fact , precise understanding of the behavior of

the fundamental solution to the wave equation on

C(N) is encoded in terms of the behavior of the

solution operator to the wave equation on the base

N. This is discussed in further detail in the section

on harmonic analysis. In the case where C(N)is

given by [15], we are dealing with the wave

equation on an interval [a, b], whose behavior is

elementary.

One can use analysis of [15] together with finite

propagation speed to get a good qualitative picture of

diffraction of waves in R

by a polygonal obstacle. A

variation of this argument allows one to understand

the behavior of the wave equation on a ‘‘polygonal’’

domain N in S

, that is, one whose boundary consists

of a finite number of geodesic segments in S

. Going

from there to C(N), one can then analyze diffraction

of waves in R

by a polyhedr on.

It is worth remarking how the ‘‘shadow region’’

for such an obstacle as a wedge in R

differs from

that in [12]–[14]. For example, if one considers M

given by [16] and u( t, x) = (x

 t), for t < 0, then

the region

S¼fx ¼ðx

; x

Þ : x

; x

> 0g½19

is the ‘‘shadow region,’’ in the sense that rays either

missing or reflecting off the obstacle {(x

,0):x

> 0}

do not enter the region [19]. However, unlike the

case [13], the solution u(t, x) is not smooth in the

region [19] for t > 0. There is a singularity there,

although it is weaker than the singularity of the

main wave.

Taking Cartesian products of spaces of the form

[15] with R

yields spaces with k-dimensional

edges. There are also spaces with curvy edges.

Rather than continuing with further general

description, one more particular case is discussed

next, which has had a historical significan ce.

Namely, we consider the reflection of waves in R

off a disk, that is, take

M ¼ R

n D; D ¼ðx

; x

; 0Þ : x

þ x

 1



½20

Consider a wave given for t < 0byu(t, x) = (x

 t).

This wave hits D = @M at t = 0, giving off a diffracted

wave, traveling away from  = {(x

, x

,0):x

= 1} at speed 1 for t > 0. This diffracted wave

carries a singularity that blows up like the 1=2power

of the distance to the torus of points of distance t from

,fort 2 (0, 1). For t > 1, there is a focusing effect

along the x

-axis, producing a stronger singularity for

u(t, x)there.

This sort of phenomenon was understood, at

least from a heuristic point of view, in the

nineteenth century, and it played a role in an

important argument of Poisson. At the time, there

was a debate about whether the propagation of

light was a wave phenomenon. Poisson did not

think it was, and he noted that if it were, the light

waves propagated past such an obstacle should

produce a bright spot along the axis normal to the

disk and through its center. The experiment was

performed and the bright spot was observed.

This is now called the Poisson spot, and its

occurrence convinced many physicists, including

Poisson, that the propagation of light is a wave

phenomenon.

Harmonic Analysis and the Wave

Equation

The wave equation [1] with Cauchy data [3] can be

regarded as an operator differential equation, with

solution

uðt; xÞ¼cos t

ﬃﬃﬃﬃﬃﬃﬃﬃ



f ðxÞþ

sin t

ﬃﬃﬃﬃﬃﬃﬃﬃ



ﬃﬃﬃﬃﬃﬃﬃﬃ



gðxÞ½21

This brings one to investigate functions of the self-

adjoint operator .IfM = R

, one can do this using

the Fourier transform, which is give n by

Ff ðÞ¼

f ðÞ¼ð2Þ

n=2

f ðxÞe

ix

dx ½22

One defines F



by changing e

ix

to e

ix

in [22],

and the Fourier inversion formula says F and F



are

404 Wave Equations and Diffraction

inverses of each other on various function spaces,

including L

). Then one has

’ð

ﬃﬃﬃﬃﬃﬃﬃﬃ



Þf ðxÞ¼ð2Þ

n=2

’ðj jÞ

f ðÞe

ix

d ½23

Note that [23] is equal to

ðx  yÞf ðyÞdy ¼   f ðxÞ½24

where

ðxÞ¼ð2Þ

n

’ðj jÞe

ix

d ½25

In particular, [21] becomes

uðt; xÞ¼

 f ðxÞþR

 gðxÞ½26

where

ðxÞ¼ð2Þ

n

sin tjj

jj

ix

d ½27

is the fundamental solution to the wave equation.

The integral [27] is not an easy integral when

n > 1, but the answer can be derived by analytic

continuation from the Poisson kernel, that is,

y

ﬃﬃﬃﬃﬃﬃ



f ðxÞ¼P

 f ðxÞ

ðxÞ¼C

yðjxj

þ y

ðnþ1Þ=2

½28

where C

= 

(nþ1)=2

((n þ 1)=2). One gets

ðxÞ¼lim

"&0

n  1

Imðjxj

ðt  i"Þ

ðn1Þ=2

½29

Taking this limit for n = 2, 3 yields the formulas

[8]–[9]. There are several ways to derive [28]. One,

which is flexible and useful for other situations,

derives it from the formula for the heat kernel,

t

f ðxÞ¼H

 f ðxÞ; H

ðxÞ¼ð4tÞ

n=2

jxj

=4t

½30

via the subordination identity:

yA

2

1=2

y

=4t

tA

3=2

A > 0; y > 0 ½31

with A =

ﬃﬃﬃﬃﬃﬃﬃ



. The heat kernel can be computed via

[23], which becomes a well-known Gaussian inte-

gral. The identity [31] can be proved using the fact

that the Fourier integral formula for P

(x)is

elementary to compute when n = 1.

To understand functions of the Laplace operator

on a cone C(N), one uses

 ¼

n  1



½32

where 

is the Laplace operator on N, which

follows from [2] and [18].Heren = dim C(N). This is

a modified Bessel operator. We define the operator

 ¼ð

þ 

1=2

;¼

n  2

½33

For each 

in the spectrum of , we consider the

Hankel transform



gðÞ¼

gðrÞJ



ðrÞr dr ½34

where J



is the Bessel function of order 

. The

Hankel inversion formula says H



is unitary

on L

, r dr), and is its own inverse. Conse-

quently, we can write the action of ’(

ﬃﬃﬃﬃﬃﬃﬃ



)on

(C(N)) as

’ð

ﬃﬃﬃﬃﬃﬃﬃﬃ



Þgðr; xÞ¼

’

ðr; s;Þgðs; xÞs

n1

ds ½35

where K

’

(r, s,) is a family of operators on L

(N),

given by

’

ðr; s;Þ¼ðrsÞ



’ðÞJ



ðrÞJ



ðsÞ d ½36

To obtain the wave kernel on C(N), one can

analytically continue formulas for the Poisson

kernel, for e

y

ﬃﬃﬃﬃﬃ



. Such formulas arise from the

Lipschitz–Hankel identity:

y



ðrÞJ



ðsÞd



ðrsÞ

1=2

1=2

þ s

þ y

2rs



½37

Here Q

1=2

() is a Legendre function. The identity

[37] is one of the more difficult identities in the

theory of Bessel functions. It is useful to know that it

can be derived by applying a slight variant of the

subordination ident ity [31] to the more elementary

identity

t



ðrÞJ



ðsÞ d ¼

ðr

þs

Þ=4t





½38

(where I



(y)=e

i=2



(iy)fory > 0), which describes

the behavior of the heat kernel on C(N).

Carrying out the analytic continuation of [37]

to imaginary y yields results stated in the section

on ba sic propagat ion phenome na, once one

Wave Equations and Diffraction 405

understands the behavior of families of functions of

the operator  so produced. An approach taken by

Cheeger and Taylor (1982) to this was to synthesize

these operators from e

is

, s 2 R, and deduce their

behavior from the behavior of the solution operator

to the wave equation on the base N.

One can apply similar considerations to M = R

nB,

which is the truncated cone [1, 1)  S

n1

, with

metric tensor [18], where g is the metric tensor on

n1

, and Laplace operat or given by [32], with 

the Laplace operator on S

n1

. The problem of

diffraction of wave s by the ball B can be recast as

solving

 u ¼ 0onR  M

R@M

¼ f ; uðt; xÞ¼0 for t  0

½39

with f compactly supported on R  @M. Taking the

partial Fourier transform with respect to t yields the

reduced wave equation

ð þ 

Þv ¼ 0 for jxj > 1; v

n1

¼ gðx;Þ

½40

and the condition u(t, x) = 0 for t  0 yields for v

the outgoing radiation condition

ðn1Þ=2

 iv



! 0asr !1 ½41

The solution is

vðx;Þ¼r

ðn2Þ=2

ð1Þ



ðrÞ

ð1Þ



ðÞ

gðx;Þ½42

with  as in [33] and H

(1)

the Hankel function.

The behavior of H

(1)



(r)=H

(1)



()as

,  !1

with ratio in a small neighborhood of 1 can be

shown to control the behavior of the solution u to

[39] near grazing rays. There is an asymptotic

formula for this, which is one of the most delicate

analytical results in the theory of Bessel functions.

The result is that, uniformly for z near 1, as

 !1,

ð1Þ



ðzÞ2e

i=3

4

1  z



1=4

 A

ð

2=3

Þ

k0

ðÞ

1=32k

(

þA

ð

2=3

Þ

k0

ðÞ

5=32k

)

½43

Here

ðÞ¼Aið e

2i=3

Þ½44

where Ai is the Airy function. The coefficients a

()

and b

() are smooth functions of their argument,

 = (z), which is defined by



3=2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1  t

½45

Making use of [43] in [44], one can obtain a

parametrix for u (i.e., a solution modulo a C

error)

whose form is a special case of the formula [50],

which we will present in the next section.

Geometrical Optics and Extensions

By results of the last section, the solution to [1]

when M = R

has the form

uðt; xÞ¼



itjjþix



ðÞd ½46

where the functions h



are produced from the initial

data via simpler transformations. For a general

metric tensor, one can produce a parametrix (i.e.,

an approximation to u(t, x) with a C

error) in the

following form:

uðt; xÞ¼



ðt; x;Þe

i’



ðt;x;Þ



ðÞd ½47

Here the phase functions ’



(t, x,) are smooth for

 6¼ 0 and homogeneous of degree 1 in  . The

amplitudes a



(t, x,) are smooth and have asympto-

tic expansions as jj!1:



ðt; x;Þ

k0



ðt; x;Þ½48

with a



(t, x,) homogeneous of degree k in . One

applies @

  to both sides of [47], and obtains an

operator of a similar form, with new amplitudes



(t, x,) 



(t, x,). Setting the terms in this

asymptotic expansion equal to zero yields, first for

’



(t, x,), a partial differential equation known as

the eikonal equation:

@’



¼ jr

’



j½49

where jvj is the norm of a vector v 2 T

determined by the metric tensor. Sett ing b



(t, x,)

= 0 for k  1 yields linear differential equations for

the amplitude terms in [48], known as transport

equations.

Operators of the form [47] are special cases

of Fourier integral operators. Seminal works of

Keller (1953) and Lax (1957) gave an important

stimulus to work on these operators, and work of

Ho¨ rmander (1971) turned this into a systematic and

powerful theory. A particular advance regards

406 Wave Equations and Diffraction

producing a parametrix valid for all t.Generally,one

can solve [49] and the associated transport equations

for t in some interval, past which the eikonal

equation might break down. Ho¨ rmander’s theory

treats products of Fourier integral operators, yielding

global constructions. This facilitates the treatment of

caustics mentioned earlier. Stationary-phase methods

can be brought to bear to relate the singularities of

Th to those of h,whenT is a Fourier integral

operator.

To construct parametrices for waves reflecting off

a boundary, one can again reduce the problem to

one of the form [39]. Waves that reflect transver-

sally are given by parametrices of the form [47],

although with the role of the variables changed, so

that t in [47]–[49] is replaced by a coordinate that

vanishes on R  @M.

A parametrix that treats grazing rays can be written

in the form of a Fourier–Airy integral operator:

uðyÞ¼

ðÞþijj

1=3

ðÞ

 A

ð

1

i

FðÞd ½50

Here y = (y

, ..., y

nþ1

) denotes a coordinate system

on a neighborhood of a boundary point of R  M,

with y

nþ1

= 0onR  @M. We have a pair of phase

functions (y, ) and (y, ), homogeneous in  of

degree 1 and 2/3, respectively, and a pair of

amplitudes a(y, ) and b(y, ), each having asympto-

tic expansions of the form [48]. The function A

the Airy function [44]. The phase functions satisfy a

coupled pair of eikonal equations:

; r

iþhr

;r

i¼0

; r

i¼0

½51

where h, i denotes the Lorentz inner product on

(R  M) given by dt

 g. More precisely, [51] is

to hold in the region where   0, and also to

infinite order at y

nþ1

= 0, for   0. One requires

@=@

to have linearly independent y-gradients, for

j = 1, ..., n, and

ðy;Þ¼

ðÞ¼

1=3



for y

nþ1

¼ 0 ½52

The terms in the asymptotic expansions of a(y, )

and b(y, ) satisfy coupled systems of transport

equations. One can arrange that b(y, ) = 0 for

nþ1

= 0. Then uj

R@M

= TF, where T is a Fourier

integral operator, which can be inverted, modulo a

smooth error, by Ho¨ rmander’s theory, producing a

parametrix for [39].

The construction of solutions to [51] satisfying

[52] is due to Melrose. Thi s followed earlier works

of Ludwig (1967), Melrose (1975), and Taylor

(1976), which produced solutions satisfying [52] to

infinite order at 

= 0. This earlier construction is

adequate to produce a grazing ray parametrix, but

the sharper result [52] is extremely va luable for

constructing a gliding ray parametrix. This has the

form

uðyÞ¼

½a AiðÞþijj

1=3

b Ai

ðÞ

 Aið

1

i

FðÞd ½53

It differs from [50] in the use of Ai rather than A

Since Ai has real zeros, it is also convenient to pick

T > 0 and evaluate , , a, and b at (

, ..., 

n1



þ iT), and take 

= 

1=3

(

þ iT). The treatment

of the eikonal and transport equ ations is as above,

though the Fourier–Airy integral operator [50] has a

different behavior from [53], reflecting the differ-

ence between how singularities in solutions to the

wave equation are carried by grazing and by gliding

rays.