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Wave Propagation
JOSEPH B. KELLER
Department of Mathematics, Stanford University
Stanford, CA 94305, USA
1.
Introduction
The mathematical theory of wave propagation is the study of partial differential
equations, or systems of such equations, with wave-like solutions. An example of
such an equation is the wave equation
Au(x,t) - ——uttfat) = 0 .
cA[x)
(1.1)
Here the unknown u(x, t) is a real- or complex-valued function of a point x in Rn
and of the time t. The given positive function c(x) is the speed of propagation of
waves at the point x.
Equations like (1.1) are satisfied by the acoustic pressure in a gas or liquid,
by each component of an electromagnetic field, by each component of the displacement of a string or membrane from its rest position, and by many other physical
quantities. In these examples, the number n of space dimensions is one for a string,
two for a membrane, and three for acoustic and electromagnetic fields. In these
cases, the physical quantity is real, so it is represented by a real solution, or the
real part of a complex-valued solution.
In quantum mechanics, the wave function u(x, t) is complex valued and the
number of space dimensions is three times the number of particles in the physical
system under consideration, so it can be very large. The differential equation for
u is the Schrödinger equation. It differs from (1.1) in having utt replaced by iut,
and in containing terms in which u is undifferentiated.
When c(x) is a constant, (1.1) has the plane wave solution
u(x,t) = A e < k - x - " 0 .
(1.2)
The constants A, k, and CJ are called respectively the amplitude, the wavevector,
and the angular frequency of the wave. In order for (1.2) to satisfy (1.1), the
angular frequency UJ must be related to the wavenumber A; = |k| by the equation
UJ2 = k2c2. The solution (1.2) is called a plane wave because at each value of t it
is constant on the planes k • x = constant. These planes are called wavefronts, and
they propagate in the direction of k with the velocity c. The wave is periodic in
the direction of k with the period À = 2-iï/k, which is called the wavelength. We
note that A is small when k is large.
Proceedings of the International Congress
of Mathematicians, Zürich, Switzerland 1994
© Birkhäuser Verlag, Basel, Switzerland 1995
Wave Propagation
107
Because (1.1) is linear in u, more general solutions can be formed by superposition of plane wave solutions:
u(x,t) = f [ ^ ( k ) ^ * - ^ ) + Aa(k)ei(k,x+W*)] dk .
(1.3)
In fact the two amplitude functions AiQa) and A2(k) can be adjusted to make u,
given by (1.3), and its derivative ui, take on any specified values at t = 0. Then
(1.3) is the solution of the initial value problem, also called the Cauchy problem,
for (1.1) with c = constant.
More interesting problems occur when (1.1) holds in a subdomain of Rn,
with conditions imposed upon u at the boundary of the sub domain. We shall now
describe the two classical boundary value problems that arise in this way. Similar
„problems „occur when„c(x) is_ not „constant, _as_we_shall_see_.
2.
Boundary Value Problems
First we consider a solution of (1.1), with c = constant, in a bounded domain D
in Rn. On the boundary 3D of D we require that u = 0. Furthermore, we require
that u be the product of a function v(x) multiplied by a function of t. We find at
once that the function of t must be e±lUJt, so that u must have the form
u(x,t) = v(x)e±iujt
.
(2.1)
Now the problem is to find a constant UJ and a function v(x) not identically zero,
satisfying the following equations:
Aw + k2v
v
=
=
0
0
,
,
xeD
xedD .
(2.2)
(2.3)
In the Helmholtz equation (2.2), kr = ur jc2. This problem is called an eigenvalue
problem. The values of k2 for which it has solutions are called eigenvalues, and
the corresponding solutions v(x) are called eigenfunctions.
In one dimension, each eigenvalue determines a possible frequency of vibration of a vibrating string with fixed endpoints. The corresponding eigenfunction
determines the amplitude of vibration at each position x along the string. Similarly
in two dimensions, the solutions determine the frequencies and modes of vibration
of a membrane, such as a drumhead, held fixed at 3D. In three dimensions, the solutions determine the frequencies and modes of vibration of fluid in a soft container
with the surface 3D. At 3D, the acoustic pressure is forced to vanish.
The second problem we consider is called the scattering problem. In it we
seek a solution u of (1.1) with c = constant, in the exterior of a bounded domain
D. This solution must consist of the incident plane wave (1.2), which is specified,
and a scattered wave of the form (2.1) with the negative sign, which is to be found.
Thus, u has the form
u(x, t) = yle'C"-"*) + v(x)e~iujt
.
(2.4)
On the boundary of D we require that u — 0, and at infinity we impose a condition
that guarantees that v(x) does indeed represent a wave travelling outward from D.
108
Joseph B. Keller
The problem for v can be formulated as follows: given k and A, find the
scattered wave v(x) satisfying
Av-rk2v
=
v
T L - l
lim r7 ^ (vT -ikv)
i—»oo
0
xeRn-D
,
=
-Ae*
=
0.
kx
,
(2.5)
xedD
(2.6)
(2.7)
In (2.7), which is called the Sommerfeld radiation condition, r denotes distance
from x to an origin in D and vr denotes the radial derivative of v.
The eigenvalue problem determines a discrete spectrum of positive real values
of k2 that depend upon the size and shape of D and upon the dimension n. In the
scattering problem however, the spectrum consists of all positive real values of k2.
3.
Classical Methods of Solution
Most of the classical methods of solving the eigenvalue problem, the scattering
problem, and other boundary value problems, are based upon separation of variables. In this procedure, the point x is represented in terms of curvilinear coordinates and the differential equation (2.2) is expressed in terms of these coordinates.
For example, in n = 2 dimensions we write x = (£,77) and express A in terms of £
and 77. Then we seek a solution that is a product of a function of £ times a function
of 77, v = f(Ç)g(rj).
If there are product solutions, the variables separate in (2.2). This leads to
two ordinary differential equations, one for /(£) and another for g(n):
i i / ( 0 = ßf(0
, L2g(V) = - W f a ) .
(3.1)
Here Li and L2 are second order ordinary differential operators in £ and 77 respectively, and ß is a constant called the separation constant. All the special functions
of classical mathematics and mathematical physics arise as the solutions of these
ordinary differential equations: Bessel, Hankel, Mathieu, Legendre, Hermite, Laguerre, and other functions.
To complete the specification of the solutions of (3.1), we must impose the
boundary condition (2.3), the radiation condition (2.7), regularity conditions at
singularities of the coordinates, periodicity conditions if coordinates are cyclic, etc.
These conditions determine a spectrum of values of fi and corresponding solutions
of (3.1).
If the fi spectrum is discrete, say J.LI,JI2,. .., we denote the solutions corresponding to ßj by /j (£) and gj(r]),j = 1, 2 , . . . . Then we can write the solution v
as a series of product solutions with undetermined coefficients Cj :
00
^rì)
= YJCjm)gj(rì).
(3.2)
i=i
When the spectrum of /j, is continuous, we denote the solutions by /(£,/.£) and
#(77, fi). Then we represent v as an integral of product solutions with the undeter-
Wave Propagation
109
mined coefficient function c(fi):
v(^v)= JcMffarìgfarìdiJ,
.
(3.3)
The coefficients c7- or c(/i) must be determined from the inhomogeneous data
of the problem, such as the prescribed value of v on the boundary, a source term
in the equation, initial data in the case of the wave equation (1.1), etc. Various
discrete and continuous transforms have been developed for this purpose: Fourier,
Laplace, Hankel, Mellin, Lebedev, etc.
Once Cj or c(ji) is determined, (3.2) or (3.3) provides a representation of the
solution v as an infinite series or as an infinite integral. It is then necessaiy to
evaluate the series or integral in order to determine the quantitative behavior of
~v(x). Whe^hTc^is^ smällT^öThat~the wavelength~is~large compärecTtö tlië~dimensions""
of D, the series (3.2) converges rapidly and the integral (3.3) can be evaluated
readily. However, when k is large, which corresponds to a high frequency, the
wavelength is small compared to the dimensions of D. Then the terms in the
series (3.2) are oscillatory, as is the integrand in (3.3). In that case, there is a great
deal of cancellation in the series and in the integral, and it is difficult to evaluate
them.
To overcome this difficulty, methods for the asymptotic evaluation of series,
such as the Euler-Maclaurin sum formula and the Poisson summation method,
have been adapted. A new method, the Watson transformation, has also been
developed. For asymptotic evaluation of the integral, Laplace's method has been
used, Kelvin's method of stationary phase was developed, the saddle point method
has been used, etc.
In addition to the methods described above, there are other classical methods
of solution based upon Green's functions, integral equations, images, etc.
4.
Limitations of the Classical Methods
At the end of the nineteenth century and in the beginning of the twentieth century,
it was discovered that separation of variables is possible in only very few coordinate
systems. This was shown by expressing the Laplacian in (2.2) in general curvilinear
coordinates and then deducing conditions on the metric coefficients in order that
separation be possible. In the plane, for example, (2.2) is separable only when the
coordinate lines are linear or quadratic curves, i.e. conic sections, and a similar
conclusion holds in R?.
This limitation on the coordinate systems also limits the domains D to which
separation of variables is applicable. This can be seen by noting that when a
product solution v = f(^)g(i]) vanishes at one point £0,770 then either /(£o) = 0
or
#(?7o) = 0- Thus, v vanishes all along the coordinate line £ = £0 or along the
line 77 = 770. It is only when 3D consists of one or more such lines that a product
solution can vanish on it. The consequence of this fact is this: the method of
separation of variables can be used to solve a boundary value problem only when
the boundary consists of portions of coordinate lines (or surfaces) of a coordinate
system in which the differential equation is separable.
110
Joseph B. Keller
Within these limits, a large number of scattering problems have been solved.
It is from these solutions that our physical insight has developed, and it is from
them that we have obtained many quantitative results.
Some of the objects (or domains) for which the scattering problem has been
solved are listed below, together with the name of the solver and the approximate
date:
circular cylinder
sphere (electromagnetic waves)
parabolic cylinder
Rayleigh
Mie
Epstein
1881
1908
1914
,., . ,
}
slit in plane screen J
circular disc
^
, , , . ,
\
circular hole in plane screen J
circular cone (electromagnetic)
plane angular sector
Morse and Rubinstein
1938
r> ,
Bouwkamp
^AA
1944
Hansen and Schiff
Kraus and Levine
1945
1963
Asymptotic evaluation of the solutions for high frequency or short wavelength
was done for the following cases:
circular cylinder
sphere
parabolic cylinder
elliptic cylinder
Debye
Nussenzweig
Watson
Fock
Rice
Keller
Levy and Keller
1908
1970
1912
1946
1950
1956
1960
A new modification of the classical method, useful for short wavelengths, was
developed by Sommerfeld to solve the problem of diffraction by a half plane, i.e.
a thin screen with a straight edge:
half plane
wedge
Sommerfeld
McDonald
1896
1902
During the 1940s, another new modification was developed based upon the
Wiener-Hopf method for solving integral equations with convolution kernels. This
was done by Schwinger and by Vainstein, and the method has been used by many
others. This work is described in the book of Noble (1965) on this method.
5.
Geometrical Optics
In addition to the classical methods described above, there is a different classical
method called geometrical optics, which is used for describing the propagation of
light. It was known to the ancient Greeks, and is presented in the works of Euclid.
It is called geometrical optics because it determines the paths, called rays, along
which light travels. This method is based upon the following three laws.
Wave Propagation
111
Law of Propagation (Euclid). In a homogeneous medium light travels in a straight
line.
Law of Reflection (Euclid). A ray that hits a reflecting surface produces a reflected
ray. The reflected ray makes the same angle with the normal to the surface as does
the incident ray, and the two rays lie on opposite sides of the normal. Alhazcn, in
the eleventh century, noted that the two rays and the normal are coplanar.
Law of Refraction (Snell 1626, Descartes 1637). A ray that hits the interface between two different media produces a refracted ray on the other side of the interface
from the incident ray. The angle ß between the refracted ray and the normal is
related to the angle a between the incident ray and the normal by Snell's law:
m sin a = n2sinß. Here m is the refractive index of medium one containing the
incident ray and n2 is that of medium two containing the refracted ray. We now
._know_that_ni_=_co/c^-wliere co-is a_reference-speed, usually-that of-light-in avac-uum, and Q is the speed in medium i. The two rays are coplanar with the normal
and lie on opposite sides of it. Ptolemy, in the first century, gave an approximation
to Snell's law that is valid for small angles, namely nia = n2ß.
These three laws suffice for the calculation of the focal lengths of mirrors and
lenses, and for the design of optical instruments. They were used, for example, by
Gauss in the early 1800s in his analysis of imaging in axially symmetric optical
systems.
There is a different formulation of the laws of optics, based upon the calculus
of variations. For propagation it was known to Euclid, because he knew that a
straight line is the shortest distance between two points. Thus, the ray from P to
Q is the shortest path from P to Q. For reflection, Heron of Alexandria, in the
second century, showed that the incident ray from P to a plane mirror, plus the
reflected ray from the mirror to Q, is the shortest path from P to Q with one point
on the plane mirror.
Finally Fermât (1661) found a variational formulation of the law of refraction.
This led to his principle of least time, which states that in a medium with the
refractive index n(x), the light ray from P to Q minimizes the integral L:
rQ
rQ ds
r
n [x(s)] ds = cQ /
= c0 / dt .
(5.1)
Jp
Jp c[x(a)]
J
The last integral in (5.1) shows that L is Co times the time required for light to
travel from P to Q along the path x(s). L is called the optical length of the path.
We now know that L is stationary at the light ray, but not necessarily a minimum.
Fermat's Principle yields the laws of propagation and refraction, and it determines the rays in inhomogeneous media. It also yields the law of reflection when it
is applied to paths with one point on a given (mirror) surface. Thus, it determines
all the rays of geometrical optics.
Hamilton (1833) gave various alternative formulations of the laws of optics.
One is the system of six first order ordinary differential equations for the rays in R?.
These are now known as Hamilton's equations, and are most familiar in mechanics.
Another is the eiconal equation, a nonlinear first order partial differential equation
for the eiconal function S(x)\
L=
(VS)2 = n2(x) .
(5.2)
112
Joseph B. Keller
The surfaces S(x) = constant are orthogonal to a normal congruence of rays.
These surfaces are called wavefronts. The difference between the values of S on
two wavefronts is equal to the optical length L of any ray of the congruence between
them.
Geometrical optics does not suffer from the limitation to special geometries
that restricts the method of separation of variables. However, it has its own limitation, namely that it describes only the rays along which light travels, but it gives
no quantitative information about the amplitude or phase of the light field.
6.
Modern Developments
The modern theory of partial differential equations began to develop in the last
half of the nineteenth century, and it has flourished in the twentieth century. It has
focused on the questions of existence and uniqueness of solutions of boundary value
problems, of initial value problems, of initial and boundary value problems, etc. In
addition it has considered whether the solutions depend continuously on the data
of the problem. A further concern has been regularity of solutions, i.e. how many
continuous derivatives they have and the Holder continuity of those derivatives.
It also deals with the singularities of solutions, the locus of such singularities, the
Hausdorff dimension of the set of singularities, etc. For linear partial differential
equations, this theory is presented in an elegant and general form in the treatise
of Hörmander (1983).
In addition to these mainly qualitative developments, there have been two
quantitative developments. One is the use of computers and the related numerical
analysis. The other is the development of asymptotic analysis, which is the topic
of the remainder of this survey.
An asymptotic expansion of a wavelike solution of the Helmholtz equation
(2.2) is an expression of the form
oo
^fcJ^e^W^-i-A^x).
(6.1)
j=o ^ >
This expression is not assumed to be a convergent series, but rather an asymptotic
expansion of v, valid as k tends to infinity. This means that as k —» oo,
j
v(x,k) - eikSW Y^W-jAj(x)
= o (k~J)
.
(6.2)
j=o
In other words, the difference between v and any partial sum vanishes more rapidly
than the last term retained, as k —> oo. Therefore such an expansion can be
useful in representing v when the wavenumber k is large, or equivalently when the
wavelength 2ir/k is small.
Carlini (1815) introduced an expression somewhat like (6.1) to solve Bessel's
equation, which is an ordinary differential equation of second order. Jacobi (1817)
explained and elaborated upon this work. Then Liouville and Green in the 1830s
used expansions of the form (6.1) to solve general linear second order ordinary
Wave Propagation
113
differential equations. However, despite repeated attempts, no one succeeded in
showing that these series converged. Then Poincaré (1886) suggested that they
were not convergent but asymptotic, and he introduced the definition (6.2). Steltjes
(1886) introduced the same concept to deal with certain formal series that arose in
the evaluation of integrals. Finally, in 1893 Korn showed, for certain linear second
order ordinary differential equations, that the formal scries were indeed asymptotic
to solutions.
Shortly after the introduction of quantum mechanics in 1925, this method
of solution was rediscovered by Wentzel, by Kramers, and by Brillouin. Therefore
physicists call it the WKB method, and sometimes the WKBJ method because
Jeffreys had used it in the early 1920s.
Other asymptotic expansions, different from (6.1), have also been used to
„represent solutions. „Often _a_number_ofjdifTerent_expansions_are_used_to_represent_
a single solution, each valid in a different region of space. These expansions must
match together where these different regions overlap. The use of such combinations
of expansions is called the method of matched asymptotic expansions. It is widely
used in wave propagation, and it is a basic tool of modern applied mathematics.
7.
Asymptotic Expansions and Geometrical Optics
In 1916 Sommerfeld and Runge attempted to show the connection between geometrical optics and the wave equation. They began with the Helmholtz equation
in an inhomogeneous medium with the refractive index n(x):
Av + k2n2(x)v = Q .
(7.1)
They represented v by the first term of the expansion (6.1), substituted it into
(7.1), and equated to zero the coefficient of k2, which was the highest power of fc.
This yielded the eiconal equation for the exponent, or phase function, 5(x):
(VS)2=n2(x) .
(7.2)
This is just the equation (5.2) obtained by Hamilton for the eiconal function of
geometrical optics, so a connection between the wave equation and geometrical
optics was established.
By equating to zero the coefficient offc,they obtained an equation for AQ(X):
2V5 • VA 0 + A0AS = 0 .
(7.3)
Luneberg (1944) and Friedlander (1947) used the full expansion (6.1) and obtained
equations for the other An:
2V5 • VAn + AnAS
= -AAn_i
,
n> 1.
(7.4)
The equation for AQ is called the (first) transport equation, and the equations for
the other An are called the higher transport equations.
The eiconal equation can be solved by the method of characteristics, which
Hamilton developed for this purpose. The characteristics are just the rays, and the
114
Joseph B. Keller
solution for S at the point x on the ray x(r) is
S(x) = 5(x 0 ) + fT n [x(r')] dr1 .
(7.5)
I TQ
Here xo = X(TO) is some point on the ray and 5(x 0 ) is the value of S there. The
integral in (7.5) is just the optical length (5.1), which occurs in Fermât 's principle.
Each transport equation can be written as a first order ordinary differential
equation along a ray, and then it can be solved explicitly. The solution for AQ can
be expressed in the form
n(x)Al(x)do(x)
= n(x0)Al(x0)da(x0)
.
(7.6)
Here da(x) denotes the normal cross-sectional area at x of a narrow tube of rays,
and dcr(xo) denotes the corresponding area at xo- Thus, the relation (7.6) expresses
conservation of energy in a tube of rays: the flux of energy nA2da at x is the same
as that at XQ. From (7.6) we can solve for A(x)\
AQ(x)=AQ(x0)
n(xQ) do-(x0)
n (x) da(x)
1/2
(7.7)
Upon using (7.5) for S(x) and (7.7) for AQ(x) in the first term of (6.1), we
obtain
n(x 0 ) do-(x 0 )] 1 / 2
J
v(x,k) ~ e
-o
A 0 (x 0 )
L n[x) da(x)
This expression for v involves only quantities determined by geometrical optics,
in addition to k and the initial value AQ(XQ). Therefore we call it the geometrical
optics field. It is the field associated with a particular ray through x.
As Luneberg (1944) observed, the total field at x is the sum of the fields on all
the rays through x. They are the direct rays from the source to x, the reflected rays,
the refracted rays, and the multiply reflected and/or multiply refracted rays, if
there are any. Therefore the solution v(x, k) is represented by the sum of asymptotic
expansions of the form (6.1), with one such expansion associated with each ray
through x. We call each of these expansions a wave. Then we can say that v is
represented as a sum of direct, reflected, refracted, and multiply reflected and/or
refracted waves.
Upon examining asymptotic expansions of various exact solutions of scattering problems, we find that they do indeed contain all these waves. However, they
usually contain additional waves, called diffracted waves, which are not associated
with the rays of geometrical optics. We shall explain how the preceding theory
can be extended to include those waves. This requires an extension of geometrical
optics, which we shall present first.
8.
The Geometrical Theory of Diffraction
Geometrical optics does not specify what happens when a ray hits an edge or
vertex on a boundary or interface. This is so because there is no unique normal to
Wave Propagation
115
the boundary or interface at such places, so the laws of reflection and refraction
do not apply. Therefore we have introduced the following two new laws to describe
what happens in such cases, and a third law to describe what happens when a ray
is tangent to a boundary or interface (Keller 1953).
Law of Edge Diffraction. A ray that hits the edge of a boundary or interface
produces a one parameter family of ra}'S that we call edge diffracted rays. The
angle ß between an edge diffracted ray and the tangent to the edge at the point of
diffraction is equal to the angle a between the incident ray and the edge, if they
lie in the same medium. If they lie in different media then m cos a = n2 cos ß. The
edge diffracted rays and the incident ray lie on opposite sides of the plane normal
to the edge.
Law of Vertex Diffraction. A ray that hits a vertex of a boundary or interface
-produces a ^wo_paraineter_farnily_of vertex_diffracted_rays._They_leave jthe_vertex_
in all directions.
Law of Surface Diffraction. A ray that hits a boundary or interface tangentially
produces a surface diffracted ray that is a geodesic on the surface in the metric
n(x)ds. It sheds rays along its tangent at each point. At each point on an interface
it also sheds rays into the second medium in accordance with Snell's law.
In addition to these diffracted rays, we have introduced complex rays, which
are complex-valued solutions of the ray equations. They can be defined when n(x)
is analytic or piecewise analytic.
The principle of "least" time can be extended to yield all these new rays. To
extend it we introduce various classes of curves Cj, j = 1,2,.... Then we define
the rays from P to Q in Cj to be the curves that make the optical length L from
P to Q stationary among all curves in Cj. If C\ is the class of curves with no
points on boundaries or interfaces, it yields the direct rays from P to Q. If C2
consists of curves with one point on a boundary or interface it yields the reflected
and refracted rays from P to Q. The class C3 of curves with one point on an
edge yields the edge diffracted rays and the law of edge diffraction. The class C4
of curves with a point at a vertex yields the vertex diffracted rays. The class of
curves C5 with an arc on a boundary or interface yields surface diffracted rays.
Multiply reflected, refracted, and diffracted rays can be obtained in a similar way.
The study and use of these new rays, the diffracted rays and the complex rays,
is called the geometrical theory of diffraction. There are normal congruences of
such rays, and the wavefronts orthogonal to them are called diffracted wavefronts.
There is an eiconal or phase function S(x) associated with each family of diffracted
wavefronts, and it satisfies the eiconal equation (7.2). The initial value problem
for the wavefronts associated with edge diffracted rays involves specification of the
value of S(x) on the edge. Similarly, for the phase function associated with vertex
and surface diffracted rays, the initial values must be specified at the vertex or on
the surface.
We have now presented three equivalent ways of determining the diffracted
rays and the corresponding diffracted wavefronts. The first way is to use the laws
governing the rays. The second way is to use the extended form of Fermât's principle. The third way is to solve the eiconal equation, including the multiple branches
of the solution that branch at the edges, vertices, and surfaces. From normal con-
116
Joseph B. Keller
gruences of rays we can construct the wavefronts and the solution of the eiconal
equation, and conversely from the wavefronts we can determine the rays. All three
of these methods yield the usual rays and wavefronts of geometrical optics as well
as the diffracted rays and diffracted wavefronts. Thus, all three formulations of the
geometrical theory of diffraction are extensions of the corresponding formulations
of geometrical optics.
9.
Asymptotic Expansions of Diffracted Waves
Next we shall indicate how to use this geometrical theory to obtain asymptotic
solutions of boundary value problems for (7.1). Just as in the case of geometrical optics, we construct asymptotic expansions of the form (6.1) by the method
described in Section 7. We construct one such expansion for each wave in the problem, i.e. for each normal congruence of rays or equivalently for each branch of the
solution of the eiconal equation, or for each family of wavefronts. As before, we
represent the solution v(x,k) as a sum of asymptotic expansions vp(x,k) of the
form (6.1), with one expansion associated with each ray through x. Thus, we write
v(x, k)~Y,**&k) ~ EeikSp{x) E jàj
p
p
j=Q
*
A
^ )
•
(91)
'
Now the sum includes expansions corresponding to all the diffracted rays or waves
through x, as well as to all the usual rays of geometrical optics.
The initial values SI(XQ) and Aji(x§) on incident rays are specified, if the rays
come from infinity. If the rays come from a source, the initial values are determined
by the solution of a local problem containing the source. The initial values on a
reflected or transmitted ray are obtained from the values of S(x) and Aj(x) on the
incident ray that produces it, by means of a reflection or transmission coefficient.
This coefficient is also determined by the solution of a local problem. Similarly,
the initial values on a diffracted ray are determined from the values of S(x) and
Aj(x) on the corresponding incident ray by means of a diffraction coefficient.
There are edge diffraction coefficients, vertex diffraction coefficients, and surface diffraction coefficients. Each of them is determined by the solution of a suitable
problem, called a canonical problem, which incorporates the local geometry of the
boundaries or interfaces near the point of diffraction, and the corresponding local
properties of n(x). These solutions often can be found by the classical methods of
separation of variables, etc., described in the previous sections. The use of local solutions is an instance of the method of matched asymptotic expansions, which was
mentioned earlier. It describes the solution near the point of diffraction, where the
ray expansion (9.1) becomes singular because of the vanishing of the cross-sectional
area da(x) in the denominator of the expression (7.7) for AQ(X). By matching the
local solution to the ray expansion, the diffraction coefficients can be determined.
This procedure shows that any edge diffraction coefficient D is proportional
to A; -1 / 2 , whereas any vertex diffraction coefficient C is proportional to k~x. These
same conclusions also follow from dimensional analysis. They show that the field
on both edge diffracted and vertex diffracted rays vanishes as k —• oo. That on
Wave Propagation
117
a vertex diffracted ray vanishes more rapidly; in fact it vanishes as fast as the
field on a ray doubly diffracted by edges. Furthermore, the value of v on a surface
diffracted ray decays exponentially with distance along the surface, with a decay
rate proportional to k1/3. The field on a complex ray has a complex phase S(x),
so it decaj's at a rate kImS(x) proportional to k. For simplicity, the dependence
upon k described in this paragraph is not shown in (9.1).
These results show that all diffracted fields, i.e. the values of v associated with
diffracted waves, vanish as k —> oo. The values of v on the ordinary geometrical
optics rays do not vanish as k —> oo, so they are stronger than the diffracted fields
for large values of k. However, there are no geometrical optics rays in shadows, so
the diffracted fields are the only fields present there.
The method described in Sections 6-9 can be used to solve both scattering
. and„eigenvalue„problems. Jt„can also be used to solve other partial differential,
equations. For scattering problems, the method is relatively straightforward, although there are of course, many complications. But for eigenvalue problems some
further considerations are needed, as we shall now explain.
10.
Eigenvalue Problems
In an eigenvalue problem in a domain D, there are no incident rays, and the value of
k is unknown, so it is not clear how to start using the preceding method. Therefore,
we must determine all the rays or waves in the solution simultaneously. Thus, in
the domain D we seek an asymptotic expansion of the form (9.1) consisting of a
sequence of waves p = 1,... , N. The pth wave is defined in a domain Dp, which
may be all of D or a sub domain of D. On each part of the boundary of Dp, either
the pth wave is produced by some other wave or it produces some other wave.
Therefore, its phase Sp(x) must equal that of the other wave on the corresponding
part of the boundary. Within Dp, Sp(x) must satisfy the eiconal equation (7.2).
It is convenient to introduce an TV-sheeted space E consisting of the Af domains Dp. The domains Dp and Dpt are joined together at those parts of their
boundaries where one of these waves produces the other. On this space E, the
function VS(x) is single valued, with the value VSp(x) on the sheet Dp. When we
integrate VS(x) to obtain S(x), we obtain a multiple-valued function. The corresponding amplitude function AQ(X), given by (7.7), may also be multiple valued.
But the solution v(x,k) must be single valued. This gives rise to the condition
kAS ~ iA log AQ = 27T77, + o(l).
(10.1)
Here AS(x) denotes the difference between two values of S(x) at a point x on the
multi-sheeted space E, and similarly for AAQ(X), while o(l) denotes terms which
tend to zero as k —> oo, and n is an integer.
We can write AS(x) as a line integral along a closed curve C on E in the
form
AS(x) = I WS-d£ .
(10.2)
c
Every C on E is a linear combination with integer coefficients, of basis curves
Ci, C2,..., CB of the fundamental group E. Here B, the first Betti number of E,
118
Joseph B. Keller
is the order of the fundamental group. Therefore (10.1) will hold for every C if the
following B conditions hold with integers nj :
ké VS-di-i(AlogAQ)Cj=2Tvnj
,
j = 1,2,... ,B .
(10.3)
The function log AQ(X) changes its value by —itr/2 as a wave passes through
a caustic surface, i.e. an envelope of rays, as can be shown by a local analysis.
Therefore, we denote by Vj the number of times the curve Cj crosses a caustic.
Then — z(Alog AQ)CJ = ~\vj and (10.3) becomes
k&VS-d£
= 2iv{nj + ^A
,
j = 1,2,... ,B .
(10.4)
Cj
This result applies when v satisfies the boundary condition 3nv = 0 on 3D. For
other boundary conditions, (AlogAQ)cj
contains an additional term associated
with each time Cj crosses 3D.
To use (10.4) we must find TV solutions Sj of the eiconal equation that join
together pairwise along portions of the boundary of D, or along caustic curves or
caustic surfaces inside D. These solutions must depend upon B — 1 parameters.
Then k and these B — 1 parameters must be determined to satisfy (10.4) for each
choice of the nonnegative integers rij, j = 1,... ,B. Finding the TV solutions Sj is
equivalent to finding TV families of wavefronts, each of which reflects into another
family at 3D, or that join pairwise at caustics. Alternatively we could find TV
normal congruences of rays which reflect into one another at 3D or which join one
another at caustics.
In quantum mechanics, conditions like (10.4) are called quantum conditions.
Bohr (1913) presented the first such condition for the motion of an electron in a
hydrogen atom, with B = 1 and vi = 0. Sommerfeld (1916) and Wilson (1916)
presented B such conditions for separable systems with B degrees of freedom, and
Einstein (1917) gave these conditions for nonseparable systems, all with Vj = 0 .
After the Schrödinger equation was formulated, the Sommerfeld-Wilson conditions
for separable systems were rederived from it by separation of variables. In some
cases in which the separated equations had turning points, this derivation yielded
corrected Sommerfeld-Wilson conditions with Vj =^0. The present author's derivation, described above, which does not require separability, led to the widespread
use of this method by chemists and physicists. They call it the EBK method, for
Einstein, Brillouin (who used the single-valuedness argument), and the author.
The conditions (10.4) were later derived independently by Maslov (1965).
Then Arnold (1970) showed that the Vj are invariants (i.e. independent of the
curves Cj). The integer Vj is sometimes called the Keller-Maslov index.
11.
Conclusion
Many authors have contributed to the further development of the asymptotic
methods described above. Some of them are R. K. Luneberg, F. G. Friedlander,
Wave Propagation
119
M. Kline, I. Kay, R. M. Lewis, R. Buchal, B. Levy, B. Seckler, N. Bleistein, J. Cohen, R. Handelsman, B. Matkowsky, D. Ahluwalia, F. Karal, L. Felsen, D. Ludwig,
L. Hörmander, C. Morawetz, C. Bloom, W. Miranker, N. Kazarinoff, P.D. Lax,
R.M. Phillips, W. Straus, J. Ralston, A. Majda, M. Taylor, F . Ursell, Y. Kravtsov,
L. Babich, P. Ufimtsev, R. Melrose, M. Zworski, J. Rauch, V. Lazutkin, V. Maslov,
and M. Berry. T h e y have proved the validity of these m e t h o d s in many cases,
extended t h e m to other equations and systems of equations, applied t h e m to special problems, developed computer programs to determine t h e various kinds of
diffracted rays and to calculate the corresponding phases and amplitudes, etc. In
addition, some of these methods have been extended to weakly nonlinear waves.
Some of the early work on these methods is described in the papers listed
below, which also contain references to many other works.
References
[1] J. B. Keller, A geometrical theory of diffraction, calculus of variations and its applications, Proc. Sympos. Appi. Math., 8 (1958), 27-52; Math. Revs., 20 (1959),
103.
[2] J. B. Keller, Corrected Bohr-Sommerfeld quantum conditions for nonseparable systems, Annals Physics, 4 (1958), 180-188; Math. Revs., 20 (1959), 934.
[3] J. B. Keller, Rays, waves and asymptotics, Bull. Amer. Math. Soc. 84 (1978), 727750.
[4] J. B. Keller, One hundred years of diffraction theory, IEEE Trans. Antennas and
Propagation, AP-33 (1985), 200-214.
[5] J. B. Keller, Semiclassical mechanics, SIAM Rev., 27 (1985), 485-504.
[6] J. B. Keller and S. I. Rubinow, Asymptotic solution of eigenvalue problems, Annals
Physics, 9 (1960), 24-75.