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INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERAL
J. Phys. A: Math. Gen. 34 (2001) 6249–6257 PII: S0305-4470(01)19778-7
Elastic and sound orthonormal beams and localized
fields in linear mediums: I. Basic equations
George N Borzdov
Department of Theoretical Physics, Belarusian State University, Fr. Skaryny avenue 4, 220050
Minsk, Belarus
Received 22 November 2000, in final form 13 June 2001
Published 3 August 2001
Online at stacks.iop.org/JPhysA/34/6249
Abstract
Basic definitions and general relations for elastic and sound fields, defined
by a given set of orthonormal functions on a real manifold, are presented.
The proposed mathematical formalism makes it possible to obtain families of
orthonormal beams and localized fields in both isotropic and anisotropic linear
elastic mediums as well as the similar sound fields in an ideal liquid. All
these fields are described as superpositions of plane waves whose intensities
and phases are specified by a set of orthonormal scalar functions on a two- or
three-dimensional manifold. By way of illustration, the fields defined by the
spherical harmonics are considered.
PACS numbers: 62.30.+d, 43.20.+g, 02.30.Nw
1. Introduction
In recent decades, electromagnetic, elastic and sound beams have been studied extensively
and many interesting solutions of the wave equation, such as fractional solutions [1], non-
diffracting—Bessel and Bessel–Gauss—beams [2], and various localized fields (see, for
example, [3–12] and references therein), have been suggested. At the beginning of the 1980s,
Brittingham [3] proposed the problem of searching for specific electromagnetic waves—focus
wave modes (FWMs)—having a three-dimensional pulse structure, being non-dispersive for
all time, and moving at light velocity in straight lines. A number of packet-like solutions have
been presented [3–7]. Although the FWM has an infinite total energy, a superposition of the
FWMs can produce finite-energy pulses that exhibit extended ranges of localization [4–7]. It
was shown by Ziolkowski et al [5] that such pulses can be excited from finite apertures. In
1985, Wu [8] introduced a conception of electromagnetic missiles moving at light velocity and
having a very slow rate of decrease with distance. Recently, Kiselev and Petel [9] presented
packet-like solutions with Gaussian localization in both longitudinal and transverse directions.
The Fourier transform and plane-wave expansions play a very important role in the analysis
of localized fields. By using the Fourier transform, a method for obtaining separable and
0305-4470/01/326249+09$30.00 © 2001 IOP Publishing Ltd Printed in the UK 6249
6250 G N Borzdov
non-separable, localized solutions of constant coefficient homogeneous partial differential
equations was developed by Donnelly and Ziolkowski [6]. In the frame of this approach,
the evanescent fields and the causality of the FWMs have been studied comprehensively
elsewhere [7]. In particular, it was shown that the source-free FWMs are composed solely
of backward- and forward-propagating homogeneous plane waves.
By using expansions in plane waves, we have introduced [10–12] a specific type of linear
field—beams defined by a set of orthonormal scalar functions on a two-dimensional or three-
dimensional manifold (beam manifold B). The proposed approach enables one to obtain a
set of orthonormal beams, normalized to either the energy flux through a given plane (beams
with two-dimensional B) or the total energy transmitted through this plane (beams with three-
dimensional B), and various families of localized fields: three-dimensional standing waves,
moving and evolving whirls, etc. This can be applied to any linear field, such as electromagnetic
waves in free space, isotropic, anisotropic and bianisotropic mediums, elastic waves in isotropic
and anisotropic mediums, sound waves, weak gravitational waves, etc.
On this basis, we have found [10–12] unique families of exact solutions of the
homogeneous Maxwell equations for electromagnetic waves in isotropic mediums and free
space. The families of orthonormal beams [10, 11] form convenient functional bases
for more complex electromagnetic fields. This provides a means to generalize the free-
space techniques [13] for characterizing complex mediums and the covariant wave-splitting
technique [14] to the case of incident beams. The families of localized electromagnetic
fields (electromagnetic storms, whirls and tornadoes) [10–12] also possess very interesting
properties. A small (about several wavelengths) clearly defined core region with maximum
intensity of field oscillations is an inherent feature of these fields. For an electromagnetic
storm, the time average energy flux vector Sis identically zero at all points. For both whirls
and tornadoes, the radial SR, the azimuthal SA, and the normal (to a given plane) SNcylindrical
components of Sare independent of the azimuthal angle ψ, besides SR=SN≡0 for whirls.
As a result, whirls and tornadoes have circular and spiral energy flux lines, respectively.
The solutions, which describe electromagnetic whirls moving without dispersion with speed
0<V <c, finite-energy evolving electromagnetic whirls [10, 11], and weak gravitational
orthonormal beams and localized fields [12], have been found as well.
The mathematical formalism proposed in [11] can be applied to scalar, vector and tensor
plane-wave superpositions. However, the special features of its application to obtain sets of
exact solutions to one or another of the wave equations, as well as the properties of the resulting
solutions, substantially depend on the properties of the partial eigenwaves. The corresponding
mathematical framework for two types of transverse waves—vector (electromagnetic) and
tensor (weak gravitational)—has been presented in [11,12]. In this paper, to treat sound fields
in liquids and elastic fields in solids, this framework is augmented to include also scalar, vector
and tensor plane waves describing the sound pressure fields in liquids, the displacement vector
fields for longitudinal (quasi-longitudinal) and transverse (quasi-transverse) waves in isotropic
(anisotropic) solids, and the corresponding deformation and stress tensor fields.
The objectives of the current series of papers are twofold:
(1) We extend the proposed formalism to the cases of elastic waves in isotropic and anisotropic
mediums and sound waves in an ideal liquid.
(2) On this basis, we obtain unique solutions which describe orthonormal beams and localized
fields in an isotropic elastic medium and an ideal liquid. They illustrate both similarities
of and distinctions between scalar (sound waves), vector (radial displacement vector for
longitudinal elastic waves, meridional and azimuthal displacement vectors for transverse
elastic waves) and tensor (deformation and stress tensors) plane-wave superpositions
defined by the same set of functions, such as the spherical harmonics.
Orthonormal beams and localized fields: I 6251
Accordingly, in this paper basic definitions and general relations for the plane-wave
superpositions, defined by a given set of orthonormal functions on a real manifold, are presented
in section 2. In section 3, we discuss the properties of elastic and sound harmonic plane waves
(eigenwaves) and supply the relations that are necessary to apply the general theory to both
isotropic and anisotropic mediums. A mathematical formalism convenient for analysis of
orthonormal beams and localized fields is briefly outlined in section 4. Three subsequent
papers will deal with the superpositions of longitudinal (paper II) and transverse (paper III)
elastic plane waves in an isotropic medium, and sound plane waves in an ideal liquid (paper IV).
2. Fields defined by a set of orthonormal functions on a real manifold
To compose a field from eigenwaves in a linear medium, we must specify all eigenwave
properties: propagation directions, frequencies or wave numbers, polarizations, intensities
and phases. Of course, in the case of elastic waves in an anisotropic medium, eigenwave
polarizations are specified by the medium itself, and we have to set just propagation directions,
intensities and phases.
The fields (beams) defined by a set of orthonormal functions (un)on a two- or three-
dimensional real manifold Bucan be written as [11]
Wn(r,t) =B
exp {i[r·k(b) −ω(b)t]}un(b)ν(b)W(b) dB(1)
where beam manifold Bis a subset of Buwith non-vanishing values of function W=
ν(b)W(b). In such a manner intensities and phases of all eigenwaves are specified by the
same complex scalar function unsatisfying the condition
um|un≡Bu
u∗
m(b)un(b) dB=δmn (2)
where u∗
mis the complex conjugate function to um, and δmn is the Kronecker δfunction.
Amplitude function W=W(b) must be given in an explicit form for each specific field (see
section 3).
There are four key elements defining the properties of these fields: (1) functions (un),
(2) beam manifold B, (3) beam base, i.e., a set of eigenwaves forming the field and specified
by wavevectors k=k(b), angular frequencies ω=ω(b) and amplitudes W=W(b), and
finally (4) beam state function ν=ν(b). By setting these elements in various ways, one can
obtain a multitude of specific fields [10–12], among them orthonormal beams satisfying the
condition
Wm|Q|Wn≡σ0
W†
m(r,t)QWn(r,t)dσ0=NQδmn (3)
where σ0is either a two- or a three-dimensional manifold, Qis some Hermitian operator, and
W†
m(r,t)is the Hermitian conjugate of Wm(r,t);NQis the normalizing constant.
Time-harmonic beams with two-dimensional manifold Bcan be written as
Wn(r,t) =exp(−iωt ) B
exp[ir·k(b)]un(b)ν(b)W(b) dB.(4)
They become orthonormal, provided the following conditions are met [11]:
(1) σ0is a plane with unit normal q, passing through the point r=0.
(2) The tangential component
t(b) =k(b) −q[q·k(b)] (5)
of k(b) is real for all b∈B, and the mapping b→t(b) is one–one (injective).
6252 G N Borzdov
(3) B=Bu, and the function ν(b) is given by
ν(b) =1
2πNQJ(b)
g(b)W†(b)QW(b) .(6)
Here, J(b) =D(tj)/D (ξi)is the Jacobian determinant of the mapping b→t(b), calculated
in terms of the local coordinate systems (ξi,i=1,2)and (tj,j=1,2), and dB=
g(b)dξ1dξ2.
In the current series of papers, we illustrate the general theory by applying it to time-
harmonic fields given by
Ws
j(r,t) =exp(−iωt ) 2π
0
dϕθ2
θ1
exp[ir·k(θ, ϕ)]Ys
j(θ, ϕ)ν(θ , ϕ)W(θ , ϕ) sin θdθ. (7)
They are defined by the spherical harmonics
Ym
l(θ, ϕ) =NlmP|m|
l(cos θ)exp(imϕ) (8)
where
Nlm =(2l+1)(l −|m|)!
4π(l +|m|)!(9)
and Pm
l(cos θ)is the spherical Legendre function [15, 16]. For Ws
j(r,t)(7), the beam manifold
Bis the spherical zone (θ∈[θ1,θ
2] and ϕ∈[0,2π]) of the unit sphere Bu=S2, and
dB=sin θdθdϕ.
3. Eigenwave properties
To apply the presented general relations, it is necessary first to calculate parameters of
eigenwaves. In this section, we present the corresponding relations for elastic and sound
waves.
3.1. Elastic waves
A linear elastic medium is described by the Hooke law [17]
σij =cij lm
∂um
∂xl(10)
where σis the stress tensor, uis the displacement vector, (cij l m;i, j , l, m =1,2,3)are the
elastic modules, and summation over repeated indices is carried out from 1 to 3.
3.1.1. Wavevectors and amplitudes. For an eigenwave, the elastodynamics equation [17]
%∂2ui/∂ t2=∂σij /∂xjbecomes
(k·c·k)u=%ω2u(11)
where %is the medium density, and a·c·bdenotes the dyadic with the components
(a·c·b)im =ajcij lm bl. The wavevector surface is defined by the dispersion equation
|k·c·k−%ω21|=0 (12)
which is of sixth order in k. Here, 1is the unit dyadic, and |'|denotes the determinant of
a dyadic '. If the unit wave normal ˆ
k=k/k is given, equation (12) reduces to the bicubic
equation
v6
p−v4
p't+v2
p't−|'|=0 (13)
Orthonormal beams and localized fields: I 6253
where vpis the phase velocity (k=(ω/vp)ˆ
k), '=ˆ
k·c·ˆ
k/%,'is the adjoint tensor
(''='' =|'|1), and 'tis the trace of '.
The eigenwave amplitude uis given by [17]
u=χpχ='−v2
p1(14)
where pis an arbitrary vector. Ifχis a dyad, i.e. χ=0 and χ=cu⊗nu, the amplitude subspace
becomes two-dimensional, and uis an arbitrary vector normal to nu=pχ. To compose a
family of orthonormal beams, we shall use the six-dimensional eigenwave amplitude
W0=u
ff=σq=i(q·c·k)u=ieicij lm qjklum(15)
where kand uare specified by equations (12) and (14).
3.1.2. Wavevector surface parametrization by the tangential component tof k.Substituting
k=t+ξq(t·q=0) in equation (12), we obtain the sixth-order equation in ξ[17]:
ξ2A+ξB +C≡
6
n=1
anξn+|C|=0 (16)
where
a1=(CB)ta2=(BC +CA)t(17)
a3=|B|+(ABC +CBA +ABtCt−AtBC −BtCA −CtAB)t(18)
a4=(AC +BA)ta5=(AB)ta6=|A|(19)
A=q·c·qC=t·c·t−%ω21(20)
B=B1+B2B1=t·c·qB2=q·c·t.(21)
The roots (ξj,j=1,2,...,6)of this equation specify all six wavevectors kj=t+ξjq,
which have the same given tangential component t.
3.1.3. Amplitude orthogonality in a non-dissipative medium. Substituting kj=t+ξjqin
equations (11) and (15), we obtain
RWj=ξjWjWj=uj
fj(22)
where
R=−A−1B2−iA−1
i(B1A−1B2−C) −B1A−1.(23)
If the dyadics A, B1,B
2and Chave the properties A†=A, C†=C, B†
1=B2, the
block matrix R(23) satisfies the identity
R†=Q0RQ0Q0=0−i1
i10.(24)
Hence, at ξj= ξ∗
i, equation (22) results in the orthogonality relation W†
iQ0Wj≡i(f∗
i·uj−
u∗
i·fj)=0. This is true in a non-dissipative medium at real values of t, since cijl m has the
properties [17] cij lm =cjilm =cij ml =clmij =c∗
ij lm .
As in the case of time-harmonic electromagnetic beams [11], the condition Wn|Q|Wn=
NQwith Q=(ω/4)Q0normalizes the time average elastic beam energy flux NQthrough the
plane σ0:
Wn|Q|Wn=−1
4σ0
(v∗·f+v·f∗)dσ0=NQ(25)
6254 G N Borzdov
where the interrelation v=−iωubetween the velocity vand the displacement uhas been
taken into account.
3.2. Elastic beams in an isotropic medium
In an isotropic elastic medium, the Hooke law (10) becomes [17]
σ=λLdiv u1+2µLγ(26)
where λLand µLare the Lam´
e modules, γis the deformation tensor with the components
γij =1
2∂ui
∂xj
+∂uj
∂xi.(27)
We shall characterize elastic beams by the time average kinetic and elastic energy densities [17]
wK=1
4%ω2|u|2wE=1
4Re (σikγ∗
ik)(28)
and the time average energy flux density vector
S=ω
2Re (iσ∗u). (29)
Hence, for an eigenwave with wavevector k=kˆ
k, from equations (11), (15) and (26) it follows
that
γ=ik
2(ˆ
k⊗u+u⊗ˆ
k)(30)
σ=ik[λL(ˆ
k·u)1+µL(ˆ
k⊗u+u⊗ˆ
k)] (31)
f=ik[λL(ˆ
k·u)q+µL(u·q)ˆ
k+µL(ˆ
k·q)u] (32)
'=v2
21+(v2
1−v2
2)ˆ
k⊗ˆ
k(33)
where v1=(λL+2µL)/% and v2=√µL/% are the velocities of longitudinal and transverse
elastic waves, respectively.
3.3. Sound waves
The general relations presented in section 2 can also be applied to scalar waves. By way of
example let us consider sound waves in an ideal liquid. In the linear approximation, the velocity
vof fluid particles is assumed to be far less than the sound velocity c0, and the variations of
pressure p=p−p0and density %=%−%0are assumed to be far less than the equilibrium
values p0and %0. Therefore, for an eigenwave, the continuity equation and the Euler equation
reduce to [18]
ω%=%0k·vω%0v=pk(34)
where p=c2
0%. The dispersion equation k2−ω2/c2
0=0 has two different solutions
kj=t+ξjqj=1,2ξ1,2=±
ω2/c2
0−t2(35)
given the tangential component tof k(t·q=0). From equations (34), (35) we obtain
RWj=ξjWjWj=p
j
q·vjj=1,2 (36)
where
R=0ω%0
A0A=1
%0ω
c2
0−t2
ω(37)
Orthonormal beams and localized fields: I 6255
i.e. the amplitude Wjis specified by ξjand p
jas follows:
Wj=p
j
p
jξj/(ω%0).(38)
At real values of t,Rsatisfies the identity
R†=Q0RQ0Q0=01
10
.(39)
Equation (36) results in the orthogonality relation [18] W†
1Q0W2=q·(p∗
1v2+p
2v∗
1)=0
at ξ1= ξ∗
2. The condition Wn|Q|Wn=NQwith Q=Q0/4 normalizes the time average
sound energy flux NQthrough the plane σ0:
Wn|Q|Wn=1
4σ0
(p∗v+pv∗)·qdσ0=NQ.(40)
We assume below that q=e3.
4. Field parametrization and representation
In an anisotropic medium, substitution of the Rayleigh formula [16]
eik·r=4π
+∞
l=0
iljl(kr )
l
m=−l
Ym
l∗(ˆ
k)Y m
l(ˆr)(41)
in equation (7) yields [11]
Ws
j(r,t) =e−iωt
+∞
l=0
il
l
m=−l
Ym
l(ˆr)Wm
l(r) (42)
where
ˆ
k=sin θ(e1cos ϕ+e2sin ϕ)+e3cos θ(43)
ˆ
r=r/r =sin γ(e1cos ψ+e2sin ψ) +e3cos γ(44)
Ym
l(ˆ
k)≡Ym
l(θ,ϕ
),Ym
l(ˆr)≡Ym
l(γ , ψ),jl(kr ) is the spherical Bessel function [15, 16], and
vector coefficients Wm
ldepend only on radius r. In an isotropic medium, this expansion
becomes [11]
Ws
j(r,t) =e−iωt
+∞
l=0
iljl(kr )
l
m=−l
Ym
l(ˆr)Wm
l.(45)
In this case, the field is completely characterized by coordinate-independent vector coefficients
Wm
l.
To set the function k=k(θ , ϕ), one can use either the normal k=k(θ,ϕ
)ˆ
k(θ,ϕ
)
or the tangential k=t(θ,ϕ
)+ξ(θ,ϕ
)qparametrization (see section 3) with some given
functions θ=θ(θ, ϕ) and ϕ=ϕ(θ, ϕ ). By setting these functions in various ways, one
can obtain diverse orthonormal beams and localized fields [10–12]. It is essential that, in the
general case, the coordinates θand ϕon Bdo not coincide with the spherical coordinates
θand ϕof ˆ
k. We shall restrict our consideration to the fields with θ=κ0θand ϕ=ϕ,
where parameter κ0satisfies the condition 0 <κ
01. Such fields comprise plane waves with
wave normals ˆ
klying in the same solid angle 7=2π(cos κ0θ1−cos κ0θ2). There are two
basically different ways to obtain a family of orthonormal beams [11]. One possibility, applied
in this series of papers, is that beams are composed of eigenwaves with different tangential
6256 G N Borzdov
components t. To this end, we use two sets of parameters (θ1=0 in both cases): θ2=π/2
and κ0=1, and θ2=πand 0 <κ
01/2, respectively. To specify the amplitude functions,
we use the spherical basis vectors
er(θ,ϕ)=sin θ(e1cos ϕ+e2sin ϕ) +e3cos θ(46)
eθ(θ,ϕ)=cos θ(e1cos ϕ+e2sin ϕ) −e3sin θ(47)
eϕ(ϕ) =−e1sin ϕ+e2cos ϕ. (48)
To find the coefficients Wm
l, we shall extensively use functions Ism
j[f] defined as
follows [11]:
2π
0
dϕπ/2
0
exp {i[kr·er(θ , ϕ) +nϕ]}Ys
j(θ, ϕ)f (θ ) sin θdθ
=exp[i(s +n)ψ]Iss+n
j[f](r, γ ) (49)
Ism
j[f]=Ism
j[f](r, γ ) =
+∞
l=|m|
iljl(kr )P |m|
l(cos γ)Psm
jl [f] (50)
Psm
jl [f]=8π2NjsN2
lm π/2
0
P|s|
j(cos θ)P|m|
l(cos θ)f(θ)sin θdθ(51)
where f=f(θ)is a scalar function of the polar angle θ, and nis an integer. Function Ism
j[f]
at fixed rand γas well as coefficient Psm
jl [f] are functionals regarding f. For any given f,
Ism
j[f] is a function of rand γ, whereas Psm
jl [f] is a constant. We omit the arguments (r, γ )
where appropriate. The real and imaginary parts of Ism
j[f] can be separated as
Ism
j[f]=i|m|(J sm
j0[f]+iJsm
j1[f])(52)
where
Jsm
jp [f]=Jsm
jp [f](r, γ ) =
+∞
ν=0
(−1)νj|m|+2ν+p(kr )P |m|
|m|+2ν+p(cos γ)Ps|m|
j|m|+2ν+p[f].(53)
Additional information on these functions can be found in [11].
5. Conclusion
The relations presented in this paper provide means to extend the formalism proposed in [11]
to elastic beams in isotropic and anisotropic mediums and sound beams in an ideal liquid.
This formalism makes it possible to obtain exact solutions of linear field equations, which
describe families of orthonormal beams and various types of localized fields. Owing to the
orthonormality conditions, the families of orthonormal beams constitute convenient functional
bases for complex elastic and sound fields and can be applied for modeling the beams now in
use. In the subsequent three papers of the current series, it will be shown that the localized
elastic and sound fields also possess interesting properties. In particular, they have a very
small (about several wavelengths) core region with maximum intensity of field oscillations
and unique energy transport.
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