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ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS
Editorial Board
P. Flajolet, M.E.H. Ismail, E. Lutwak
Volume 107
Multiple Scattering
The interaction of waves with obstacles is an everyday phenomenon in science and engineering,
cropping up for example in acoustics, electromagnetism, seismology and hydrodynamics.
The mathematical theory and technology needed to understand the phenomenon is known as
multiple scattering, and this book is the first devoted to the subject. The author covers a variety
of techniques, for example separation of variables,T-matrix and integral equation methods,
describing first the single-obstacle method and then extending it to the multiple-obstacle case.
A key ingredient in many of these extensions is an appropriate addition theorem: a coherent,
thorough exposition of these theorems is given, and computational and numerical issues around
them are explored.
The application of these methods to different types of problems is also explained. In
particular, sound waves, electromagnetic radiation, waves in solids and water waves. A com-
prehensive reference list of some 1400 items rounds off the book, which will be an essential
reference on the topic for applied mathematicians, physicists and engineers.

ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS
All the titles listed below can be obtained from good booksellers or from Cambridge Univer-
sity Press. For a complete series listing visit
http://publishing.cambridge.org/stm/mathematics/eom/
88. Teo MoraSolving Polynomial Equation Systems I
89. Klaus BichtelerStochastic Integration with Jumps
90. M. LothaireAlgebraic Combinatorics on Words
91. A.A. Ivanov & S.V. ShpectorovGeometry of Sporadic Groups, 2
92. Peter McMullen & Egon SchulteAbstract Regular Polytopes
93. G. Gierz et al.Continuous Lattices and Domains
94. Steven R. FinchMathematical Constants
95. Youssef JabriThe Mountain Pass Theorem
96. George Gasper & Mizan RahmanBasic Hypergeometric Series 2nd ed.
97. Maria Cristina Pedicchio & Walter TholenCategorical Foundations
98. Mourad IsmailClassical and Quantum Orthogonal Polynomials in One Variable
99. Teo MoraSolving Polynomial Equation Systems II
100. Enzo Olivieri & Maria Eulalia VaresLarge Deviations and Metastability
101. A. Kushner, V. Lychagin & V. RoubtsovContact Geometry and Nonlinear Differential
Equations
102. R.J. Wilson & L. BeinekeTopics in Algebraic Graph Theory
103. Olof Johan StaffansWell-Posed Linear Systems
104. John Lewis, S. Lakshmivaraham, Sudarshan DhallDynamic Data Assimilation
105. M. LothaireApplied Combinatorics on Words

Multiple Scattering
Interaction of Time-Harmonic Waves withNObstacles
P.A. MARTIN
Colorado School of Mines

cambridge university press
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge CB2 2RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521865548
© Cambridge University Press 2006
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without
the written permission of Cambridge University Press.
First published 2006
Printed in the United Kingdom at the University Press, Cambridge
A catalogue record for this publication is available from the British Library
ISBN-13 978-0-521-86554-8 hardback
ISBN-10 0-521-86554-9 hardback
Cambridge University Press has no responsibility for the persistence or accuracy of URLs for
external or third-party internet websites referred to in this publication, and does not guarantee that
any content on such websites is, or will remain, accurate or appropriate.

To Ruth, Richard, Frances and, last but not least, Ann

Contents
Preface pagexi
1 Introduction 1
1.1 What is ‘multiple scattering’? 1
1.2 Narrowing the scope: previous reviews and omissions 10
1.3 Acoustic scattering byNobstacles 11
1.4 Multiple scattering of electromagnetic waves 16
1.5 Multiple scattering of elastic waves 19
1.6 Multiple scattering of water waves 23
1.7 Overview of the book 27
2 Addition theorems in two dimensions 29
2.1 Introduction 29
2.2 Cartesian coordinates 30
2.3 Hobson’s theorem 31
2.4 Wavefunctions 34
2.5 Addition theorems 36
2.6 The separation matricesSand

S 40
2.7 Use of rotation matrices 43
2.8 Two-centre expansions 44
2.9 Elliptical wavefunctions 46
2.10 Vector cylindrical wavefunctions 53
2.11 Multipoles for water waves 54
3 Addition theorems in three dimensions 62
3.1 Introduction 62
3.2 Spherical harmonics 62
3.3 Legendre’s addition theorem 65
3.4 Cartesian coordinates 66
vii

viii Contents
3.5 Hobson’s theorem 67
3.6 Wavefunctions and the operator
m
n
69
3.7 First derivatives of spherical wavefunctions 73
3.8 Axisymmetric addition theorems 75
3.9 A useful lemma 81
3.10 Composition formula for the operator
m
n
83
3.11 Addition theorem forj
nY
m
n
87
3.12 Addition theorem forh
1
n
Y
m
n
89
3.13 The separation matricesSand

S 91
3.14 Two-centre expansions 95
3.15 Use of rotation matrices 98
3.16 Alternative expressions forSbˆz 99
3.17 Vector spherical wavefunctions 106
3.18 Multipoles for water waves 107
4 Methods based on separation of variables 122
4.1 Introduction 122
4.2 Separation of variables for one circular cylinder 122
4.3 Notation 125
4.4 Multipole method for two circular cylinders 126
4.5 Multipole method forNcircular cylinders 128
4.6 Separation of variables for one sphere 132
4.7 Multipole method for two spheres 135
4.8 Multipole method forNspheres 138
4.9 Electromagnetic waves 140
4.10 Elastic waves 141
4.11 Water waves 143
4.12 Separation of variables in other coordinate systems148
5 Integral equation methods, I: basic theory and applications 152
5.1 Introduction 152
5.2 Wave sources 153
5.3 Layer potentials 154
5.4 Explicit formulae in two dimensions 159
5.5 Explicit formulae in three dimensions 162
5.6 Green’s theorem 164
5.7 Scattering and radiation problems 166
5.8 Integral equations: general remarks 168
5.9 Integral equations: indirect method 169
5.10 Integral equations: direct method 172
6 Integral equation methods, II: further results and applications179
6.1 Introduction 179
6.2 Transmission problems 179

Contents ix
6.3 Inhomogeneous obstacles 180
6.4 Electromagnetic waves 187
6.5 Elastic waves 196
6.6 Water waves 203
6.7 Cracks and other thin scatterers 206
6.8 Modified integral equations: general remarks 209
6.9 Modified fundamental solutions 209
6.10 Combination methods 217
6.11 Augmentation methods 219
6.12 Application of exact Green’s functions 222
6.13 Twersky’s method 229
6.14 Fast multipole methods 235
7 Null-field andT-matrix methods 242
7.1 Introduction 242
7.2 Radiation problems 242
7.3 Kupradze’s method and related methods 243
7.4 Scattering problems 246
7.5 Null-field equations for radiation problems: one obstacle 247
7.6 Null-field equations for scattering problems: one obstacle 249
7.7 Infinite sets of functions 251
7.8 Solution of the null-field equations 257
7.9 TheT-matrix for one obstacle 268
7.10 TheT-matrix for two obstacles 274
7.11 TheT-matrix forNobstacles 281
8 Approximations 284
8.1 Introduction 284
8.2 Small scatterers 284
8.3 Foldy’s method 297
8.4 Point scatterers 304
8.5 Wide-spacing approximations 306
8.6 Random arrangements of small scatterers; suspensions 312
Appendices 323
A Legendre functions 323
B Integrating a product of three spherical harmonics;
Gaunt coefficients 325
C Rotation matrices 329
D One-dimensional finite-part integrals 333
E Proof of Theorem 5.4 339
F Two-dimensional finite-part integrals 343
G Maue’s formula 350

x Contents
H Volume potentials 351
I Boundary integral equations forG
E
355
References 357
Citation index 421
Subject index 433

Preface
It has been said that ‘a true scholar publishes only one book in his lifetime and that
posthumously’ [105, p. 435]. In fact, this book has not taken a lifetime to complete,
although it has had a very long gestation period: there have been many distractions,
most of which have not been unpleasant.
The book is concerned with the scattering of time-harmonic waves by obsta-
cles; the words ‘multiple scattering’ in the title signify that there are at least two
obstacles. The problems considered come from acoustics (sound waves, Helmholtz
equation), electromagnetics (Maxwell’s equations), elastodynamics (waves in solids)
and hydrodynamics (surface water waves). The book describes a variety of mathe-
matical techniques for solving such problems: the main techniques involve separation
of variables, integral equations andT-matrices. Most of the book is concerned with
exact methods, although the last chapter discusses several effective approximate
methods. There are also two chapters on addition theorems; these are useful in other
contexts as well as for multiple scattering. For detailed information on the topics
covered, see the Table of Contents and Section 1.7; for a list of topicsnotcovered,
see Section 1.2.
The mathematics used is classical: separation of variables, special functions,
Green’s functions, Fourier methods, asymptotics. The reader should also have some
familiarity with simple uses of boundary integral equations. Operator notation is used
when convenient.
As far as I know, there is no other book that treats all four of the main phys-
ical domains: acoustics, electromagnetics, elastodynamics and hydrodynamics. The
reader will see many connections between these domains. As far as I know, there
is no other book that focuses on multiple scattering. (Of course, it is inevitable that
we begin by considering scattering by one obstacle.) In addition, the book has an
extensive reference list. This should be useful to future workers, and may help reduce
duplication of effort.
xi

xii Preface
The book contains a sprinkling of quotations from the literature. These are intended
to be illuminating, amusing or both. Some may help the reader to not lose heart when
the calculations become complicated; some may hint that great men are human; and
some may simply reflect the author’s English sense of humour!
I cannot give thanks individually to everybody who has helped me over the
years, perhaps by sending me reprints or answering my questions. However, I must
thank four people for shaping my outlook and taste in applied mathematics: Ralph
Kleinman, Frank Rizzo, Fritz Ursell and Gerry Wickham. I also thank Chris Linton
for his detailed comments on an early draft. No doubt some errors remain, for which
I must take sole responsibilty. Please let me know if you find errors. In particular, if
you think that I should have cited one of your papers, send me a copy!

1
Introduction
Privately,[Rayleigh]often quoted with relish a saying attributed to Dalton when
in the chair at a scientific meeting: “Well, this is a very interesting paper for
those that take any interest in it”.
(Strutt [1157, p. 320])
1.1 What is ‘multiple scattering’?
The mathematics of the full treatment may be altogether beyond human power
in a reasonable time; nevertheless
(Heaviside [489, p. 324])
‘Multiple scattering’ means different things to different scientists, but a general
definition might be ‘the interaction of fields with two or more obstacles’. For example,
a typical multiple-scattering problem in classical physics is the scattering of sound
waves by two rigid spheres. Further examples, such as the scattering of spherical
electron waves by a cluster of atoms, can be found in condensed-matter physics [1379,
168, 422, 424, 423]. Many other examples will be discussed in this book.
The waves scattered by a single obstacle can be calculated in various well-known
ways, such as by the method of separation of variables,T-matrix methods or integral-
equation methods. All of these methods will be discussed in detail later.
If there are several obstacles, the field scattered from one obstacle will induce
further scattered fields from all the other obstacles, which will induce further scattered
fields from all the other obstacles, and so on. This recursive way of thinking about
how to calculate the total field leads to another notion of multiple scattering; it can
be used to actually compute the total scattered field – each step is called anorder of
scattering. In 1893, Heaviside [489, p. 323] gave a clear qualitative description of
this ‘orders-of-scattering’ process.
1.1.1 Single scattering and independent scattering
In his well-known book on electromagnetic scattering, van de Hulst [1233, §1.2]
considers two classifications, namelysingle scatteringandindependent scattering.
Let us review his definitions of these ideas.
1

2 Introduction
1.1.1.1 Single scattering
This is the simplest approximation, in which the effects of multiple scattering are
ignored completely: ‘the total scattered field is just the sum of the fields scattered
by the individual [obstacles], each of which is acted on by the [incident] field in
isolation from the other [obstacles]’ [111, p. 9]. This approximation is used widely; it
is only expected to be valid when the spacing is large compared with both the size of
the obstacles and the length of the incident waves. Indeed, with these assumptions,
higher-order approximations can be derived [1382, 1383, 1381] and these can be
effective [511]. However, there are many instances where multiple scattering is
important; for some natural examples, see Bohren’s fascinating book [109] and his
related paper [110]. Thus, in atmospheric physics, the single-scattering approximation
is not justified, ‘for example, by clouds, wheremultiple scatteringcan be appreciable’
[111, p. 9].
1.1.1.2 Independent scattering
When waves interact with several obstacles, a ‘cooperative effect’ may occur. This
could be constructive interference, leading to unexpectedly large fields, such as
can happen with a periodic arrangement of identical scatterers as in a diffraction
grating or a crystal lattice. Alternatively, there could be destructive interference,
leading to unexpectedly small fields, such as can happen with a random arrangement
of scatterers. These are examples ofdependent scattering: in theory, one ‘has to
investigate in detail the phase relations between the waves scattered by neighboring
[scatterers]’ [1233, p. 4]. Thus, the ‘assumption of independent scattering implies
that there is no systematic relation between these phases’ [1233, p. 5].
The notions of single scattering and independent scattering need not be separated.
For example, the authors of [866] consider
only independent scattering, randomly positioned particles. This means that par-
ticles are separated widely enough, so that each particle scatters light in exactly
the same way as if all other particles did not exist. Furthermore, there are no
systematic phase relations between partial electromagnetic waves scattered by
different particles, so that the intensitiesof the partial waves can be added
without regard to phase. In other words, we will assume that each particle is in
the far-field zone of all other particles, and that scattering by different particles
is incoherent.
(Mishchenko et al. [866, p. 4])
The authors go on to quantify what ‘separated widely enough’ means: ‘Exact scat-
tering calculations for randomly oriented two-sphere clusters composed of identical
wavelength-sized spheres suggest that particles can scatter independently when the
distance between their centers is as small as four times their radius [868]’ [866, p. 5].
This is consistent with van de Hulst [1233, p. 5]: ‘Early estimates have shown that
a mutual distance of 3 times the radius is a sufficient condition for independence’.

1.1 What is ‘multiple scattering’? 3
1.1.2 Scattering byNobstacles
Suppose that we haveNdisjoint obstacles,B
i,i=12N. The boundary ofB
i
isS
i. A given wave is incident upon theNobstacles, and the problem is to calculate
the scattered waves.
We assume that we know everything about every obstacle: location, shape, orienta-
tion and boundary condition; if the obstacles are penetrable, so that waves can travel
through them, we assume that we know the internal composition. There are many
situations where all of this information is not available; for example, the obstacles
might be located randomly.
Mathematically, the exact (deterministic
formulated: it is an exterior boundary-value problem (with a radiation condition at
infinity) where the boundary is not simply-connected. However, the problem is not
easy to solve, due mainly to the complicated geometry: hence Heaviside’s pessimistic
comment. Another comment, in a similar vein, was made by van de Hulst [1233]:
Multiple scattering does not involve new physical problems,≡≡≡≡Yet the problem
of finding the intensities inside and outside the cloud[ofNscatterers]is an
extremely difficult mathematical problem.
(van de Hulst [1233, p. 6])
This attitude led naturally to single-scattering approximations, as mentioned above.
One scatters the incident wave from theith obstacle (ignoring the presence of the
other obstacles), and then sums overi. Indeed, van de Hulst’s book and [276] are
devoted entirely to single scattering.
At the other extreme, one may attempt to solve theN-body scattering problem
directly, perhaps by setting up a boundary integral equation over
S=
N
≡
j=1
S
j← (1.1)
see Chapter 5. Analytically, although ‘it would be esthetically preferable to treat
the [N -body] configuration as a unit, this approach seems limited to certain special
problems’ [1196, p. 42]. Computationally, this direct approach can be expensive,
especially for problems involving many three-dimensional obstacles.
In the first comprehensive review of the literature on multiple scattering, Twersky
opined that
it is convenient in considering multiple scattering, to assume that solutions for the
component scatterers when isolated are known, and that they may be regarded
as “parameters” in the more general problem.
Thus, one seeks representations for scattering by many objects in which the effects
of the component scatterers are “separated” from the effects of the particular con-
figuration (or statistical distribution of configurations) in the sense that the
forms of the results are to hold independently of the type of scatterers involved.
(Twersky [1198, p. 715])

4 Introduction
Similarly, in the context of hydrodynamics (where water waves interact with immersed
structures, such as neighbouring ships, wave-power devices or elements of a single
larger structure), Ohkusu wrote:
For the purpose of calculating hydrodynamic forcesit is essential that only
the hydrodynamic properties of each element be given. A method having such a
merit will facilitate the calculation for a body having many elements and may be
applied to the design arrangement of the elements.
(Ohkusu [927, p. 107])
In other words, assuming that we know everything about scattering by each obstacle
in isolation, how can we use this knowledge to solve the multi-obstacle problem?
The best way is to use a ‘self-consistent’ method. In the next section, we describe
such a method in general terms.
1.1.3 Self-consistent methods
A self-consistent method
assumes that a wave is emitted by each scatterer of an amount and directionality
determined by the radiation incident on that scatterer (the effective field). The
latter is to be determined by adding to the incident beam the waves emitted
by all other scatterers, and the waves emitted by those scatterers are in turn
influenced by the radiation emitted by the scatterer in question.≡≡≡The self-
consistent procedure is not an expansion in primary, secondary, tertiary waves,
etc. The field acting on a given scatterer, or emitted by it includes the effects of
all orders of scattering.
(Lax [687, pp. 297–298])
Specifically, write the total field as
u=u
inc+
N

j=1
u
j
sc
(1.2)
whereu
incis the given incident field andu
j
sc
is the field scattered (‘emitted’
jth scatterer. Define the ‘effective’ or ‘external’ or ‘exciting field’ by
u
n≡u−u
n
sc
=u
inc+
N

j=1
j=n
u
j
sc
← (1.3)
it is the ‘radiation incident on [thenth] scatterer’ in the presence of all the other
scatterers.
Now, as the problem is linear, it must be possible to write
u
j
sc
=≡
ju
j (1.4)

1.1 What is ‘multiple scattering’? 5
where≡
jis an operator relating the field incident on thejth scatterer,u
j, to the field
scattered by thejth scatterer,u
j
sc
. Hence, (1.3
u
n=u
inc+
N

j=1
j=n
≡
ju
jn=12N (1.5)
or, equivalently,
u
n
sc
=≡
n
←
u
inc+
N

j=1
j=n
u
j
sc

n=12N (1.6)
If one could solve (1.5u
nor (1.6u
n
sc
,n=12N, the total field would
then be given by
u=u
inc+
N

j=1
≡
ju
j (1.7)
or (1.2
The derivation of (1.5
Chapter 7, §2]. Its simplicity is somewhat illusory, because we have not clearly
defined the operator≡
j; also, we have not indicatedwhere(1.5
to hold in space. Nevertheless, we have given an abstract framework within which a
variety of concrete methods can be developed.
The general scheme leading to (1.5 Foldy–Lax self-
consistent method. Foldy [354] used a special case of the method for ‘isotropic point
scatterers’; see Section 8.3 for a detailed description. Lax [687] used the general
scheme, with a certain prescription for≡
j; see [687, §III]. We will see several
specific realisations later, including theT-matrix methods developed in Chapter 7.
For simple geometries, such as circular cylinders or spheres, a self-consistent
method is easily realised. One combines separated solutions of the Helmholtz equa-
tion (multipoles
tipoles centred at one origin in terms of similar multipoles centred on a different
origin. This old but useful method will be developed in detail in Chapter 4. The
method itself goes back to a paper of Lord Rayleigh, published in 1892; we discuss
this next.
1.1.4 Rayleigh’s paper of 1892
In his paper ‘On the influence of obstacles arranged in rectangular order upon
the properties of a medium’ [1009], Rayleigh considered potential flow through a
periodic rectangular array of identical circular cylinders. As a special case of his
analysis, let us consider an infinite square array of rigid cylinders of radiusawith
centres atx y=mb nb, where mandnare integers andb>2a; see Fig. 1.1.

6 Introduction
P
Q
x
y
Fig. 1.1. Rayleigh’s problem: an infinite square array of circles.
The ambient flow has potentialV=Hx. In Rayleigh’s words [1009, p. 482]: ‘If we
take the centre of one of the cylinders P as origin of polar coordinates, the potential
external to the cylinder may be expanded in the series
V=A
0+A
1r+B
1r
−1
cos+A
3r
3
+B
3r
−3
cos 3 +···’ (1.8
whereis measured from thex-axis. Symmetry implies thatV−A
0must be an odd
function ofxand an even function ofy; these conditions lead to the form of the
expansion (1.8 ≤V/≤r=0onr=agives
B
n=a
2n
A
nn=135 (1.9)
Next [1009, p. 483]: ‘The values of the coefficientsA
1,B
1,A
3,B
3≡≡≡are neces-
sarily the same for all the cylinders, and each may be regarded as a similar multiple
source of potential. The first termA
0, however, varies from cylinder to cylinder, as
we pass up or down the stream’.
At this stage, we have obtained one condition relatingA
nandB
n, namely (1.9
we need another. To find it, Rayleigh begins as follows [1009, p. 483]: ‘The potential
Vat any point may be regarded as due to external sources at infinity (by which the
flow is caused) and to multiple sources situated on the axes of the cylinders. The
first part may be denoted byHx’.
Then, Rayleigh proceeds [1009, p. 484] ‘by equating two forms of the expression
for the potential at a pointx ynear P. The part of the potential due toHxand to the
multiple sources Q (P not included) is
A
0+A
1rcos+A
3r
3
cos 3 +≡≡≡←

1.1 What is ‘multiple scattering’? 7
or, if we subtractHx, we may say that the potential atx ydue to multiple sources
at Q is the real part of’
A
0+A
1−H⎫z+A
3z
3
+A
5z
5
+···withz=x+iy≡ (1.10)
Continuing: ‘But ifx

y

are the coordinates of the same point when referred to the
centre of one of the Q’s, the same potential may be expressed by’
⎭B
1z
−1
+B
3z
−3
+··· withz

=x

+iy

(1.11)
‘the summation being extended over all the Q’s. If be the coordinates of a Q
referred to P,x

=x−,y

=y−; so that’B
nz
−n
=B
nz−z
0⎫
−n
withz
0=+i.
Then, the binomial theorem gives
z
−n
=−z
0⎫
−n
←
1+nz/z
0⎫+
1
2
nn+1z/z
0⎫
2
+···

≡ (1.12)
Hence [1009, p. 484]: ‘Since (1.10
x+iy[=z], we obtain, equating term to term,’
H−A
1=B
1
2+3B
3
4+5B
5
6+···
−3!A
3=3!B
1
4+
1
2
5!B
3
6+···
−5!A
5=5!B
1
6+
1
2
7!B
3
8+···
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭
(1.13)
‘and so on, where

2n= +i⎫
−2n
’ (1.14)
‘the summation extending over all the Q’s.’ (As
2s+1=0, we also obtainA
0=0.)
Thus, the system comprising (1.9
A
nandB
n. Note that, for two-dimensional potential flow (Laplace’s equation), the
addition theorem amounts to an application of the binomial theorem, (1.12
also that the situation becomes more complicated when the periodicity is destroyed,
because then the coefficentsA
nandB
nwill vary from cylinder to cylinder.
Rayleigh [1009] also considered flow past a rectangular three-dimensional array
of identical spheres, and (briefly
arrays of rigid cylinders or spheres. For further comments, see [1205, §(6
§2] and [888, §3.1].
1.1.5 Kasterin, KKR and the electronic structure of solids
Shortly after Rayleigh’s paper[1009]was published, a graduate student at
Moscow University, N.P. Kasterin, set out to apply[Rayleigh’s]ideas to a gen-
uine scattering problem. He chose the relatively simple phenomenon of reflec-
tion and refraction of low-frequency sound by an orthorhombic grid of hard

8 Introduction
spheres.≡≡≡Kasterin’s results were published in his 1903 Moscow thesis. A pre-
liminary report≡≡≡came out in 1898.
(Korringa [648, p. 346])
As a special case of Kasterin’s analysis, consider an infinite planar square array of
spheres of radiusawith centres atx y z=mb nb0⎫, wheremandnare integers
andb>2a. We take the incident field as a plane wave at normal incidence to the
array,u
inc=e
ikz
.
Generalising (1.8 rnear thejth sphere can be expanded as
ur⎫=

nm

d
m
njˆ≥
m
n
r
j⎫+c
m
nj
≥
m
n
r
j⎫

(1.15)
wherer=r
j+b
jandr=b
jis the sphere’s centre. Here,≥
m
n
r
j⎫are outgoing multi-
poles (separated solutions of the Helmholtz equation in spherical polar coordinates),
singular atr
j=0(r=b
j) andˆ≥
m
n
r
j⎫are regular spherical solutions. (Precise defini-
tions will be given later.) The coefficientsd
m
nj
andc
m
nj
correspond to Rayleigh’sA
n
andB
n, respectively. Evidently, the periodic geometry and the simple incident field
imply that
d
m
nj
≡d
m
n
andc
m
nj
≡c
m
n

the coefficients are the same for every sphere.
Applying the boundary condition⎬u/⎬r=0onr=ayields one relation between
d
m
n
andc
m
n
, namely
d
m
n
=
nc
m
n
(1.16)
where
nis a known constant (see Section 4.6).
The effective field incident on thejth sphere is

nm
d
m
nˆ≥
m
n
r
j (1.17)
This must be the same as the sum of the actual incident field and the scattered fields
emitted by all the other spheres, namely
e
ikz
+

l
l=j

nm
c
m
n
≥
m
n
r
l (1.18)
Equating (1.17 jth sphere gives a second
relation betweend
m
n
andc
m
n
. This solves the problem, in principle.
To proceed further, suppose that we have the expansions
e
ikz
=

nm
e
m
nˆ≥
m
n
r
j⎫ (1.19)
and
≥
m
n
r
l⎫=

S
m
n
b
j−b
l⎫ˆ≥

r
j (1.20)

1.1 What is ‘multiple scattering’? 9
we will discusse
m
n
andS
m
n
b⎫shortly. Then, equating (1.17
(1.19

nc
m
n
−

c

l
l=j
S
m
n
b
j−b
l⎫=e
m
n
≡
In this linear system of algebraic equations, we can takej=0 without loss of
generality.
The Rayleigh–Kasterin method, described above, is rigorous, and it can be gener-
alised in various ways. It has been used to obtain numerical solutions for many related
problems with (infinite
tering by a single periodic row of circles, and see [984] for two-dimensional elastic
waves around a square array of circular cavities; see also [888, Chapter 3].
The Rayleigh–Kasterin method was also adapted to problems in solid-state physics.
In that context, it is known as theKKR(Korringa–Kohn–Rostoker)method; see, for
example, [1379, §10.3], [775, 424] or [423, §6.8]. For a clear presentation of the
two-dimensional KKR method (for sound waves around an infinite square array of
soft circles), see [90].
For historical background, including a detailed description of Kasterin’s work,
see [648].
To generalise Rayleigh’s method to anon-periodicconfiguration, consider the
problem of acoustic scattering by two spheres (see Section 1.3 for background
information). Suppose that the spheres are centred atO
1andO
2. Write the scattered
fieldu
scas a superposition of outgoing multipoles≥
m
n
, one set singular atO
1and the
other set singular atO
2:
u
sc=

nm
a
m
n
≥
m
n
r
1⎫+b
m
n
≥
m
n
r
2⎫≡
Then, determine the coefficientsa
m
n
andb
m
n
by applying the boundary condition on
each sphere in turn: this requires the expansion of≥
m
n
r
2⎫in terms of regular spherical
solutions centred onO
1,ˆ≥
m
n
r
1⎫. Thus, we need theaddition theorem
≥
m
n
r
2⎫=

S
m
n
b⎫ˆ≥

r
1
which is valid forr
1<b, wherer
1=⎫r
1⎫,r
2=r
1+bandb=⎫b⎫is the distance
betweenO
1andO
2. The matrixS=S
m
n
⎫is called theseparation matrixor the
translation matrixor thepropagator matrix. It is an important ingredient in several
exact theories of multiple scattering. We will give much attention to various methods
for calculatingS, with emphasis on acoustic problems (Helmholtz equation) in two
(Chapter 2) and three (Chapter 3) dimensions.
We also need expansions of the incident field, similar to (1.19
derived too.
Kasterin did not have explicit expressions for the matrixS: we can see that the
expansions (1.19 r
j=0, and

10 Introduction
so the coefficients could be obtained by applying appropriate differential operators.
This is one of several methods for constructingSthat we shall develop later.
1.2 Narrowing the scope: previous reviews and omissions
Multiple scattering is a huge subject with a huge literature. For an extensive review up
to 1964, see [1198] and the supplement [163]. For a collection of articles surveying
many aspects of scattering (including theory, computation and application), see the
957-page volume edited by Pike & Sabatier [977].
There is a 1981 survey by Oguchi on ‘multiple scattering of microwaves or
millimeter waves by an assembly of hydrometeors’ [924, p. 719]. Two approaches
are reviewed. One is theFoldy–Lax–Twersky integral equation method, introduced
by Foldy in 1945 [354] and generalised by Lax [687, 688] and Twersky [1200, 1201].
The second approach is based on theradiative transfer equation; this may be regarded
as the final stage in a larger calculation:
the treatment of light scattering by a cloud of randomly positioned, widely sepa-
rated particles can be partitioned into three steps:(i)computation of the far-field
scattering and absorption properties of an individual particle≡≡≡(ii)computa-
tion of the scattering and absorption properties of a small volume element
containing a tenuous particle collection by using the single-scattering approx-
imation; and(iii)computation of multiple scattering by the entire cloud by
solving the radiative transfer equation supplemented by appropriate boundary
conditions.
(Mishchenkoet al.[870, p. 7])
We do not consider radiative transfer further, but see [1190] for more information.
In 2000, Tourinet al.[1182] reviewed a variety of applications, including theory
and experiment: in one example of note, sound waves in water are scattered by a
random collection of 1000 identical parallel steel rods.
Major areasnotcovered in this book include the following.
(i
manner, such as in a row or in a regular lattice. For plane-wave scattering, problems
of this type can be reduced to a problem in a single ‘unit cell’ (for lattices) or to
waveguide problems (for a row of equally-spaced obstacles). The prototype for
this reduction, of course, is Rayleigh’s paper [1009], discussed in Section 1.1.4.
Larsen [684] gave an early review of scattering by periodic rows of identical
cylinders. For scattering by a semi-infinite periodic row of cylinders, see [859,
502, 501, 861, 729].
(ii
a governing partial differential equation (such as the Helmholtz equation) in
the regiony > fx, where y=fx,?<x<, is the rough surface with

1.3 Acoustic scattering byNobstacles 11
0≤fx≤h, say, andfandhare given. There is an extensive literature on
this topic; see, for example, the reviews [280, 1286, 1049, 279] or the books
by Ogilvy [923] and Voronovich [1268].
(iii
dimensions (such as spheres) or cylindrical scatterers with bounded cross-
sections in two dimensions. Thus, we do not consider problems such as diffrac-
tion by parallel semi-infinite rigid planes or by wedges and cones. For early
work on such problems with one scatterer, see [135].
(iv
discussed in Section 8.6, although we do not develop probabilistic techniques
in detail. In particular, we do not consider thelocalisationof waves by random
arrays of identical scatterers; see, for example, [508, 615, 228, 1132, 1100].
(v
sient problems may be reduced to time-harmonic problems using Fourier trans-
forms. Laplace transforms may be used for initial-value problems. There is
also an extensive (and classical) literature on the use of retarded potentials,
leading to integral equations of Volterra type. For more information, see the
books by Friedlander [365] and by Baker & Copson [53]. Numerical aspects
are discussed in [60, §10.5], [116, Chapter 7], and [367, 773, 1040].
(vi
they are ignored.
(vii
particular, we do not discuss finite-element methods (except that certain hybrid
methods are mentioned) or the use of Sobolev spaces. For some information in
this direction, see [526, 836, 903, 879].
1.3 Acoustic scattering byNobstacles
May not Music be described as the Mathematic of sense, Mathematic as Music of
the reason? the soul of each the same! Thus, the musician feels Mathematic, the
mathematician thinks Music, – Music the dream, Mathematic the working life –
each to receive its consummation from the other when the human intelligence,
elevated to its perfect type, shall shine forth glorified in some future Mozart–
Dirichlet or Beethoven–Gauss – a union already not indistinctly foreshadowed
in the genius and labours of Helmholtz!
(J.J. Sylvester, quoted in [331, p. 133])
Sound waves in a homogeneous, compressible fluid are governed by the wave
equation,

2
U=
1
c
2
≤
2
U
≤t
2

where
2
is the Laplacian,cis the constant speed of sound,tis time, andU
is a physical quantity such as the (excess

12 Introduction
for example, [885, Chapter 6], [714, Chapter 1] or [974, Chapter 1]. We consider
time-harmonic motions exclusively, so that
U=Re

ue
−it

(1.21)
whereis the radian frequency. Hence, the complex-valued functionusatisfies the
Helmholtz equation,

2
+k
2
⎫u=0inB
e (1.22)
whereB
eis the unbounded exterior region occupied by the fluid andk=/c.We
assume that the wavenumberkis real and positive. Then, the problem is to solve
(1.22 Sand a radiation condition at infinity.
Nowadays, the use of complex-valued functionsu, as in (1.21
it usually goes unremarked. However, the following quotation suggests that this was
not always the case.
There was Lamb, who had only recently gone to Adelaide[in 1875],and whose
book on hydrodynamics[682, 514](then a slight volume, being an exposition
of lectures he had given at Trinity) was the first English book that revealed a
use of the complex variable in mathematical physics: let me add that it was an
age when the use of
√
−1was suspect at Cambridge even in trigonometrical
formulae.
(Forsyth [357, p. 164])
1.3.1 The Sommerfeld radiation condition
The radiation condition serves two purposes. First, it ensures that the scattered waves
are not incoming at infinity. Thus, imagine enclosing theNobstacles by a large
sphere, and then sending an incident wave into this sphere; then, we suppose that
scattered waves can leave the large sphere but not enter. This is physically reasonable.
Second, the radiation condition (usually
reasonable boundary conditions, the radiation condition ensuresuniqueness:inthe
absence of an incident wave, the only solution of the boundary-value problem is
u≡0inB
e.
For example, in three dimensions, the scattered fieldu
scis required to satisfy
r
≥
⎬u
sc
⎬r
−iku
sc

→0asr⎪ (1.23)
uniformly in⎪and, wherer,⎪andare spherical polar coordinates with respect
to a chosen origin in the vicinity of the scatterers. Equation (1.23Sommerfeld
radiation condition; it implies that ru
scisO1⎫asr⎪ [223, Remark 3.4].
A consequence of (1.23
u
scr ∼
e
ikr
ikr
f asr⎪ (1.24)

1.3 Acoustic scattering byNobstacles 13
wheref is known as thefar-field pattern[223, Corollary 3.7]. The function
fis not known but it may be calculated by solving the boundary-value problem
foru
sc. Note that the factor ofik⎫
−1
on the right-hand side of (1.24
absorbed intof; the definition (1.24 u
scis dimensionless,
then so isf.
In two dimensions, the Sommerfeld radiation condition is
r
1/2
≥
⎬u
sc
⎬r
−iku
sc

→0asr⎪ (1.25)
uniformly in⎪, whererand⎪are plane polar coordinates.
For further discussion on the Sommerfeld radiation condition, see, for example,
[1149, §9.1], [1144], [564, §1.27], [53, Chapter I, §4.2] and [974, §4-5].
1.3.2 Boundary conditions
For a typical scattering problem, we have a given incident fieldu
inc. We then write
the total field as
u=u
inc+u
scinB
e
whereu
scis the (unknown
fied in terms ofu, but it can then be rewritten in terms ofu
sc, if convenient. There
are three standard boundary conditions:
Dirichlet condition(‘sound-soft’ or ‘pressure-release’)
u=0oru
sc=−u
inconS≡
Neumann condition(‘sound-hard’ or ‘rigid’)
⎬u
⎬n
=0or
⎬u
sc
⎬n
=−
⎬u
inc
⎬n
onS≡
Here,⎬/⎬ndenotes normal differentiation onS. The (unknown
values ofucan be used to calculate the far-field pattern directly; see (5.81
Robin condition[457] (impedance condition)
⎬u
⎬n
+u=0onS
whereis given;may be real or complex, it may be a constant, or it may
vary with position onS.
Uniqueness theorems for these three problems are proved by Colton & Kress [223,
Theorems 3.13 and 3.37]. Their proofs are for one obstacle, but the number of
obstacles is irrelevant to the argument.
For multiple-scattering problems, there is the possibility that we have a different
boundary condition on each scatterer. For example, we could haveN
1sound-hard

14 Introduction
obstacles andN
2sound-soft obstacles, withN
1+N
2=N. We shall refer to these as
mixture problems. For more information and a uniqueness theorem, see [657].
1.3.3 Transmission problems
The three standard boundary conditions given above all suppose that the obstacles
B
iare impenetrable: there is no transmission through the interfacesS
i. There are
situations where this is unrealistic. For example, consider the scattering of a wave in
a compressible fluid by blobs of another fluid; the wave will be partially reflected
and partially transmitted throughSinto the blobs. Then, mathematically, we have to
solve the followingtransmission problem: given u
inc, finduandu
0, whereusatisfies
(1.22),u
0satisfies

2
+k
2
0
⎫u
0=0inB=
N
≡
j=1
B
j
u−u
incsatisfies the radiation condition (1.23uandu
0satisfy a pair oftrans-
mission conditionson the interfaces,
u=u
0and
⎬u
⎬n
=
⎬u
0
⎬n
onS≡ (1.26)
Here,k
0=/c
0is the wavenumber inB,c
0is the speed of sound inBandmay be
interpreted as a density ratio (whenuis regarded as the excess pressure); see (1.31
and [262, §1.A.3].
This is the simplest transmission problem, wherek,k
0andare all constants.
There are situations wherek
0andare different constants for each obstacle. There
are also important situations wherek
0varies with position, in which case the obstacles
areinhomogeneous; see Section 1.3.4.
Ifk,k
0andare all constants, we have uniqueness provided that their values are
restricted suitably [627]; in particular, we have uniqueness ifk,k
0andare all real
and positive [223, Theorem 3.40].
1.3.4 Inhomogeneous obstacles
Time-harmonic waves in a compressible, inhomogeneous fluid are governed by [885, p. 408]
div

−1
gradp

+
2
p=0 (1.27)
whereis the density,pis the acoustic pressure,=/CandCis the speed of
sound; in general,andCare functions of position. According to Pierce [975],
(1.27
If the density is constant, (1.27

2
p+k
2
nr⎫p=0 (1.28)

1.3 Acoustic scattering byNobstacles 15
wherenr⎫=c/Cr
2
is the (square of the)refractive indexat positionr,k=/c
andcis a constant sound speed. For our discussion here, we assume thatnr⎫→1as
r⎪in all directions; this excludes media that are layered at infinity, for example.
There is a considerable literature on (1.28
mechanics; some of this will be mentioned below. In addition, several point-source
solutions (Green’s functions) are known for various functional forms ofnr⎫; see
[711] for a review.
For scattering problems in acoustics, there are essentially three cases, depend-
ing on properties ofn. First, suppose that1−nrhas compact support, so that
nr⎫≡1 forr=⎫r⎫>a, say. Suppose further thatnr⎫is smooth for allrin
three-dimensional space. Then, one can reduce the scattering problem to an integral
equation overB, the finite region in whichnr⎫≡1. One such is the Lippmann–
Schwinger equation; see Section 6.3.3 and, for example, [14], [225, §8.2] and [911,
§10.3]. Asymptotic approximations are available forka1 [14, 660]; the paper
[660] also discusses (1.27
Second, we could have situations in which1−nrdoesnothave compact
support, but is such thatnr⎫→1asr ⎪. The corresponding scattering problems
are uncommon in acoustics.
Third, we could havenr⎫≡1 outsideBwithndiscontinuous acrossS, the
boundary ofB. The corresponding scattering problem will require transmission con-
ditions across the interfaceS. (If the material inBis actually homogeneous, so that
nr⎫=n
0, a constant, inB, we recover the standard transmission problem discussed
in Section 1.3.3.) Here, we are mainly concerned with this third class of problem:
acoustic scattering byNbounded inhomogeneous obstaclesB
j,j=12N, sur-
rounded by an unbounded homogeneous fluid occupyingB
e.InB
e, we denote the
density, speed of sound and pressure by
e,candp
e, respectively, where
eandc
are constants. For scattering problems, we write
p
e=p
inc+p
scinB
e
wherep
incis the given incident field andp
scis the corresponding scattered field.
From (1.27 p
scis

2
+k
2
⎫p
sc=0inB
e (1.29)
wherek=/cis a positive constant. We assume thatp
incsatisfies (1.29
except possibly at some places inB
e. We require thatp
scsatisfies the Sommerfeld
radiation condition (1.23
Inside the inhomogeneous obstacles, we denote the density, speed of sound and
pressure by
0,c
0andp
0, respectively, so that (1.27

0div

−1
0
gradp
0

+k
2
0
p
0=0inB (1.30)
wherek
0=/c
0; in general,
0andk
0vary withinB. Across the interfaceS,we
require continuity of pressure and normal velocity. These transmission conditions

16 Introduction
reduce to
p
e=p
0and
1

e
⎬p
e
⎬n
=
1

0
⎬p
0
⎬n
onS≡ (1.31)
This defines the scattering problem for inhomogeneous obstacles.
1.3.4.1 Reduced equations
We can reduce Bergmann’s equation to an equation without first derivatives by
introducing a new dependent variable [87]; thus, define
p
0=
1/2
0
v
0 (1.32)
whencev
0is found to satisfy

2
v
0+k
2
0
+K⎫v
0=0 (1.33)
where
K=
1
2

−1
0

2

0−
3
4

−2 0
⎫grad
0⎫
2
(1.34)
=−
1/2
0

2

−1/2
0

≡ (1.35)
Equations (1.32
In a homogeneous region,
0andc
0are constants. Then,K≡0,k
0is a constant and
(1.33
0is constant butc
0is not, we
still haveK≡0 and then (1.33
Chapter 8].
Finally, we can write (1.33

2
v
0+k
2
−V⎫v
0=0 (1.36)
wherek
2
is a constant andV=k
2
−k
2
0
−K. In this form, (1.36
Schrödinger’s equationwith potentialV[911, eqn (10.59
For scattering problems, it is natural to mimic (1.32B
e, definingv
ebyp
e=

1/2
e
v
e. Then, the transmission conditions (1.31

1/2
v
e=v
0and
−1/2
⎬v
e
⎬n
=
⎬v
0
⎬n
+
v
0
2
0

0
⎬n
onS
where=
e/
0evaluated onS.
1.4 Multiple scattering of electromagnetic waves
In the past 25 years, numerical techniques for scattering and absorption by
variously-shaped objects have proliferated like weeds.
(Wiscombe & Mugnai [1330, p. 119])

1.4 Multiple scattering of electromagnetic waves 17
The scattering of electromagnetic waves is governed by Maxwell’s equations. Thus,
in the exterior domainB
e, we have
curlE−iH=0and curlH+iE=0 (1.37)
whereEis the electric field,His the magnetic field,is the electric permittivity,
is the magnetic permeability and the time-dependence of e
−it
is suppressed. We
always assume thatandare constants. EliminatingH, say, from (1.37
curl curlE−k
2
E=0
wherek=
√
;Hsatisfies the same equation. Also, (1.37 E=0
and divH=0. It follows that

2
+k
2
E=0and
2
+k
2
H=0
so that all the Cartesian components ofEandHsatisfy the same Helmholtz equation.
A given electromagnetic wave,E
incH
inc, is incident upon the obstaclesB
i,
i=12N. Write
E=E
inc+E
scandH=H
inc+H
scinB
e (1.38)
whereE
scH
scis the scattered field. Then, the problem is to solve Maxwell’s
equations (1.37B
esubject to appropriate boundary conditions onSand a radiation
condition at infinity.
1.4.1 The Silver–Müller radiation conditions
The generalisation of the Sommerfeld radiation condition to electromagnetics was given by Silver and Müller in the late 1940s. There are two conditions, namely
√
ˆr×H
sc+
√
E
sc→0 (1.39)
and
√
ˆr×E
sc−
√
H
sc→0 (1.40)
asr=r→, uniformly inˆr, wherer=rˆr. In fact, it is sufficient to impose just
one of (1.39 Maxwell’s equations can be used to deduce radiation conditions forE
scandH
sc
separately; for example, (1.39
ˆr×curlE
sc+ikE
sc→0asr→≡

18 Introduction
In particular, we can deduce thatˆr·E
scandˆr·H
scvanish asr→. For more
information, see [223, §4.2] or [262, §1.EM.4].
1.4.2 Boundary conditions
In electromagnetic problems, the most important boundary condition is that corre-
sponding to a perfectly-conducting surface. This condition is
n×E=0orn×E
sc=−n×E
inconS (1.41)
wherenis the unit normal pointing fromSintoB
e; see [223, §4.1] or [225, §6.4].
1.4.3 Transmission problems
If the regionsB
i,i=12N, are filled with a different material, we are led to an
electromagnetic transmission problem: givenE
incH
inc, findEHandE
0H
0,
whereEHsatisfies (1.37E
0H
0satisfies
curlE
0−i
0H
0=0and curlH
0+i
0E
0=0inB (1.42)
E−E
incH−H
incsatisfies the Silver–Müller radiation conditions, andEHand
E
0H
0satisfy
n×E=n×E
0andn×H=n×H
0onS≡ (1.43)
Here,
0and
0are the electric permittivity and the magnetic permeability, respec-
tively, of the material inB. For more information on this problem, see [482, 799].
For the transmission problem when the material inBischiral, see [47, 48].
1.4.4 Inhomogeneous obstacles
Suppose that the obstaclesBare inhomogeneous, with electric permittivity
0rand
magnetic permeability
0rat positionr. Maxwell’s equations, (1.42
hold. EliminatingH
0gives

0curl
−1
0
curlE
0−k
2
0
E
0=0 (1.44)
wherek
2
0
r=
2

0
0. Similarly, eliminatingE
0gives

0curl
−1
0
curlH
0−k
2
0
H
0=0≡
These equations are discussed in [196, §1.3], together with their generalisation to
anisotropic media for which
0rand
0rare replaced by (invertible
For problems where
0is constant, see [225, Chapter 9].

1.5 Multiple scattering of elastic waves 19
1.4.5 Two-dimensional problems
For electromagnetic scattering by perfectly-conducting cylinders, where the fields,
boundary conditions and geometry do not vary withz, say, we can reduce the problem
to two uncoupled problems for the scalar Helmholtz equation [564, §1.34]. They are

2
+k
2
⎫E
z=0 withE
z=0onS (1.45)
and

2
+k
2
⎫H
z=0 with⎬H
z/⎬n=0onS≡ (1.46)
Solutions of (1.45 TM wavesandTE waves, respec-
tively [549, §8.2].
1.5 Multiple scattering of elastic waves
In this section, we formulate some problems involving the scattering of elastic waves
by obstacles. Thus, we begin by supposing that the exterior domainB
eis filled with
homogeneous elastic material, with elastic stiffnessesc
ijkland mass density.We
assume, as usual, that
c
ijkl=c
jikl=c
klij≡ (1.47)
For an isotropic solid,
c
ijkl=
ij
kl+

ik
jl+
il
jk

(1.48)
whereandare the Lamé moduli. Hooke’s law gives the stresses as

ij=c
ijkl/x
k⎫u
l
whereu
iare the components of the total displacementuand we have used the
standard Einstein summation convention. The equations of motion foruare
⎬
⎬x
j

ij+
2
u
i=0i=123inB
e≡ (1.49)
For an isotropic material, these reduce to
k
−2
grad divu−K
−2
curl curlu+u=0inB
e (1.50)
where the two wavenumberskandKare defined by

2
=+2k
2
=K
2
≡ (1.51)
We assume thatkandKare real and positive, and, as usual, the time dependence
e
−it
is suppressed.

20 Introduction
A given stress wave,u
inc, is incident upon the obstaclesB
i,i=12N. Write
u=u
inc+u
scinB
e
whereu
scis the scattered field. Then, the problem is to solve (1.49
to a boundary condition onSand a radiation condition at infinity.
There are many good textbooks on elastodynamics. See, for example, [2, 429, 520,
83, 250].
1.5.1 The Kupradze radiation conditions
Kupradze generalised the Sommerfeld radiation to elastodynamics, assuming that
the (homogeneous B
eis isotropic; see [671, Chapter III, §2], [673,
pp. 124–130] or [520, §6.5]. Thus, decompose the scattered field as
u
sc=u
P
sc
+u
S
sc
inB
e
where
u
P
sc
=−k
−2
grad divu
scandu
S
sc
=K
−2
curl curlu
sc≡
Then, in three dimensions, the elastodynamic radiation conditions are
r
≥
⎬u
P
sc
⎬r
−iku
P
sc

→0andr
≥
⎬u
S
sc
⎬r
−iKu
S
sc

→0
asr⎪, uniformly inˆr. Note that the fieldsu
P
sc
andu
S
sc
are the longitudinal and
transverse parts, respectively, of the scattered field; they satisfy

2
+k
2
⎫u
P
sc
=0and
2
+K
2
⎫u
S
sc
=0
and correspond to radiatedP-waves andS-waves, respectively. Note also that
curlu
P
sc
=0and divu
S
sc
=0.
1.5.2 Boundary conditions
In elastodynamics, the most important boundary condition is that corresponding to a
cavity or hole. This condition is
Tu=0orTu
sc=−Tu
inconS (1.52)
where the traction operator is defined by
Tu⎫
m
=n
ic
imkl
⎬u
k
⎬x
l
=n
m
⎬u
j
⎬x
j
+n
j
≥
⎬u
m
⎬x
j
+
⎬u
j
⎬x
m

(1.53)
andnis the unit normal vector pointing fromSintoB
e.

1.5 Multiple scattering of elastic waves 21
Equation (1.52
condition. The corresponding Dirichlet condition is
u=0oru
sc=−u
inconS≡
This condition is usually said to characterise a rigid surfaceS. In fact, ifBcontains
a rigid material (which cannot deform, by definition), it will move, and this should
be taken into account [936].
1.5.3 Transmission problems
If the regionsB
i,i=12N, are filled with a different homogeneous isotropic
elastic material, we are led to an elastodynamic transmission problem (orinclusion
problem): given u
inc, finduandu
0, whereusatisfies (1.50u
0satisfies
k
−2
0
grad divu
0−K
−2
0
curl curlu
0+u
0=0inB
u−u
incsatisfies the Kupradze radiation conditions, anduandu
0satisfy
u=u
0andTu=T
0u
0onS≡ (1.54)
Here,k
0,K
0andT
0are defined by (1.51 ,andreplaced
by
0,
0and
0, respectively, which are the material constants for the elastic solid
occupyingB. More generally, each inclusion could be made of a different anisotropic
elastic material.
The transmission conditions (1.54
They can be modified to account for various imperfect interfaces [790].
The acoustic transmission problem (Section 1.3.3) can be modified so as to rep-
resent the problem offluid–solid interactions: given the acoustic pressure field u
inc,
find the acoustic scattered fieldu
scinB
eand an elastic displacementu
0inB, subject
to
≤u
≤n
=
f
2
n·u
0and−un=T
0u
0onS
whereu=u
inc+u
scand
fis the density of the compressible inviscid fluid occupying
B
e. For more information on this problem, see [750].
1.5.4 Inhomogeneous obstacles
Suppose that the obstacles are inhomogeneous and anisotropic, with elastic stiffnesses c
0
ijkl
rand density
0rat positionr. The equations of motion will now involve
derivatives of the stiffnesses:
≤
≤x
j

0
ij
+
0
2
u
0
i
=0i=123inB

22 Introduction
where Hooke’s law gives the stresses as

0
ij
=c
0
ijkl
/x
ku
0
l

andu
0
i
are the components ofu
0, the displacement inB. The traction vector isT
0u
0
with components
0
ij
n
j.
If the stiffnesses are only piecewise-smooth functions of position withinB, one
may sometimes make progress by partitioningBinto subregions, within each of
which the stiffnesses are smooth. Then, one has to impose transmission conditions,
similar to (1.54
example, if each scatterer is layered like an onion or the trunk of a tree.
1.5.5 Two-dimensional problems
Letx y zbe Cartesian coordinates and suppose that eachB
iis a cylinder with genera-
tors parallel to thez-axis. Then, if the incident field is independent ofz, the
three-dimensional scattering problems formulated above can be decomposed into two
subproblems in a cross-sectional plane. Thus, letu=u v w, where ux yandvx y
are the in-plane components ofuandwx yis the anti-plane component. The problem
of findingwreduces to an acoustic problem, as
2
+K
2
w=0; solutions forware usu-
ally referred to asSH-waves. Finding uandvremains as a genuinely coupled problem.
For more information, see, for example, [2, §2.7] or [520, §1.8]. The two-dimensional
Kupradze radiation conditions are discussed in [69, 1167] and [520, §6.9].
In-plane problems are often reduced to problems involving acoustic potentials.
Thus, withu
2≡u vbeing the in-plane displacement, we can write
u
2=grad+curlˆz (1.55)
where

2
+k
2
=0
2
+K
2
=0
andˆzis a unit vector in thez-direction [520, §2.7]. The representation (1.55
venient for constructing valid displacement fields. However, althoughand≥sat-
isfy separate Helmholtz equations, they are inevitably coupled through the boundary
conditions onS, and this fact makes elastic problems more difficult to solve.
1.5.6 Half-space problems
There are several areas of application for which the most relevant problems involve
obstacles in (or attached to) an elastic half-space, rather than an unbounded ‘full-
space’. Examples come from seismology [1123], soil-structure interaction [585, 269]
and non-destructive evaluation.
Suppose that the half-space occupies the regiony>0. The boundary is usually
taken to be free from tractions, so thatTu=0on (all or most of)y=0. The obstacles

1.6 Multiple scattering of water waves 23
could be grooves, cracks, canyons or other indentations in the free surface, they could
be buried cavities, cracks or inclusions, or they could be elastic structures extending
intoy<0. The flat free surface extends to infinity, and can support Rayleigh (surface
waves; see, for example, [1265], [2, §5.11] or [520, §3.4].
For problems of this kind, it is convenient to writeu=u
inc+u
scinside the
half-space, whereu
incnow consists of the total displacementin the absence of the
scattering obstacles. If the incident wave is a plane P-wave, for example, thenu
inc
would consist of this wave together with the reflectedP- andS-waves: in other
words,u
incis constructed so thatTu
inc=0ony=0.
A uniqueness theorem for two-dimensional half-plane problems has been proved
by Gregory [440].
1.6 Multiple scattering of water waves
Consider an infinite ocean of constant depthh. Specifically, letx y zbe Cartesian
coordinates, chosen so thatz=0 is the undisturbed free surface withzincreasing
with depth: the rigid bottom is atz=h. The water is assumed to be incompressible
and inviscid. The motion is assumed to be irrotational, whence a velocity potential
exists. For time-harmonic motions, we have=Ree
−it
, where

2
=0inB
e (1.56)
which is the region occupied by the water.
We suppose that there areNobstaclesB
i(i=12N) immersed in the water;
these may be completely submerged or they may pierce the free surface. The wetted
boundary ofB
iisS
i.
A given wave with velocity potential
incis incident upon the obstacles. Write
=
inc+
sc
where
scis the scattered potential. Then, the problem is to solve Laplace’s equation
subject to boundary conditions onS(defined by (1.1
surface, and a radiation condition.
The standard references on water waves are the books of Lamb [682] and Stoker
[1145], and the encyclopaedic review by Wehausen & Laitone [1303]. More recent
books include [102, 842, 731, 675].
For reviews on the interaction of water waves with multiple bodies, see [816, 909,
832].
1.6.1 Boundary conditions
In hydrodynamics, there is usually no flow throughSor the bottom, whence

≤n
=0or

sc
≤n
=−

inc
≤n
onS (1.57)

24 Introduction
and

⎬z
=0 on the bottomz =h≡ (1.58)
For ‘deep water’ (h =⎪), the bottom boundary condition (1.58
⎫grad⎫0as z⎪≡
On the free surface, we have a Robin condition,
K+

⎬z
=0 on the free surfaceF
whereK=
2
/gandgis the acceleration due to gravity. The boundaryFis all
ofz=0 if the obstacles are all submerged, otherwise it is the portion ofz=0 not
intersected by the obstacles.
The boundary condition (1.57 fixedobstacles.
More generally, we could consider floating, movable obstacles. The solution of such
problems (withNthree-dimensional obstacles) can be written as a linear combination
of 6N (in general) radiation problems (with no incident wave) and the solution of the
scattering problem (with all obstacles held fixed). The appropriate linear combination
is determined using the equations of motion of the moving obstacles; see, for example,
[753]. So, for one obstacle, one has to solve six radiation problems, in general; see,
for example, [1303, §19]. In this context, the scattering problem is sometimes
known as thediffraction problem[1303, §18 ], [906].
1.6.2 Radiation conditions
Water waves are surface waves with gravity as the restoring force. Typical velocity
potentials for wave-like solutions are
e
ikx
coshkh−z⎫andJ
0krcoshkh−z (1.59)
wherekis the unique positive real solution of
K=ktanhkh (1.60)
J
0is a Bessel function, andr=x
2
+y
2
⎫
1/2
. For deep water,k=K.
The free-surface elevation is given by
−
1
g

⎬t

z=0
=Re

x ye
−it

so thatx y=i/g x y0⎫. Thus, for three-dimensional water waves, the
waves propagate over a two-dimensional plane (the free surface,z=0). Also, we see
that the surface elevations corresponding to the wave-like solutions (1.59
dimensional Helmholtz equations so that it is natural to impose the two-dimensional
Sommerfeld radiation condition, (1.25
scx y z; this

1.6 Multiple scattering of water waves 25
condition is to hold uniformly in all horizontal directions and for all depthsz. For
more information on radiation conditions for water-wave problems, see [731, §1.3.1]
or [675, p. 13].
1.6.3 Two-dimensional problems
For two-dimensional scattering problems, it is usual to express the far field (and
hence, the radiation condition) in terms of reflection and transmission coefficients.
Thus, we identify two problems, corresponding to waves incident from the right or
the left. For deep water and a wave incident fromx=+⎪, we have

incx z=e
−Kz−i Kx
=
−
inc

say, and then≡
+satisfies

+x z∼
←
e
−Kz
e
−iKx
+R
+e
iKx
⎫asx⎪
T
+e
−Kz−i Kx
asx?≡
Similarly, for deep water and a wave incident fromx=?, we have

incx z=e
−Kz+i Kx
=
+
inc

say, and then≡
−satisfies

−x z∼
←
T
−e
−Kz+i Kx
asx⎪
e
−Kz
e
iKx
+R
−e
−iKx
⎫asx?≡
The constantsR
±andT
±are the (complex
respectively. They satisfy the following relations:
T
+=T
−=T⎫R
+⎫=⎫R
−⎫=⎫R⎫
⎫R⎫
2
+⎫T⎫
2
=1 andT
R
++TR
−=0
where the first two equations defineTand⎫R⎫, respectively, and the overbar denotes
complex conjugation. Proofs of these relations and further references can be found
in [787] and [842, §7.6.2]. Furthermore, if one applies Green’s theorem to
+and

+
inc
in a region bounded byF,Sand straight lines atx=X,x=−Xandz=Z
(whereX⎪andZ⎪), the result is a formula forT
+; repeating the calculation
for the other three combinations of signs in
±and
±
inc
gives formulae for the other
three coefficients. Thus, we have
R
±=−i

S

±
⎬
⎬n

∓
inc
ds (1.61)
and
T
±=1−i

S

±
⎬
⎬n

±
inc
ds≡ (1.62)

26 Introduction
Such applications of Green’s theorem (and other reciprocity relations) often yield
useful formulae with minimal work; for a careful exposition, with many examples,
see Achenbach’s book [3].
The formulae (1.61
They are useful because they express the reflection and transmission coefficients in
terms of the boundary values of the total potentials,
±, and these may be computed
by solving a boundary integral equation, for example.
1.6.4 Vertical cylinders
Suppose that every obstacleB
iis a vertical cylinder and that the water has constant
finite depth,h. For the scattering problem, we start with an incident plane wave,
given by
incx y z=e
ikx
coshkh−z⎫. Then, provided each cylinder extends from
the bottom atz=hthrough the free surface, we can separate out the dependence on
z, writing
scx y z=!x ycoshkh−z⎫.As
scsatisfies Laplace’s equation in
three dimensions, we find that the reduced potential!satisfies the two-dimensional
Helmholtz equation,

2
+k
2
⎫!=0≡
Hence, the problem reduces to a two-dimensional acoustic scattering problem. For
radiation problems (with prescribed normal velocity onS), we can still separate the
variables, writingx y z=!x y Zz, say, but now, in general, we obtain an
infinite set of eigenfunctions forZ. Explicitly, we obtain
x y z=!x y Z
0z+
⎪

n=1
!
nx y Z
nz (1.63)
where
Z
0z=coshkh−z Z
nz=cosk
nh−z⎫forn=12 (1.64)
k
n(n=12) are positive real solutions of
K=−k
ntank
nh (1.65)
and!
nsolves
2
−k
2
n
⎫!
n=0, the two-dimensional modified Helmholtz equation.
Notice that the vertical eigenfunctions are orthogonal:

h
0
Z
nz Z
mzdz=0ifn=mwithn≥0 andm≥0≡ (1.66)
1.6.5 Uniqueness
Although the basic problems for the scattering and radiation of linear water waves
were formulated long ago, the first general mathematical analysis was given by John
in 1950 [559]. In particular, he studied the question of uniqueness. He proved that

1.7 Overview of the book 27
the boundary-value problems for bodies that are not completely submerged do have
at most one solution, provided the bodies satisfy a certain geometrical condition.
In the same year, Ursell [1225] proved uniqueness for a single horizontal circular
cylinder, completely submerged beneath the free surface of deep water. Subsequently,
there was much work in which the geometrical conditions were weakened: it was
widely believed that the geometrical conditions were superfluous, and that a general
uniqueness proof would be found one day. All this changed when a counter-example
was found by McIver in 1996 [826]. She constructed a symmetric catamaran-type
structure in two dimensions for which the corresponding homogeneous boundary-
value problem has a non-trivial solution at one value ofK. Further examples have
been found since; for a good discussion, see [675].
1.7 Overview of the book
In this chapter, we have delineated the scope of the book. The basic problem to
be studied is the scattering of time-harmonic waves by a finite number of bounded
obstacles: let us call this thebasic multiple-scattering problem. As a prototype, we
consider acoustic scattering (governed by the Helmholtz equation) in detail. Results
for two and three dimensions are given. Many extensions are also described, including
those for elastic waves, electromagnetic waves and water waves.
The oldest methods for solving the basic multiple-scattering problem combine
separated solutions (for example, in cylindrical or spherical polar coordinates) with
appropriate addition theorems. Indeed, the general idea of combining multipole
expansions with addition theorems has been used in many physical contexts, involv-
ing configurations of cylinders and spheres. Thus, we begin in Chapter 2 with two-
dimensional separated solutions, including their associated addition theorems. The
corresponding three-dimensional results are given in Chapter 3, where the emphasis
is on the use of spherical polar coordinates. These two chapters are rather technical,
but the material developed there is used extensively in the subsequent chapters.
Apart from cylindrical wavefunctions (involving Bessel and Hankel functions),
Chapter 2 also contains new results for elliptical wavefunctions (Mathieu functions)
and a discussion of multipole solutions for water-wave problems. Chapter 3 contains
a wealth of formulae for spherical wavefunctions. The main justification for this
development is that three-dimensional addition theorems are complicated, and new
methods continue to be devised for the efficient computational use of these theorems.
For example, we derive various formulae that have been used in ‘fast multipole
methods’ (see Section 6.14). Chapter 3 also includes new addition theorems for
water-wave multi-pole potentials.
Chapter 4 is dedicated to methods based on separation of variables. Results are
given for circular cylindrical coordinates, elliptic cylindrical coordinates and spherical
polar coordinates. These methods are exact, in principle, and they are fast in practice.

28 Introduction
However, they are limited to scatterers with simple shapes. Consequently, more
general methods are developed in the subsequent chapters.
Methods based on integral equations (mainly boundary integral equations) are
developed in Chapters 5 and 6. The first of these covers the basic theory with simple
acoustic applications. Chapter 6 considers many other problems and physical applica-
tions: penetrable and inhomogeneous scatterers; electromagnetic problems, including
inhomogeneous obstacles and thin wires; elastodynamic problems, including inclu-
sions and cracks; hydrodynamic problems; modified integral equations, designed to
eliminate ‘irregular frequencies’; and the use of exact Green’s functions. We also
discuss Twersky’s method for the basic multiple-scattering problem. The chapter
ends with a discussion of some analytical aspects of fast multipole methods.
T-matrix and related methods are developed in Chapter 7. These methods may
be viewed as generalisations of the method of separation of variables, in which
spherical multipoles are used for non-spherical scatterers, for example. They may
also be viewed as a method for solving certain integral equations. Chapter 7 also
considers several related topics: Kupradze’s method and the ‘method of fundamental
solutions’; complete sets of functions, as used to represent functions defined on a
curve or surface; and various properties of theT-matrix.
Chapter 8 is concerned with approximations. How do the exact methods in the rest
of the book simplify if the scatterers are small or widely separated? Detailed results
are given for acoustic scattering by one small obstacle in two and three dimensions;
these lead to new low-frequency approximations for theT-matrix. An exact method
due to Foldy is described: it assumes that each obstacle scatters isotropically, and so is
appropriate for small soft obstacles, for example. We extend Foldy’s method to non-
isotropic scatterers by including acoustic dipoles of unknown strength and orientation.
Various wide-spacing approximations are discussed in Section 8.5. Finally, we also
give a brief discussion of problems in which the scatterers are located randomly. This
is a large topic, with a large literature; we outline some of this work, with emphasis
on multiple scattering.
The book ends with nine appendices and a list of references. The reference list is
extensive, from Abramowitz & Stegun [1] to Zurket al.[1386]. It is not intended
to be exhaustive because, first, the subject is too old and too broad, and second, we
only cite material that we have actually seen: according to one study [1103], ‘only
about 20% of citers read the original’!

2
Addition theorems in two dimensions
Bessel functions are not algebraic functions,they are not simply periodic func-
tions, andthey are not doubly periodic functions. Consequently, in accordance
with a theorem of Weierstrass, it is not possible to expressJ
≥Z+z≡as an alge-
braic function ofJ
≥ZandJ
≥z. That is to say, that Bessel functions do not
possess addition theorems in the strict sense of the term.
(Watson [1298, p. 358])
2.1 Introduction
Watson began his chapter on ‘Addition Theorems’ for Bessel functions with the
remarks above. Nevertheless, he went on to give proofs of various formulae which
are ‘commonly described as addition theorems’.
An example of an addition theorem in the strict sense is
e
ix+y≡
=e
ix
e
iy

In the wider sense, we have the formula
J
0x+y≡=

n=?
−1≡
n
J
nx J
ny (2.1)
whereJ
nis a Bessel function. Further examples are given by Askey [46, Lecture 4].
Addition theorems are at the heart of most theories of multiple scattering. Roughly
speaking, they are used to transform one expansion about some point in space into
a similar expansion about a different point. We shall prove several theorems of
this type, in both two and three dimensions. This chapter is concerned with two-
dimensional addition theorems; these results are mainly classical. Analogous results
in three dimensions are proved in Chapter 3. These are typically more complicated,
and they are still the subject of current research.
29

30 Addition theorems in two dimensions
2.2 Cartesian coordinates
Letx
1→x
2x
q≡be Cartesian coordinates. In practice, we are usually interested in
q=2orq=3; when convenient, we writex,yandzforx
1,x
2andx
3, respectively.
Consider the general linear homogeneous partial differential equation
q

i=1
q

j=1
a
ij
∼
2
u
∼x
i∼x
j
+
q

i=1
b
i
∼u
∼x
i
+cu=0→ (2.2)
wherea
ij,b
iandcare all constants. Typical examples are Laplace’s equation,

2
q
u=0→
and the Helmholtz equation,

2
q
+k
2
≡u=0→
where
2
q
is theq-dimensional Laplacian,

2
q
=
∼
2
∼x
2
1
+
∼
2
∼x
2
2
+···+
∼
2
∼x
2
q

Solutions of Laplace’s equation are calledharmonic functions, whereas solutions of
the Helmholtz equation are calledwavefunctions.
It is obvious that ifusolves (2.2
tives ofuwith respect to anyx
k. Thus, given one solution of (2.2
can be found by differentiation. The use of this idea for constructing harmonic func-
tions in three dimensions (by repeated differentiation of 1/r ) is described at length
by Thomson & Tait [1175, Appendix B].
In order to define multipole solutions, we introduce a radial coordinate,
r=
≥

q→where
q=x
2
1
+x
2
2
+···+x
2
q

Hobson’s theorem, several variants of which are proved below, is convenient for applying differential operators in Cartesian coordinates to functions of
q.Tobe
precise, letf
nx
1→x
2x
q≡be a homogeneous polynomial of degreen, where
n≥0 is an integer. This means that
f
nx
1→≤x
2x
q≡=
n
f
nx
1→x
2x
q≡→
for any, and implies that
f
nx
1→x
2x
q≡=
n

j=0
x
j
1
g
n−jx
2→x
3x
q≡→
whereg
mis a homogeneous polynomial of degreemin theq−1 variables
x
2→x
3x
q. Givenf
n, we define the differential operator of ordern,f
n
q≡by
f
n
q≡=f
n

∼
∼x
1
→
∼
∼x
2

∼
∼x
q
≡
(2.3)

2.3 Hobson’s theorem 31
Hobson’s theorem gives the result of applyingf
n
q≡toF
q≡, whereFis any
sufficiently smooth function of one variable. The result simplifies iff
nis aharmonic
homogeneous polynomial,
nsay, so that

2
q

nx
1→x
2x
q≡=0
In this chapter, we takeq=2. In this case, there are exactly two linearly independent
harmonic homogeneous polynomials of degreenfor each positiven; they can be
taken as

±
n
x y=x±iy≡
n
=r
n
e
±in
→ (2.4)
whererandare plane polar coordinates (x =rcosandy=rsin).
2.3 Hobson’s theorem
We start with a simple one-dimensional version of Hobson’s theorem; according to
Hobson [507, p. 125], it is due to Schlömilch.
Theorem 2.1Let
1=x
2
. Letnbe a positive integer. Then
d
n
dx
n
F
1≡=

l=0
2
n−2l
l!
F
n−l≡

1≡
d
2l
dx
2l
x
n
→
whereF
m
is themth derivative ofFand the summation has a finite number of
terms. ProofWe give an inductive proof. The casen=1 is trivial. Assuming the result
forn, we consider
d
n+1
dx
n+1
F
1≡=
d
dx

l=0
2
n−2l
l!
F
n−l≡

1≡
d
2l
dx
2l
x
n
→
=

l=0
2
n−2l
l!
→
2xF
n+1−l≡
d
2l
dx
2l
x
n
+F
n−l≡
d
2l+1
dx
2l+1
x
n
∼
=2
n+1
x
n+1
F
n+1≡
+

l=1
2
n+1−2 l
l!
F
n+1−l≡

l
n
→
where

l
n
=x
d
2l
dx
2l
x
n
+2l
d
2l−1
dx
2l−1
x
n
=
d
2l
dx
2l
x
n+1

the last equality can be proved by direct calculation or by takingP=x
n
1
in Lemma 2.2.
Thus, the result is true forn+1, and so the theorem is proved.

32 Addition theorems in two dimensions
Lemma 2.2Let
2
q
be theq-dimensional Laplacian, letx
1→x
2x
q≡be Carte-
sian coordinates, and letlbe a positive integer. Then

2l
q
x
1P≡−x
1
2l
q
P=2l
2l−2
q
∼P
∼x
1
for any sufficiently smooth functionPx
1→x
2x
q≡, where, by definition,
2l
q
=

2
q
≡
l
.
ProofThe result is trivial forl=1:

2
q
x
1P≡−x
1
2
q
P=2
∼P
∼x
1
(2.5)
Assuming the result forl, we consider

2l+2
q
x
1P≡−x
1
2l+2
q
P=
2l
q

2
q
x
1P≡

−x
1
2l
q

2
q
P
=
2l
q
→
x
1
2
q
P+2
∼P
∼x
1
∼
−x
1
2l
q

2
q
P→
using (2.5
2
q
P, we see that the right-hand
side becomes
2l
2l−2
q
∼
∼x
1

2
q
P+2
2l
q
∼P∼x
1
=2l+1
2l
q
∼P
∼x
1
→
as required.
We generalise Theorem 2.1 to two dimensions.
Theorem 2.3 (Hobson’s theorem in two dimensions)Let
2=x
2
+y
2
. Letf
nx y
be a homogeneous polynomial of degreen, wherenis a positive integer. Then, with
f
n
2≡defined by(2.3),
f
n
2F
2≡=

l=0
2
n−2l
l!
F
n−l≡

2
2l
2
f
nx y
where the summation has a finite number of terms.
ProofAgain, we give an inductive proof. First of all, we note that the one-
dimensional theorem, Theorem 2.1, is valid for functions of
2, so that we have
∼
n
∼y
n
F
2≡=

l=0
2
n−2l
l!
F
n−l≡

2
2l
2
y
n
→ (2.6)
for every positiven.
For the inductive argument, we start by noting that the result is true forn=1. In
this case, we havef
1x y=a
0x+a
1y, and so
f
1
2F
2≡=2F

2≡f
1x y
as required (
2l
2
f
1=0 forl≥1).

2.3 Hobson’s theorem 33
Next, we assume that the result is true forn. We have
f
n+1x y=xf
nx y+ay
n+1
(2.7)
for some coefficienta. Thus,
f
n+1
2F
2≡=
∼
∼x
f
n
2F
2≡+a
∼
n+1
∼y
n+1
F
2≡
=
∼
∼x

l=0
2
n−2l
l!
F
n−l≡

2
2l
2
f
n
+a

l=0
2
n+1−2 l
l!
F
n+1−l≡

2
2l
2
y
n+1
→
using the assumed result and (2.6

l=0
2
n−2l
l!
→
2xF
n+1−l≡

2l
2
f
n+F
n−l≡
∼
∼x

2l
2
f
n
∼
→
whereas, by (2.7

l=0
2
n+1−2 l
l!
F
n+1−l≡

2l
2
f
n+1−

l=0
2
n+1−2 l
l!
F
n+1−l≡

2l
2
xf
n
Hence,
f
n+1
2F
2≡=

l=0
2
n+1−2 l
l!
F
n+1−l≡

2
2l
2
f
n+1x y+L
n→
where
L
n=

l=0
2
n−2l
l!

2F
n+1−l≡

x
2l
2
f
n−
2l
2
xf
n≡

+F
n−l≡
∼∼x

2l
2
f
n
≡

We must show thatL
n≡0 to complete the proof. To do this, we note that the
expression in square brackets vanishes whenl=0, whence
L
n=

l=1
2
n+1−2 l
l!
F
n+1−l≡
→
x
2l
2
f
n−
2l
2
xf
n≡+2l
∼
∼x

2l−2
2
f
n
∼

this vanishes by Lemma 2.2, as required.
The theorems above are special cases of a general theorem proved by Hobson
[507, §79]. Our proofs are less elegant but perhaps more transparent.
The result of Hobson’s theorem simplifies iff
nis harmonic. Thus, we obtain the
following corollary.
Corollary 2.4Letr=
≥
x
2
+y
2
. Let
nx ybe a harmonic homogeneous poly-
nomial of degreen, wherenis a positive integer. Then

n
2Fr=
nx y

1
r
d
dr
≡
n
Fr

34 Addition theorems in two dimensions
Thus, iff
nis harmonic, the (finite
A simple example follows; applications to wavefunctions will be given in the next
section.
Example 2.5TakeFr=logrand
n=
±
n
, defined by (2.4

∼
∼x
±i
∼
∼y
≡
n
logr=2
n−1
n−1≡!
e
±in
r
n

Theq-dimensional version of Corollary 2.4 can be found in [893, §12, Theorem 1].
2.4 Wavefunctions
Hobson applied his theorem to harmonic functions in three dimensions. We apply it
here to wavefunctions (solutions of the Helmholtz equation), in two dimensions.
LetZ
≥wdenote any cylinder function of order≥, that is
Z
≥w=J
≥w,Y
≥w,H
1≡
≥
worH
2≡
≥
w. (2.8)
Then, we have ([1, eqn (9.1.30

1
w
d
dw
≡
m
Z
≥w
w
≥
=−1≡
m
Z
≥+mw
w
≥+m
(2.9)
form=0→1→2and any≥. In particular, setting≥=0 andw=krgives

1
kr
d
dr
≡
m
Z
0kr=
−1≡
m
r
m
Z
mkr (2.10)
Next, we show that wavefunctions can be generated by the application of certain
differential operators.
Definition 2.6The differential operators≥
±
m
are defined by
≥
±
m
=

−1
k

∼
∼x
±i
∼
∼y
≡
m
→
form=0→1→2where≥
±
0
=I, the identity operator.
The action of these operators on cylindrical wavefunctions is described by the next
two theorems. Theorem 2.7
≥
±
m
Z
0kr=Z
mkre
±im
form=0→1→2
First proofApply Corollary 2.4 toFr=Z
0kr, using (2.10
m=
−1/k≡
m
x±iy≡
m
.
We can also give a direct proof of Theorem 2.7 using an inductive argument.

2.4 Wavefunctions 35
Second proofThe theorem is obviously true form=0. For subsequent values, we
use induction. We have, assuming the result is true form,
≥
±
m+1
Z
0=≥
±
1
≥
±
m
Z
0=≥
±
1
Z
mkre
±im

Now, in terms of plane polar coordinates, we have
≥
±
1
r≡=
−1
k
e
±i

∼
∼r
±
i
r
∼
∼∓
≡

Hence≥
±
m+1
Z
0kr=Z
m+1kre
±im+1≡∓
, since [1, §9.1.27]
Z

≥
w−/w Z
≥w=−Z
≥+1w
and the result follows.
Another interesting relation, similar to Theorem 2.7, is

mZ
0kr=i
m
Z
mkrcosm∓→ (2.11)
where
m=T
mik≡
−1
∼/∼x≡andT
mwis a Chebyshev polynomial of the first kind
[427, §8.94]. Formula (2.11
using the recurrence relation forT
n.
We can generalise Theorem 2.7, but first we state a trivial but useful property
of≥
±
m
.
Lemma 2.8Ifuis any solution of the two-dimensional Helmholtz equation,
≥
+
m
≥
−
m
u=≥
−
m
≥
+
m
u=−1≡
m
uform=0→1→2
We shall use this lemma in the proof of the next theorem.
Theorem 2.9
≥
+
m
Z
nkre
in
=Z
m+nkre
im+n≡∓
→ (2.12)
≥
−
m
Z
nkre
in
=−1≡
n
Z
m−nkre
−im−n≡∓
→
=−1≡
m
Z
n−mkre
in−m≡∓
→ (2.13)
form=0→1→2andn=0→±1→±2.
ProofLetAdenote the left-hand side of (2.12n≥0, we have
A=≥
+
m
≥
+
n
Z
0=≥
+
m+n
Z
0=Z
m+nkre
im+n≡∓

Ifn<0, setn=−l, whence
A=−1≡
l
≥
+
m
≥
−
l
Z
0
Now, there are two cases. First, ifm≥l, we use Lemma 2.8 to eliminate≥
−
l
:
A=≥
+
m−l
Z
0=≥
+
m+n
Z
0→

36 Addition theorems in two dimensions
as before. Second, ifm<l, we use Lemma 2.8 to eliminateβ
+
m
:
A=−1≡
l+m
β
−
l−m
Z
0=−1≡
l+m
Z
l−mkre
−il−m≡∓
and again we obtain the desired result, after usingZ
−n=−1≡
n
Z
n. A similar argument
succeeds for (2.13
We remark that similar results can be proved for modified Bessel functionsI
nw
andK
nw.
2.5 Addition theorems
Consider two origins,O
1andO
2. Letr
jbe the position vector of a general point
Pwith respect toO
j, forj=1→2. Letbbe the position vector ofO
1with respect
toO
2, so thatr
2=r
1+b. Letr
j=r
jcos
j→r
jsin
j≡andb=bcos bsin. See
Fig. 2.1 for a sketch of the geometry. The simplest addition theorem was proved by
C. Neumann in 1867; see [1298, §11.2], [884, p. 1371], [691, eqn (5.12.2
eqn (4.10.2
Theorem 2.10 (Neumann’s addition theorem)
J
0kr
2≡=
θ
θ
n=?θ
−1≡
n
J
nkb J
nkr
1≡e
in
1−
(2.14)
=
θ
θ
n=0

n−1≡
n
J
nkb J
nkr
1≡cosn
1− (2.15)
where
0=1and
n=2forn>0.
Note that (2.14
1=.
P
O
2
O
1
r
1
r
2
b
θ
2
θ
1
β
Fig. 2.1. Configuration for two-dimensional addition theorems.

2.5 Addition theorems 37
First proofThe Parseval integral forJ
0kr
2≡gives
J
0kr
2≡=
1
2

−
e
ikr
2cos
d =
1
2

−
e
ikr
2cos −
2≡
d
=
1
2

−
e
ikr
1cos −
1≡
e
ikbcos −
d (2.16)
Now, from the generating function for Bessel functions [1298, p. 14],
exp
→
1
2
wt−t
−1
≡
∼
=

n=?
t
n
J
nw
with the substitutiont=ie
i
, we obtain the Jacobi expansion [1298, p. 22]
e
iwcos
=

n=?
i
n
J
nwe
in
(2.17)
Use this expansion twice in (2.16
over using

−
e
im
d =2
0m (2.18)
to complete the proof.
Before generalising Neumann’s theorem, we give an alternative proof. This is
based on properties of the differential operators≥
±
m
, defined by Definition 2.6.
Second proof of Neumann’s addition theoremSinceJ
0kr
2≡is a solution of the
two-dimensional Helmholtz equation, it must have an expansion of the form
J
0kr
2≡=

n=?
C
nJ
nkr
1≡e
in
1
→ (2.19)
for some coefficientsC
n. To find these coefficients, we apply the operator≥
+
m
r
2≡
to (2.19 m≥0, keepingbfixed; this gives
J
mkr
2≡e
im
2
=

n=?
C
n≥
+
m
r
2≡

J
nkr
1≡e
in
1

Since the Cartesian axes are parallel, we have≥
+
n
r
2≡=≥
+
n
r
1≡. Hence, using
Theorem 2.9, we obtain
J
mkr
2≡e
im
2
=

n=?
C
nJ
m+nkr
1≡e
im+n≡∓
1
→ (2.20)
form=0→1→2Now, letr
1→0, so thatr
2→band
2→; sinceJ
m0≡=
0m,
we obtain
J
mkbe
im
=C
−m→ (2.21)

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deserted, but this I know not with any precision. I think we may go
from hence in 4 or 5 days. Matters are coming to a Crisis—therefore
you may be sure I shall not stay long lest we should be stopped and
not suffered to go from this city, but I expect an introduction to
Cazalès, La Fayette, and some others.
Lausanne has the honour of containing Lady Webster
[174]
before this
time. People are apt to spoil her. I desire you will not, because it
gives me a great deal of trouble to set her right afterwards. Milady
writes to her. She and her daughters are well and entertained. We
are just come from the Comédie Francoise.
Yours ever,
S.
I shall not write again before our departure.

572.
Lord Sheffield to Edward Gibbon.
Paris, July 8th, 1791.
The system of getting rid of the King will not do. The night before
last the principal men of the Committees, the Chiefs of the National
Assembly, about 100 in number, met, and after a considerable
debate, they resolved that the Constitution should be maintained,
and that the person of the King is inviolable. Only two of them,
Messrs. Dumont and Petier, were of a different opinion, and they
argued for a tryal. La Clos,
[175]
the friend of the Duke of Orleans, the
night preceding the Jacobins, vigorously insisted on a Tyral and a
Regency, but it is known that a great majority of the National
Assembly wish to restore the King and to put him in the situation he
held before his flight, and to give him a Council. The Queen not
being a part of the Constitution, they do not intend to take any
notice of her; much management is necessary to keep the people
quiet, and the Chiefs dare not as yet bring forward moderate
measures. The National Guard wish to get rid of the King. The
resolution I mention in the beginning of this letter is intended to be
kept secret as much as possible for the present. The measure will be
carried with some difficulty. I have scarce a moment. On Monday we
attend the Apotheosis of Voltaire.
[176]
On Tuesday next we go to the
National Assembly for the last time (having the President's Box), and
on Wednesday we talk of setting out for La Suisse, but I must stop a
day or two at Dijon and one or two perhaps at Besançon. We
suppose the Websters lost, because we have not heard from them.
Yours,
S.

LORD SHEFFIELD
AT THE
JACOBINS.
573.
Lord Sheffield to Edward Gibbon.
Paris, 13th July, 1791.
We are going at 8 o'clock this morning to the
Assembly, lest we should be too late for places to
hear the report and debate on the King's Flight.
[177]
It is not likely to finish in one day, but we are not
disposed to stay more than 2 or 3 days longer on any account. It will
be proper to remain here a day or two just to see whether they
really will cut one another's throats. I do not expect it. It does not
seem to be their genius to do more than Fishwomen, to scratch and
tear one or two to pieces in a cattish fury. Since my last the
Democratic enragé seems to have gained ground. I have been
placed high on a chair next the President at the Jacobins, having
been introduced by Noailles as a good English Patriot amidst much
applaudissement. Brissot de Warville
[178]
made (as on m'a dit) the
greatest speech that ever was heard. It was well calculated to
inflame Frenchmen, but he forgot to use any argument, and utterly
omitted to shew that the King had committed any crime against the
Law, and finished by moving that the King should be tried. The word
enragé does not half describe a French Democrate. The modern men
think the speech has made a great impression, and think matters are
not in so good train for the King as they were. I have not a moment
to say more than that I have seen men and things to great
advantages, and I shall write again.
Yours,
S.

574.
The Hon. Maria Holroyd to Edward Gibbon.
Berne, Oct. 7th, 1791.
The truth is, that I attempted to write to you, the day we arrived
here, & found myself unequal to a longer account of things, than
just to say—we are eighteen Leagues from Lausanne, & I have made
the family lift up their hands & eyes in astonishment, by wishing to
walk back that distance. I ought to express my Gratitude for all the
kindness and attentions we have met with, during our stay in
Switzerland, & if I was less sensible of it, I could compose a fine
speech—but I can only say, I can feel, & I hope you will never find
me ungrateful. Tuesday we slept at Avenches & arrived here at two
o'clock on Wednesday. If you wish to know how we amused
ourselves on the Road, I will tell you, by Meditation & Silence. If you
wish to know what was the Subject of our Meditations I will answer
for myself—Lausanne. Indeed, my Thoughts have not quitted that
place, for five minutes, & I begin to wonder, whether I shall ever
think of any thing else. Our horse, that had one Wooden & one
broken Leg, fell down, & rather damaged a third Leg—so that, as
Papa thought if any accident should happen to the fourth, we might
find some difficulty in proceeding on our journey, & being rather
indignant at their slow method of moving, he has dismissed them,
rather too precipitately, as we are now uncertain whether we shall
leave this Town to-morrow or a fortnight hence. No horses to be had
at present.
Yesterday we went to the Lac de Thun—the day was very fine, & we
crossed the Lake to Mr. Fischer's house, where we found his Lady
and Mother. We stopped at Mr. de Mulhinen's house in our return—&
saw Mad
e
. de Mulhinen, who is a very pleasing Woman. Papa was so

much pleased with the Lake that he lamented very much that he had
not persuaded you & Severy to come with us there & stay two or
three days. Perhaps you might have been inexorable, but I wish the
other part of the scheme had been thought of a little sooner. I liked
the expedition upon the lake very well, but it was not the Lake of
Geneva, nor was the Boat St. J. Legard's; & yet, as there was a Lake
& a Boat, there was resemblance enough, for me to make
comparisons, to the disadvantage of the present time. Mr. Coxe gave
Papa a Letter to Mr. Wyttenbach; he has been here, & walked about
the Town with us on Wednesday evening, & has made me very
happy by promising to send me a Collection of Alpine Plants. He is to
take us this morning to shew us the World. Papa is gone with Mr.
Fischer on horseback to see Farms. After dinner we are to go to the
library, & to-morrow, if we can get horses, we shall go to Bienne.
Mon
r
. Fredennick was here this morning, & every body seems to try
who shall pay most attention. I wish they would try & be
disagreeable, to make me rejoice at being on my return to England.
Mon
r
. Wyttenbach is come & prevents me adding any more, than to
assure you that I am your ever obliged & affec.
M. T. HçlêçyÇ.

SAFE IN THE
LAND OF LIBERTY
575.
The Hon. Maria Holroyd to Edward Gibbon.
Strasbourgh, Thursday, Oct. 13th, '91.
I felt a strong inclination to write from Basle, but as
you said, Berne or Basle & not Berne & Basle, I
was afraid of being troublesome. However I take
the first moment of my arrival at the next station from whence you
desired to hear from me, to tell you we are safe in the Land of
Liberty, where the People may sing Ça ira all day & all night, if they
like it. Friday, the day I wrote from Berne, we went to Wyttenbach's
house, to see his curiosities—& he has made me Wild again about
Botany, by giving me a Collection of Alpine Plants—so that now,
instead of admiring Nature in general, I have no eyes but for Weeds,
& I have made a considerable Collection in my Journey from
Lausanne here. The Advoyer came before dinner, & from his suit of
Black Powdered Wig & Gold headed cane, I began to be afraid I was
ill, & that the Physician was come to give his Opinion. Mr. Fischer
dined with us, Mr. Fredennick came after dinner, and they went with
us to the Library & walked upon the ramparts. In a part of the Ditch
we saw two of the Sovereign Lords of the Pays de Vaud. Their
Excellencies were very quiet and rather Rheumatic, but there were
two young ones very frisky and playful.
Saturday. We were rather unfortunate in a very rainy Day—& in one
of the Springs of the Carriage breaking near Arberg, which delayed
us some time. If it had been a fine day, we should have been very
disagreeable, but as it was impossible to go to the Island of St.
Pierre
[179]
that day, we made up our minds very tolerably. Sunday.
The Weather was very favorable, & we went upon the Island, wrote
our names in Rousseau's Bed chamber, returned to Bienne to dinner,

STRANGE CHARM
IN
SWITZERLAND.
& went to Moutier the same day. Mama had an opportunity of
shewing her heroism—for the last three Leagues we performed by
moonlight, & Coxe describes the Road as so narrow, that one Wheel
rubs against the Rocks & the other hangs over the Precipice—though
this description is poetical, yet there is some foundation for it. We
regretted passing thro' such picturesque scenes in the dark, but the
next day the Country we passed thro' from Moutier to Basle was
exactly the same. I thought after the Tour to the Glaciers, that I
should think nothing equal to that part of the World, but the Valley
of Munster pleased me more than anything I have seen. Tuesday, we
saw the Gardens of Arlesheim, the Library & the dance of death. Mr.
Ochs was the only one of Mr. Levade's friends who was at Basle—the
others were both in France. The higgledy-piggledy Party came to
Basle on Tuesday & were very much in our way on the Road. Papa
determined to go to Strasbourgh on the french side of the River, as
the Horses on the other side are quite knocked up by transporting
Aristocrates to Coblentz. The Craven family with their Guardians,
Lord Molyneux and Mr. Nott, took all the horses at the first Post, &
tho' they left Basle two hours before us, when we got to St. Louis
we were obliged to wait an hour and a half, for the return of the
horses—during which time we amused ourselves by walking to
Huningue—which I was very glad of, as it gave me a better Idea of
Scarps & Counter-Scarps, Ravelines & Bastions, than I should have
had without it.
This day's journey was rather long. We left Basle at
eight o'clock & we arrived at Krafft, where we
slept, at ½ past eleven—quite in despair at the dirt
of the French Inns; having met with such excellent
ones in Switzerland, we had quite forgot what a bad Inn was. We
are just arrived here to breakfast, & expect to meet with some
information as to the superiority of Navigation over Land carriage.
Papa was so much out of humour with the delay, occasioned by
want of horses upon the road yesterday, that he is very much
inclined to take a Boat here. The people at Lausanne, judging I
suppose by themselves, assured me I should forget that place by the

time I got to Basle. I am at Strasburgh, which is still farther; and I
can say from Experience, which is the only thing that ever convinces
me, that notwithstanding the variety of Scenes that I have passed
through, & the amusement I have found on the road, I still regret
the Terrace & the Pavillion. I do not know what strange Charm there
is in Switzerland that makes everybody desirous of returning there;
you know I did not go there with any prejudice in its favor. As to
Mama, she owes it such a Spite for fascinating you, that she will
never do it common Justice, till the Democrates have obliged you to
be content with our little Island; then perhaps her obligations to
them will change her sentiments.
Indeed I am ashamed of myself, to have taken up so much of your
time. You will, I am afraid, repent of our Engagement, & think that
reading my Letters is even more tremendous than answering them. I
live in hopes, that we shall hear something of, or from, you at
Coblentz. Remember us all, but me in particular, to every one of the
Severy family. Et dite à ma chère Angletine que je pense bien
souvent d'elle & du dernier jour que nous avons passées ensemble. I
dread exposing even that short sentence to your criticizing eye, but I
wish to shew her that, in promising to write to her, I have
undertaken what I am very unequal to perform, in order to keep up
some remembrance of me. I cannot bear the thoughts of being
forgot by those I love. Mama desires to be most affectionately
remembered to you.
Believe me,
Ever sincerely & affec^y yours,
Maêia T. HçlêçyÇ.
I just find I am too late for the Post to-day, & that my Letter must
wait till Saturday. To-morrow we stay here. Papa is happy in the Idea
of seeing the Troops exercise.

576.
The Hon. Maria Holroyd to Edward Gibbon.
Coblentz, Oct. 21st, 1791.
Our Adventures since I wrote from Strasbourgh have been very
numerous, & if every body had been equally disposed with myself to
be entertained with them, they would have lost much of their
unpleasant circumstances. Papa had determined to go from
Strasburgh to Manheim by Rastadt; but the Inn keeper advised us to
go on the other side of the Rhine, as we should find the Inns all full
in Germany & the Post horses very bad. The Rain was incessant all
day & had continued for two days before. We found the Roads very
bad & lost our way in a large forest; quite dark; amidst many
ejaculations from Mama. When we at last arrived at Girmenheim,
where we were to sleep, we found the Inn quite full. A Commission
was there from Manheim to keep the Rhine in order, who has heard
so much lately of Liberty on both sides, that he had a mind to make
the experiment, & has strayed over the neighbouring meadows,
unmindful of the excellent Caution given to a Brother River
—"Thames, ever while you live, keep between your banks." We were
put into a small room, where a Company had just finished supper.
Travellers are not often, I imagine, so unfortunate as to go that
road, if I may judge from the astonishment and, I hope, admiration
our Appearance caused. The Doors were opened and the Room was
lined with Spectators, who gazed at us in silence for near a quarter
of an hour—more to my amusement than Mama's. There was only
one Room where we could sleep—& we all arranged ourselves in
three Beds, after having quieted some delicate scruples of Papa's,
who proposed sleeping in the Coach—however by putting out the
Candles nobody found it necessary to blush.

COBLENTZ AND
WHITE
COCKADES.
We left this charming place very early, breakfasted at Spire and
arrived at Manheim early enough to see all the Lyons before dinner. I
was much entertained with the Gallery of Pictures in the Elector's
Palace. It was much superior to anything I had seen. The Library is
very handsome. Papa went to the Play in the evening & made an
acquaintance there, who he brought home with him, & talked
Commerce and Agriculture, till near one in the morning. The next
day we went to Mayence, & the day after saw the Castle, the
Provost's house, the Cathedral, &c., and left Mayence at two o'clock
in a very tolerable Boat. But the Wind was quite contrary, & it was
very late when we arrived at Bingen. Mama did not take a fancy to
Navigation in the least. For my part I enjoyed it very much, as the
Banks of the Rhine, particularly from Bingen to Coblentz, are very
picturesque. The great number of Castles made me imagine myself
in the Age of Chivalry, & I almost persuaded myself I was a
distressed Damsel carried away against my Will. The next thing, of
course, was to expect a brave Knight to set me free, but as none
made their appearance, I was obliged to quit my romantic Ideas, &
my Castles in the Air, of which I had plenty, as well in my head, as
around me. In plain English, I was much pleased with the day's
journey, & Mama was pretty well reconciled to seeing Water all
round her, which was at first a great grievance.
Our famous Adventures begin here. We arrived at
Coblentz
[180]
at five o'clock last Wednesday, &
found every Inn in the Town full of Panaches
blancs. After staying three hours in the Boat, with
difficulty Papa found one Room, with one Bed, without Curtains & no
other furniture of any kind in it. We preferred this to sleeping in the
Boat, the only alternative, & accordingly we females slept on
Mattresses upon the ground. As there were no curtains it was
impossible to admit Papa of the Party, & he remained all Night in the
Boat. The Account that was brought us of the Room we were to
sleep in, was that between forty and fifty Officers were in two
Rooms at each end of ours, which opened with Folding Doors. Upon
a nearer enquiry, the number was reduced between 10 & 20—but

they are tolerably quiet, considering they are Frenchmen. Yesterday
was passed enquiring for Lodgings, & by the help of the Duc de
Guiche,
[181]
the Woman of the house was prevailed upon to give us
three Garrets, perfectly unfurnished—but this we considered as
charming accommodation compared to the higgledy piggledy Style
we had been accustomed to—but the Ground is still our Bed.
Papa has found out a great deal of amusement for himself. He was
presented yesterday to the Elector,
[182]
Monsieur,
[183]
Madame, the
Comte d'Artois, the P. of Condé and his Son; to-day he dined at a
very large dinner at the Prince of Nassau's
[184]
—& is now at the Play
in Mad
e
. de Nassau's Box, who was very desirous of our Company,
but Mama is not fond of violent measures. The Comte de
Romanzov
[185]
is here, Ambassador to the Princes from the Empress
Cat. The Bishop of Arras
[186]
is not here; but the Duc de Guiche is
every thing that is delightful, & Papa has not been at a loss for the
Bishop. Our amusements may be mentioned in very few words. We
have seen the citadel. We go from here early to-morrow morning—in
our Boat. The Weather is very unfavorable to us. Only that you
might justly make the Observation, "If you are sensible of your fault,
why do you continue to offend?" I would apologize for the length of
my Letter. But while you allow me to write to you, I do not think I
have quite left Lausanne—& I never know how to leave off.
Distribute our Love and comp
ts
properly.
Believe me
Ever affec. yours,
Maêia T. HçlêçyÇ.
I forgot to say we found our Letters here. Mama desires her Love to
Severy & many thanks for his Letter. I have taken a great deal of
pains to persuade her to write to him; but she has not resolution
enough to take up her pen. I must whisper to you we were
disappointed at not hearing a few lines from you.

THE SIGHTS OF
BRUSSELS.
577.
The Hon. Maria Holroyd to Edward Gibbon.
Brussels, October 29th, 1791.
It is probable that this my fourth Letter may remain unopened in
your Pocket, but I shall leave that to Fate, & only think of convincing
you, that I still remember Lausanne & my promise. I like to let
People know how unreasonable I am, & therefore I will tell you I had
faint hopes of finding a few Lines here, either written or dictated by
you. I frequently ask Mama, do you think they are talking of us at
Lausanne? & she generally answers—I daresay not; so I should have
had a great deal of satisfaction in shewing her, that you thought so
much of us as to make a violent effort to tell us so. We have
proceeded on our journey with great success from Coblentz to
Brussells, & to-morrow go to Antwerp. We arrived here on Thursday
from Louvain—the Road was so bad & the Post-horses moved in
such a Swiss manner, that we were four days coming from Cologne.
The first day we slept at Juliers, the second at Liège, the third at
Louvain, & the fourth (as I had the honour of telling you) we came
to Brussells, & fortunately arrived at 'L'Imperatrice,' the only Hotel
where there was a single Room unoccupied, just as the Princesse de
Salms was moving off—& took possession of her apartments with
great satisfaction, as we expected Coblenz accomodations.
We stayed but one night at Cologne, as the
Maréchal de Castries was not there, & the Town
possessed no other Charms to tempt us to stay, for
it is the most dismal place I have seen. The Maréchal is here, & Papa
has had a long conference with him. Luckily for us, Papa has neither
met with a Quarter Master nor a Commercial man, nor a Farmer
here, so we have seen a great deal and been very much amused.

We saw the Palace of the Archduchess, a league out of Town,
yesterday—& it is fitted up with more Taste than any thing we have
seen in our travels. The rest of our time has been spent in Churches,
the Arsenal, & some good Collections of Pictures. I have not time to
be prolix in my narration, which you will perhaps not be sorry for, as
you are not as fond of a long letter as I am. Mama is pretty well, but
will not be sorry to find herself at her own fire side again.
Remember us to those who remember our existence—you will not
have much trouble, for I suppose you will only deliver the message
to yourself—I suspect nobody else of thinking of us. Louisa desires I
will not forget her best Comp
ts
to M. Mentrond; she does not choose
to suppose he can forget her. I expect to hear a great deal of Mrs.
Wood—or if any body else has supplied her place in your heart.
When you do write, if such an unlikely event should ever take place,
pray tell me something of everybody; I shall like to see the names of
those I was acquainted with, & while I read y
r
letter, shall fancy
myself at Lausanne.
Believe me,
Ever sincerely y
rs
,
Maêia T. HçlêçyÇ.
I write in such haste that you must excuse faults of Style, Writing,
&c.

MILITARY
FORCES ON
FRENCH
FRONTIER.
578.
Lord Sheffield to Edward Gibbon.
Calais, 5th Nov., 1791.
After various and sundry embarrassments, here we are safe. The
pleasure of the visit being over and the sorrow of departure come
on, it naturally occurred in aid of my concern what a damned Fool I
was to undertake such an operation. To correct such cogitation, the
distractions of the Rhine were some relief. The state of its
neighbourhood is at this time very interesting. It was curious to find
the Princes and prime Nobility of France thankfull to be allowed to
exist on a small angle formed by the Moselle and the Rhine. An army
of officers, but not a common man. For the sake of visiting the
Garrisons of Alsace, I went the whole length of that Province.
Levade's letters were of essential service to me. I thankfully wrote to
him when I had proceeded far down the Rhine. Probably I furnished
some details, with which he probably has furnished you. I know not
whether I mentioned that Huninge and Brisach are in good
condition. An incompleat Regiment of two battalions in each place
and some Dragoons, not sufficient garrisons, but some cantoned
Troops might be thrown in if required.
The Regulars at Strasburgh are 6760, including
1100 Horse and 1300 artillery. They say they have
some 7000 National Guards, I doubt it. They are
the best I have seen; and yet they are very poor
stuff for Soldiers, and many are not cloathed. I
dined with the Colonel of Carbineers, and saw the finest regiment of
France in detail, and also a Swiss Regiment (Viguier), one of the
best. Some of the regiments are not more soldier-like than the
National Guards. The Democrates say that the officers, being

Aristocrates, neglect the men on purpose, and wish the regiments to
be ruined; 40 officers had quitted one regiment in Alsace. In short,
only seven remained with it, including the Colonel. The Park of
Artillery at Strasburgh seems very compleat. It is the second in the
Kingdom. A Train is ready for 40 Battalions. The Province of Alsace is
at least half Aristocrate. I passed the evening and supped at the
Mayor's while at Strasburgh. He and his Lady (a clever dame)
thorough Republicans. Observe the Regulars in the two departments
of the Rhine are commanded by a German,
[187]
and the National
Guards by a Lt.-Colonel who is a Livonian. Mirabeau's Corps seems a
miserable collection; several deserters came from it while I was at
Strasburgh. Finally I flattered myself there would be a compleat
Brouillerie between the regulars and the National Guards before I
left that place. The officers of the latter are naturally disposed to be
very absurd. The double pay of these troops soldès is likely to have
an excellent effect on the Troupes de ligne. I found about 100
French officers at Manterin. Towards 900 at Worms. Bouillé and a
certain number are at Mayence.
Including upwards of 900 of the Gardes du Corps and near 20
Generals, there are 2500 Officers at Coblentz. I was very graciously
received by the Princes. They give a supper every night, where I had
the amusement of being introduced to Marshal Broglie,
[188]
&c., &c.
The Prince of Condé and Duc de Bourbon were there on a visit.
Romanzov has brought credentials and two millions of livres French
from the Empress; Calonne is at Baths not far distant for his health;
Burke's son had been sometime at Coblentz and was gone to
England with Cazalés. Our King has written a very amicable letter to
the Princes promising neutrality. I went to Cologne to see the
Marshal de Castries. He was gone to Brussels, where I found him. At
every inn I found 20 or 30 French Officers, the road is covered with
them from Brussels to Coblentz. Nothing can be worse timed than
this desertion. It is a Phrenzy and was like wildfire. They would be
much better with their regiments and ready in the country to protect
Friends and to avail themselves of circumstances. The most sensible
of the French disapprove this migration. The officers leave their

Regiments without concert with the Princes, who have not lately
encouraged it. There seems to be no particular plan at present but
to wait events. I unfortunately missed the Abbé Maury passing to
Coblentz. I wished to know him. He is a Cardinal in petto. He is to
have the Arch-Bishop of Sens's Hat. The regular regiments were long
ago ordered to be completed to the war establishment, but on an
average they have not above half their complements, and on the
frontier of the Low Countries there is not an officer left except
Soldiers of Fortune.
L'Esprit de Revolution is not likely to flourish again for some time in
the Pais Bas. I found Imperial and Electoral troops in possession of
Liège, and new taxes laid to pay expenses, viz. on Dogs, Servants,
bachelors, &c. The Discontents in the Austrian Netherlands are not
likely to be of much consequence, a great part of the country was
miserably duped and the whole thrown into such an execrable state,
that none but the lowest of the creation can wish for another
experiment at a Revolution. The leading party among the enemies of
the House of Austria being Clergy and Aristocrates cannot coalesce
easily with the Democrates of France. All the Provinces except
Brabant are content. Many say a counter-revolution in France is
impossible, because the Mass of the People are of one mind. Not
near so much as the Austrian Netherlands were. There almost every
man was a Patriot, yet the moment an army appeared there was not
a struggle. The different extent of country, &c., prevent a correct
comparison.
However, it may be observed that France has not a neighbour that is
not unfriendly to the Revolution. I have the worst opinion of the
French Army. The National Guard behaved execrably at Nancy,
where alone they have been tried. We indeed were told the contrary.
Some Swiss officers (Democrates) who acted against the regiment of
Chateau Vieux
[189]
have given me details. You may be sure that the
Regulars and National Guards will not agree. I am satisfied that of
those officers who remain with the Regiments, almost all except
soldiers of fortune are aristocrates. The soldiers do not desert, but
they say they will go to Paris. They will enter into the Gardes Nat.

THE MEETING AT
PILNITZ.
soldès, and they go where they please; nobody can stop them. Yet
with such a King the situation of the Aristocrates is very difficult.
Divisions will naturally take place, the Kingdom is en traine to be
torn to pieces. A foreign army on the Frontier or advancing to Paris
might unite them, therefore it may be better to wait events. At the
same time I have not a notion that a French army would fight under
its present circumstances.
It was a comfort to see the excellent Bohemian and
Hungarian Infantry in the Austrian Netherlands.
They are in fine order. The Treaty of the 23rd of
July last between the Emperor and the King of Prussia has been well
concealed.
[190]
It is defensive. The supposition is that Prussia is
dissatisfied with England. If Russia should accede to the Treaty
(which is not thought unlikely), we shall be compleatly left in the
lurch.
Maria has been alert and well-disposed to your correspondence. She
seemed pleased with the office, but she will expect an answer. She
has saved me from writing sooner. From Brussels we went to
Antwerp, Ghent, Bruges, Ostende, and Dunkirk to this place. Mi Ladi
continues the same. I should have stopped more than three days at
Brussels if I had not been afraid of the division of Mrs. Maynard. Say
everything kind for us to the de Severy family. If Mi Ladi does not
reply soon to the Fils, I shall. I found letters at Brussels by which I
learn that your £2000 is accepted by the Navigation Society, that Mr.
Taylor has found a mortgage for £5000 in Yorkshire. He says
somewhat of its being more convenient if the money is not paid till a
little time hence, and I also learn that my worldly affairs, and the
Navigation, have gone on as badly as might be expected from my
absence.
Remember us to Mrs. Grevers.

579.
Lord Sheffield to Edward Gibbon.
[Incomplete in original.]
The increase of Mi Ladi's woman, and apprehensions thereon, made
it necessary to shorten my visits. You have heard of the little
accident on board the Packet. You know dear Puff has a great dislike
to cats. About midnight, midway between France and England, an
hideous noise like that of a cat in the act of being strangled was
heard. Puff barked and was furious. I looked out of my den, and
beheld it was an human kitten that proceeded from Mrs. Maynard
who was prostrate on the floor. My Lady also incumbent there. Maria
contemplative and Louisa astonished. Not a creature on board the
Packet but ourselves and the crew. We never know what we may
come to, and above all we should not have guessed that Mi Ladi was
to become a mid-wife. The mother and child could not have been
better, (and have continued so,) if all the obstetric Faculty of Paris
and London had attended. The mother was so well that she
expressed the greatest anxiety to go with us the day following above
80 miles across the country to this place. We left her in good
lodgings and in good care. The want of her prevented the Ladies
from passing two days at Lord Guilford's. We found two letters from
him at Dover and a dozen messages. I went and had a pleasant
dinner with him, and returned at night to the Ladies.
I must now come to the unpleasant part, your business.
Immediately on my return I wrote to Taylor about the £5000
mortgage. I have a letter full of disappointment. The person to be
paid off has accepted low interest. He complains of being frequently
thus treated. Don't bother yourself. I still hope soon to settle the
business.

I wrote you a long letter from Calais.

DISTRESSES
UPON THE SEA.
580.
The Hon. Maria Holroyd to Edward Gibbon.
Sheffield Place, Nov. 13th, 1791.
It is with a mixture of satisfaction & regret, that I complete my part
of our engagement in writing this Letter. I find great pleasure in
being returned to dear, precious Sheffield, & in telling you so,
because I am sure you will be glad to hear we are safe and well; but
writing to you the last letter is like a second taking leave, & tho' I
have been near six weeks upon the journey from Lausanne, that
moment is still as fresh in my memory as it was the next day, & the
recollection of it as unpleasing. If I dared I would ask to be allowed
to tell you we were alive, now & then when Papa & Mama were in
an idle mood, & to be allowed to hope for an answer once in two or
three years, in your own hand or not, as you thought proper or
found convenient.
I was very glad Papa wrote a long letter from
Calais, as his information about things in general
would be more interesting to you than mine, as I
suppose he gave you a full account of our route from Brussells to
Calais. I will only tell you that I was very much amused at Antwerp,
& that I catch myself, now & then, making believe to know a good
Picture from a bad one—having seen so many excellent ones lately.
We left Calais last Sunday at 9 in the morning, having waited three
days for a favorable wind, & at last in despair, set off in a perfect
Calm, which prolonged our passage 24 hours—which was
uncommonly tedious, & very bad luck, as our Passage from Brighton
was as tiresome in coming over. However to pass the time, or to
diversify the amusements of the Passage, which from the sickness of
the Company would have preserved some degree of sameness, Mrs.

Maynard with Mama's assistance, half way between Calais & Dover,
presented the cabbin boy with a Sea Nymph; which with its Mother
are now as well as can be expected, at Cap
n
Sampson's house at
Dover. I have heard that Sailors, when they come from a voyage,
find great pleasure in talking of & recollecting Toils & Dangers past—
such is our case; for we have laughed very heartily here, at our
Distresses upon the Sea—which certainly at the time was no
laughing matter.
We found the inhabitants of S. P. in excellent preservation & quite
rejoyced to see us returned—most of them, when we left England,
being convinced we should be all massacred by the National
Assembly in a very short time. I hope the quiet life of this place will
soon restore Mama's health & spirits, who desires her kindest
remembrances to you & those we love at Lausanne. Do not let
anybody forget us, for we forget nobody.
Ever much y
rs
,
M. T. HçlêçyÇ.

581.
Lord Sheffield to Edward Gibbon.
Sheffield Place, 13th Dec., 1791.
My Lady is getting quite well. Bratts very fond of their Swiss Tour. I
have passed a week very pleasantly in London. King apparently quite
well. Lally a great favourite with Lord Loughborough. He assisted at
a copious dinner at Batt's, and said he never enjoyed one more
interesting. He saw Lord Guilford on his passage through London,
who was well pleased with him, as also is Douglas. Tell Mr. Trevor
good care will be taken of him. La Comtesse de Lally seems to
preserve a strict incognito. Lady Loughborough visits her. I can only
collect that she doth not appear to like to go out. It is said, she has
not confined her practices to Lally. Introduced him to Burke, who
says the said Lally persists in his errors, and justifies all his
mischievous conduct in the beginning, and the said Burke is as
ridiculous and as absurd as may be imagined.
[191]
I have been
presented to Cazalés (who resides with Burke), but I had not an
opportunity of seeing anything of him. England has supplied and has
been paid for 36,000 stand of arms for the Emigrants. The Corps
Diplomatique at London says that Spain has sent 5 millions of livres
to the Princes, and Portugal 4 millions—Berlin 2 millions.
I have expectation of seeing Batt and Lally here at Christmas. Tell
Levade I have only just learned where to find his son-in-law. I hope
also to see him here. People begin to talk of 3½ per Cent. for
money.

MARRIAGE,
BATTLE, FIRE,
AND SCANDAL.
582.
Lord Sheffield to Edward Gibbon.
Sheffield Place, 25th Dec., 1791.
I am obliged to write to you, otherwise it would not be proper,
because we are determined to starve you into a more decent
deportment; not a fragment from you except the medley to Maria.
She has been so zealous in your service, that she deserved more
notice. The obligation to write is that a pipe of Madeira (which has
travelled and is very good) is ordered to set out for Lausanne by the
same consignment and way as the last, ergo you must give notice to
your correspondent at Basle, &c.
Nothing extraordinary has occurred in this family
since my last. My Lady is better. We expect Batt to-
morrow, probably Lally, and also Mr. Levade's son-
in-law, with whom I have corresponded. The
Duchess of York
[192]
almost smothered the French Revolution in this
country; Lord Cornwallis's operations
[193]
almost suppressed the
Duchess of York, and I daresay the Duke of Richmond's fire
[194]
has
afforded some relief to Lord Cornwallis. I should be happy to furnish
you with as much scandal as possible, but I know of no event of the
kind worth record, unless Lady Belmore's
[195]
trip to the Continent
incontinently with Lord Ancrum should be deemed so, and Lady
Tyrconnell's
[196]
Flight to Glamis Castle with Lord Strathmore. These
amiable women have left behind them grown daughters. May the
Mamas——
I have secured for you the famous Shakespear; Boydel
[197]
is
satisfied that I subscribed for two setts. It is by far the finest book
ever printed. I have your first number and 5 large and 5 small prints

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