Page 1
ON THE SOLUTION OF WIENER–HOPF PROBLEMS INVOLVING
NONCOMMUTATIVE MATRIX KERNEL DECOMPOSITIONS∗
I. DAVID ABRAHAMS†
SIAM J. APPL. MATH.
Vol. 57, No. 2, pp. 541–567, April 1997
c ? 1997 Society for Industrial and Applied Mathematics
013
Abstract. Many problems in physics and engineering with semi-infinite boundaries or interfaces
are exactly solvable by the Wiener–Hopf technique. It has been used successfully in a multitude of
different disciplines when the Wiener–Hopf functional equation contains a single scalar kernel. For
complex boundary value problems, however, the procedure often leads to coupled equations which
therefore have a kernel of matrix form. The key step in the technique is to decompose the kernel
into a product of two functions, one analytic in an upper region of a complex (transform) plane and
the other analytic in an overlapping lower half-plane. This is straightforward for scalar kernels but
no method has yet been proposed for general matrices.
In this article a new procedure is introduced whereby Pad´ e approximants are employed to obtain
an approximate but explicit noncommutative factorization of a matrix kernel. As well as being
simple to apply, the use of approximants allows the accuracy of the factorization to be increased
almost indefinitely. The method is demonstrated by way of example. Scattering of acoustic waves
by a semi-infinite screen at the interface between two compressible media with different physical
properties is examined in detail. Numerical evaluation of the approximate factorizations together
with results on an upper bound of the absolute error reveal that convergence with increasing Pad´ e
number is extremely rapid. The procedure is computationally efficient and of simple analytic form
and offers high accuracy (error < 10−7% typically). The method is applicable to a wide range of
initial or boundary value problems.
Key words.
Pad´ e approximants
Wiener–Hopf technique, matrix Wiener–Hopf equations, scattering, acoustics,
AMS subject classifications. 30E10, 41A21, 45E10, 73D25, 76Q05, 78A45
PII. S0036139995287673
1. Introduction and difficulties in solving matrix Wiener–Hopf equa-
tions. The Wiener–Hopf technique has, since its invention in 1931 [34], proved an
immensely important method in mathematical physics. It offers one of the very few
approaches to obtaining an exact solution to a class of integral equations. These
equations are of either first or second kind, are defined over a half-line, and have
a difference kernel. An enormous variety of physically important problems can be
cast into this form of equation, and all have the characteristic feature of boundary
conditions defined on semi-infinite planes. Fields of application include diffraction of
acoustic [7], elastic [30], and electromagnetic waves; fracture mechanics; flow prob-
lems; diffusion models; and geophysical applications [16], to name but a few. The
solution method, although initially proposed to follow via construction of the inte-
gral equation, now usually proceeds directly from the boundary value problem to an
equation in the complex plane defined in an infinite strip [23]. This is accomplished
by Fourier transformation (or other appropriate transform) or Green’s theorem, and
the essential physical details are manifested in the singularity structure of the Fourier
transformed difference kernel often called the Wiener–Hopf kernel, K(α) (Noble [29]),
where α is the transform parameter. This kernel is usually arranged to be singular-
ity free in a finite-width strip of the complex α-plane which contains the real line as
∗Received by the editors June 14, 1995; accepted for publication (in revised form) February 26,
1996.
http://www.siam.org/journals/siap/57-2/28767.html
†Departmentof Mathematics, KeeleUniversity,
(i.d.abrahams@keele.ac.uk).
Keele,StaffordshireST55BG,UK
541
Page 2
542
I. DAVID ABRAHAMS
|α| → ∞. Further, this strip need not necessarily be straight (i.e., enclose the whole
real line) and is denoted henceforth as D. Solution of the Wiener–Hopf equation is
straightforward once K(α) is decomposed into a product of two functions K+(α),
K−(α), where K+(α) is regular and zero free in the region above and including the
strip D, denoted D+, and K−(α) is regular and zero free in the region below and
including D, denoted D−.
as |α| → ∞ in D±, respectively. Cauchy’s integral theorem provides a convenient
method for obtaining the explicit sum factorization of a function, e.g., of the form
g(α) = g+(α) + g−(α), where ± denotes the above analyticity properties, and so by
exponentiation we deduce (see Theorem C of Noble [29])
Further, K±(α) must have at worst algebraic growth
K±(α) = exp[g±(α)] = exp
?±1
2πi
?∞
−∞
logK(ζ)
ζ − α
dζ
?
,
(1)
where ζ is a contour in D and α lies above (below) ζ for K+(α) (K−(α)). Sometimes
a limiting procedure is necessary to ensure convergence of these integrals.
For problems in which K(α) is a scalar function, then, apart from some technical
difficulties associated with computing integrals of the above form (see Abrahams and
Lawrie [5]), the Wiener–Hopf procedure goes through without a hitch. However, in
more complex boundary value problems, it is not unusual to obtain a matrix functional
Wiener–Hopf equation with a square matrix kernel K(α). This presents two potential
obstacles to allow the solution scheme to go through. The first, which it is the aim
of this article to tackle, is the factorization procedure (1). For any matrix function,
K(α), the logarithm G(α) = log(K(α)) can be defined, for example by its power
series expansion log[I + (K(α) − I)] = K(α) − I −1
identity matrix, or otherwise. Hence the sum split of G(α) is accomplished by the
Cauchy-type integrals acting on each element of the matrix. However, the final step
to obtain the product factors, i.e.,
2(K(α) − I) + ···, where I is the
exp{G+(α) + G−(α)} = exp{G+(α)}exp{G−(α)},
is true if and only if G+(α) and G−(α) commute (Heins [20]). For general matri-
ces K(α) this will not be the case, and although no procedure is presently available
for tackling such arbitrary matrix Wiener–Hopf problems, Gohberg and Krein have
proved the existence of product factors in all cases [17]. Fortunately, many physically
interesting problems naturally give rise to kernels which, although intrinsically matrix
in form, have a commutative factorization or can be reworked (by premultiplication
and/or postmultiplication by suitable matrices) into such form. Khrapkov [26, 27], in
articles concerned with the stresses in elastostatic wedges with notches, was the first
author to express the commutative factorization in a form which indicates the subal-
gebra associated with this class of kernels. Many other authors have also examined
the commutative cases, and, in particular, reference should be made to the ingenious
approach to the simplest nontrivial commutative example by Rawlins [31], the direct
scheme of Daniele [14], and the Hilbert problem technique of Hurd [21]. The latter
approach, which transforms the factorization problem to a pair of uncoupled Hilbert
problems defined on a semi-infinite branch cut, is generalizable to an interesting range
of cases [32], one example of which is explored in [1]. It is generally accepted that
the generalized range of kernels susceptible to Hurd’s method and Khrapkov’s class
of commutative matrices suitably multiplied by entire matrices are in fact equivalent
[24].
(2)
Page 3
MATRIX WIENER–HOPF FACTORIZATION
543
The second difficulty associated with matrix kernels is that, although decomposi-
tion into matrix factors with overlapping domains of regularity may be achieved, the
growth of such factors could be exponential in the region of analyticity. This prohibits
the application of Liouville’s theorem at a later stage of analysis, and so a solution to
the matrix Wiener–Hopf equation cannot be obtained. Various techniques have been
proposed in the literature to overcome this difficulty, including a general scheme by
the author (Abrahams and Wickham [7]) and one specific to matrices of Khrapkov
form (Daniele [15]). A full bibliography illustrating the relevant physical applications
can be found in [6].
The aim of this article is to demonstrate a procedure for obtaining noncommu-
tative matrix factors which have algebraic growth.
first rather than the second difficulty outlined above. As already mentioned, many
useful and important problems have, or can be cast into, commutative factorization
form, but the most important, and by far the more diverse, range of physical cases
are intrinsically noncommutative. Matrix Wiener–Hopf equations arise in all areas
of physics, including those mentioned at the start of this introduction. Usually the
matrices yield noncommutative factorizations because they contain multiple branch
cut functions, infinite sequences of poles or zeros, or a combination of both of these
singularity structures. In certain special cases the factorization procedure can be
completed by subtraction of an infinite family of poles from both sides of the Wiener–
Hopf equation. The coefficients of each pole term are then found from the solution of
an infinite algebraic system of equations (see Idemen [22], Abrahams [2], Abrahams
and Wickham [8], and references quoted therein). The objective of this article is to
illustrate a new procedure for obtaining explicit but approximate noncommutative
matrix factors. The method should be applicable to many kernels of this previously
unsolved matrix class and is based on the replacement of certain components of the
matrix function by Pad´ e approximants. This allows the level of approximation to be
increased to very high accuracies while at the same time offering a surprisingly simple
factorization form. An alternative technique has recently been proposed by Wickham
[33] which, to the author’s knowledge, is the first to offer a computational route to fac-
torize noncommutative kernels. This involves the derivation of certain exact coupled
integral equations from which rigorous bounds on the convergence of such solutions
can be derived. However, the method described herein is a good deal simpler to use
in practice and appears much more direct than that suggested by Wickham. The
relationship between the two methods is discussed further in the concluding remarks.
The clearest and most concise method for illustrating the Pad´ e factorization pro-
cedure is to examine one specific example. We take a typical physical situation which
gives rise to a noncommutative kernel, namely a semi-infinite barrier at an interface
between two compressible media (Figure 1). This yields the simplest and therefore
the canonical matrix kernel and so will omit any extraneous or obfuscating technical
details. The plan of the paper is as follows. In section 2 the interfacial boundary value
problem is stated mathematically, and in section 3 it is reduced to a matrix Wiener–
Hopf equation. The explicit kernel factorization, employing Pad´ e approximants for
the approximate matrix kernel, is performed in section 4, and a discussion of the
derivation of an ansatz used in that section can be found in Appendix B. The formal
solution to the scattered field is given in section 5, and in section 6 the pressure in
the far-field is derived asymptotically for the upper half-plane. Section 6 also presents
numerical results and timings for different approximant numbers. A summary of the
important properties of Pad´ e approximants is written in Appendix A, and concluding
remarks are made in section 7.
Thus, we concentrate on the
Page 4
544
I. DAVID ABRAHAMS
FIG. 1. The physical configuration.
2. The boundary value problem. As discussed in the introduction, we will
study a two-dimensional problem involving a simple planar interfacial boundary be-
cause it offers the simplest form of a nontrivial matrix Wiener–Hopf kernel (with two
branch cut pairs). Defining (x,y) as a two-dimensional Cartesian coordinate system,
a semi-infinite screen lying along x < 0,y = 0 is placed between two different inviscid
compressible media. The upper medium has sound speed c1, say, and lower c2. Along
the extension of the screen (x > 0,y = 0) it is assumed that they have equal pressures
and normal velocities, and if the continua are fluids, then they do not mix. Thus, it
could be envisaged that on this line (x > 0) the two fluids are separated by a thin
membrane of negligible mass and surface tension. On the physical screen (x < 0), the
upper surface (y = 0+) is chosen to be perfectly soft (vanishing pressure p), whereas
the underside (y = 0−) is taken to be perfectly rigid (vanishing normal velocity v).
Figure 1 illustrates this geometry.
The governing equations for material motions can be expressed in terms of velocity
potentials as
?∂2
?∂2
where the pressure and velocity are related to these potentials via
p = −ρ1∂Φ1
∂x2+
∂2
∂y2−1
∂2
∂y2−1
c2
1
∂2
∂t2
∂2
∂t2
?
?
Φ1= 0,y ≥ 0,
(3)
∂x2+
c2
2
Φ2= 0,y ≤ 0,
(4)
∂t,
y ≥ 0;
p = −ρ2∂Φ2
∂t,
y ≤ 0,
(5)
u = (u,v) = ∇Φj,j = 1,2 for y > 0,y < 0.
(6)
Here ρ1,ρ2and c1,c2are the density and sound speeds in media 1, 2, respectively.
For brevity, only the case of scattering by plane waves, incident from below, will
be examined in this article, and the propagation speed c2is taken to be less than c1.
All other cases can be dealt with in an identical fashion. It is the purpose of this
paper to demonstrate the efficacy of the solution method, not to provide exhaustive
details of the scattered field. Writing the incident waves (satisfying (4)) with angular
frequency ω as
Φinc(x,y;t) = ?{eiω[(xcosθ0+y sinθ0)/c2−t]},
(7)
Page 5
MATRIX WIENER–HOPF FACTORIZATION
545
then, for steady-state motions, the total velocity potential is given by
Φ1(x,y;t) = ?{ψ1(x,y)e−iωt},
(8)
Φ2(x,y;t) = Φinc(x,y;t) + Φinc(x,−y;t) + ?{ψ2(x,y)e−iωt}.
Note that the reflected wave Φinc(x,−y;t) has been written explicitly for convenience,
and ψ1(x,y) and ψ2(x,y) are the scattered potentials which must be purely outgoing
as (x2+ y2)Substituting (8) and (9) into the governing equations and
boundary conditions gives
?∂2
?∂2
ψ1(x,0) = 0,∂ψ2
∂y(x,0) = 0,
(9)
1
2 → ∞.
∂x2+
∂2
∂y2+ω2
∂2
∂y2+ω2
c2
1
?
?
ψ1= 0,y ≥ 0,
(10)
∂x2+
c2
2
ψ2= 0,y ≤ 0,
(11)
x ≤ 0,
(12)
ρ1
ρ2ψ1(x,0) − ψ2(x,0) = 2eiωx
In order to exclude extraneous sound sources, it must also be insisted that
c2cosθ0,
∂ψ1
∂y(x,0) −∂ψ2
∂y(x,0) = 0,x ≥ 0.
(13)
ψ1,ψ2contain only outgoing waves as
?
x2+ y2→ ∞; (14)
ψ1,ψ2are bounded everywhere, including the origin (0,0).
(15)
The boundary value problem is fully specified by statements (10)–(15).
3. Reduction to a Wiener–Hopf equation. To solve the boundary value
problem defined in the previous section it is useful to introduce Fourier transforms.
As there is no intrinsic length scale in the model, it is advantageous first to scale x
and y on the length c1/ω, viz.
x = Xc1/ω,y = Y c1/ω,
(16)
and to write the sound speed ratio as
c1/c2= k.
(17)
As stated earlier, and with no real loss of generality, c1is taken to be greater than
c2, which means that k > 1. (If this is not the case, then x and y could be scaled on
c2/ω, k redefined as c2/c1in order to force it to be larger than unity, and the following
analysis repeated appropriately.) The Fourier transform of ψjcan be written as
?∞
where α can at present be taken as a real parameter, and so, assuming convergence
of transformed terms, (10) and (11) give
Ψj(α,Y ) =
−∞
eiαXψj(Xc1/ω,Y c1/ω) dX,j = 1,2,
(18)
∂2Ψ1
∂Y2− (α2− 1)Ψ1= 0,
∂2Ψ2
∂Y2− (α2− k2)Ψ2= 0,
Y ≥ 0,
(19)
Y ≤ 0.
(20)
Page 24
564
I. DAVID ABRAHAMS
and if we write the first column of [K+
column of K−
?
=cos(γ(α)s−(α))I −
N]−1as
1
r+(α)(a+(α),b+(α))Tand the first
Nas r−(α)(a−(α),b−(α))T, then
cos(γ(α)s+(α))I +
?
1
γ(α)sin(γ(α)s+(α))JN(α)
1
γ(α)sin(γ(α)s−(α))JN(α)
??
a+(α)
b+(α)
??
?
a−(α)
b−(α)
?
.
(B.135)
The left-hand side of this equation is analytic in D+except for simple poles at α = pn,
from the zeros of fN(α) in JN(α), and at α = qnfrom the poles of fN(α) (see (73)).
Similarly the right-hand side is analytic in D−except for poles at α = −pn,−qn. It
is simple to show, from (75), that the top element of the left-hand side of (B.135) has
residue
βm
2pmγ(pm)sin[γ(pm)s+(pm)]b+(pm) =Bm
2pm,
(B.136)
say, at α = pm, and the top element of the right-hand side has residue
βm
2pmγ(pm)sin[γ(pm)s+(pm)]b−(−pm) =Cm
2pm,
(B.137)
say, at α = −pm, where in (B.137) the symmetry property γ(pm) = γ(−pm), s+(pm) =
s−(−pm) has been employed. Thus, the top line of (B.135) can be written as
a+(α)cos(γ(α)s+(α)) + b+(α)γ(α)
fN(α)sin(γ(α)s+(α))
?
fN(α)sin(γ(α)s−(α))
?
−
N
?
n=1
?
Bn
α − pn
+
Cn
α + pn
1
2pn
= a−(α)cos(γ(α)s−(α)) − b−(α)γ(α)
?
where the left (right) side is now regular in D+(D−). Hence by the usual analytic
continuation arguments both sides of this equation must be equal to an entire function,
which, without loss of generality, we can choose to be a constant, C1, say (for a similar
exercise see Abrahams [2]). Similarly, the bottom equation of (B.135) must have poles
at α = ±qnsubtracted from both sides:
b+(α)cos(γ(α)s+(α)) − a+(α)fN(α)
?
= b−(α)cos(γ(α)s−(α)) + a−(α)fN(α)
−
N
?
n=1
Bn
α − pn
+
Cn
α + pn
1
2pn,
(B.138)
γ(α)
?
sin(γ(α)s+(α))
+1
2
N
?
n=1
An
α − qn
+
Dn
α + qn
γ(α)
?
sin(γ(α)s−(α))
+1
2
N
?
n=1
?
An
α − qn
+
Dn
α + qn
,
(B.139)
Page 25
MATRIX WIENER–HOPF FACTORIZATION
565
where use has been made of (74), and
Am=
αm
qmγ(qm)sin[γ(qm)s+(qm)]a+(qm),
αm
qmγ(qm)sin[γ(qm)s+(qm)]a−(−qm).
(B.140)
Dm=(B.141)
Again both sides of (B.139) are analytic in overlapping half-planes and so are equal
to the constant C2, say. Hence (B.138) and (B.139) give
?
?
2
and all that remains is to determine the coefficients An–Dn. Taking the top line of
(B.142) first, we can obtain a relation between the unknown coefficients if α → qm.
Omitting all details, we find
?
or, from (B.140),
?
Similarly, the top line of (B.143) gives, as α → −qm, the identity
αm
qmγ(qm)tan[γ(qm)s+(qm)]
a+(α)
b+(α)
?
?
= [Q+(α)]−1
C1+?N
C1+?N
n=1
?N
α−pn+
?N
?
Bn
α−pn+
?
Cn
α+pn
?
1
?
,
2pn
C2−1
2
n=1
An
α−qn+
Cn
α+pn
Dn
α+qn
,
(B.142)
a−(α)
b−(α)
= Q−(α)
n=1
?
Bn
?
?
1
?
2pn
C2−1
n=1
An
α−qn+
Dn
α+qn
(B.143)
a+(qm) =
1
cos[γ(qm)s+(qm)]
C1+
N
?
n=1
?
Bn
qm− pn
+
Cn
qm+ pn
?
1
2pn
?
(B.144)
Am=
αm
qmγ(qm)tan[γ(qm)s+(qm)]
C1+
N
?
n=1
?
Bn
qm− pn
+
Cn
qm+ pn
?
1
2pn
?
.
(B.145)
Dm=
?
C1−
N
?
n=1
?
Bn
qm+ pn
+
Cn
qm− pn
?
1
2pn
?
,
(B.146)
and the bottom rows of (B.142), (B.143) also reveal
Bm= βmγ(pm)tan[γ(pm)s+(pm)]
?
?
C2−1
2
N
?
N
?
n=1
?
?
An
pm− qn
An
pm+ qn
+
Dn
pm+ qn
??
??
,
(B.147)
Cm= βmγ(pm)tan[γ(pm)s+(pm)]
C2+1
2
n=1
+
Dn
pm− qn
.
(B.148)
The two constants C1, C2are still arbitrary and so, again without loss of generality,
we can take two cases: first C1 = 1, C2 = 0, and second C1 = 0, C2 = 1. When
C2= 0, the symmetry in equations (B.145)–(B.148) gives
Am= Dm,
(B.149)
and so we obtain the pair of coupled algebraic equations
Bm= −Cm,
Am=
αm
qmγ(qm)tan[γ(qm)s+(qm)]
?
1 +
N
?
N
?
n=1
Bn
q2
m− p2
An
p2
n
?
,
(B.150)
Bm= −pmβmγ(pm)tan[γ(pm)s+(pm)]
n=1
m− q2
n
(B.151)
Page 26
566
I. DAVID ABRAHAMS
for m = 1,...,N. These agree exactly with (81), (80). The alternative choice of
constants, C1 = 0, C2 = 1, is easily shown to give the coupled equations for¯Am
(defined here as qmAm),¯Bm(defined here as Bm/pm) found in section 4. Therefore,
forming K−
of constants, we obtain
?
n=1
α2−q2
which is the ansatz defined in (72).
N(α) from (B.143) by taking the two column vectors with the above choice
M(α) =
1 +?N
n=1
Bn
α2−p2
An
n,
n,
α?N
n=1
¯ Bn
α2−p2
An
α2−q2
n
−α?N
1 −?N
n=1
¯
n
?
,
(B.152)
Acknowledgments.
and encouragement shown by the late Professor Gerry R. Wickham, Brunel University,
over many years. He was a very good friend and close collaborator, and his many
important research studies in diffraction theory and elasticity will continue through
the activities of his coworkers.
The author wishes to acknowledge the enormous support
REFERENCES
[1] I. D. ABRAHAMS, Scattering of sound by a semi-infinite elastic plate with a soft backing; a
matrix Wiener–Hopf problem, IMA J. Appl. Math., 37 (1986), pp. 227–245.
[2] I. D. ABRAHAMS, Scattering of sound by three semi-infinite planes, J. Sound Vibration, 112
(1987), pp. 396–398.
[3] I. D. ABRAHAMS, The application of Pad´ e approximants to scalar and matrix Wiener–Hopf
factorization, Proc. Roy. Soc. London Ser. A, (1997), to appear.
[4] I. D. ABRAHAMS, Radiation and scattering of waves on an elastic half-space; a noncommutative
matrix Wiener–Hopf problem, J. Mech. Phys. Solids, 44 (1996), pp. 2125–2154.
[5] I. D. ABRAHAMS AND J. B. LAWRIE, On the factorization of a class of Wiener–Hopf kernels,
IMA J. Appl. Math., 55 (1995), pp. 35–47.
[6] I. D. ABRAHAMS AND G. R. WICKHAM, On the scattering of sound by two semi-infinite parallel
staggered plates I. Explicit matrix Wiener–Hopf factorization, Proc. Roy. Soc. London Ser.
A, 420 (1988), pp. 131–156.
[7] I. D. ABRAHAMS AND G. R. WICKHAM, General Wiener–Hopf factorization of matrix kernels
with exponential phase factors, SIAM J. Appl. Math., 50 (1990), pp. 819–838.
[8] I. D. ABRAHAMS AND G. R. WICKHAM, The scattering of water waves by two semi-infinite
opposed vertical walls, Wave Motion, 14 (1991), pp. 145–168.
[9] G. A. BAKER JR., Essentials of Pad´ e Approximants, Academic Press, New York, 1975.
[10] O. P. BRUNO AND F. REITICH, Numerical solution of diffraction problems: A method of varia-
tion of boundaries. II. Finitely conducting gratings, Pad´ e approximants, and singularities,
J. Opt. Soc. Amer. A, 10 (1993), pp. 2307–2316.
[11] P. A. CANNELL, Acoustic edge scattering by an elastic half-plane, Proc. Roy. Soc. London Ser.
A, 350 (1976), pp. 71–89.
[12] G. F. CARRIER, Useful approximations in Wiener–Hopf problems, J. Appl. Phys., 30 (1959),
pp. 1769–1774.
[13] P. H. DAHL AND G. V. FRISK, Diffraction from the junction of pressure release and locally
reacting half-planes, J. Acoust. Soc. Amer., 90 (1991), pp. 1093–1100.
[14] V. G. DANIELE, On the factorization of Wiener–Hopf matrices in problems solvable with Hurd’s
method, IEEE Trans. Antennas and Propagation, 26 (1978), pp. 614–616.
[15] V. G. DANIELE, On the solution of two coupled Wiener–Hopf equations, SIAM J. Appl. Math.,
44 (1984), pp. 667–680.
[16] A. M. J. DAVIS, Continental shelf wave scattering by a semi-infinite coastline, Geophys. As-
trophys. Fluid Dynamics, 39 (1987), pp. 25–55.
[17] I. C. GOHBERG AND M. G. KREIN, Systems of integral equations on a half-line with ker-
nels depending on the difference of arguments, Am. Math. Soc. Transl. Ser. 2, 14 (1960),
pp. 217–287.
[18] P. R. GRAVES-MORRIS, Pad´ e Approximants, Institute of Physics, London, 1973.
[19] P. R. GRAVES-MORRIS, Pad´ e Approximants and their Applications, Academic Press, London,
1973.
Page 27
MATRIX WIENER–HOPF FACTORIZATION
567
[20] A. E. HEINS, Systems of Wiener–Hopf equations, in Proceedings of Symposia in Applied Math-
ematics II, McGraw-Hill, New York, 1950, pp. 76–81.
[21] R. A. HURD, The Wiener–Hopf Hilbert method for diffraction problems, Canad. J. Phys.,
54 (1976), pp. 775–780.
[22] M. IDEMEN, A new method to obtain exact solutions of vector Wiener–Hopf equations, Z.
Angew. Math. Mech., 59 (1976), pp. 656–658.
[23] D. S. JONES, A simplifying technique in the solution of a class of diffraction problems, Quart.
J. Math., 3 (1952), pp. 189–196.
[24] D. S. JONES, Factorization of a Wiener–Hopf matrix, IMA J. Appl. Math., 32 (1984),
pp. 211–220.
[25] W. T. KOITER, Approximate solution of Wiener–Hopf type integral equations with applications,
parts I–III, Koninkl. Ned. Akad. Wetenschap. Proc., B57 (1954), pp. 558–579.
[26] A. A. KHRAPKOV, Certain cases of the elastic equilibrium of an infinite wedge with a non-
symmetric notch at the vertex, subjected to concentrated forces, Appl. Math. Mech.,
35 (1971), pp. 625–637.
[27] A. A. KHRAPKOV, Closed form solutions of problems on the elastic equilibrium of an infinite
wedge with nonsymmetric notch at the apex, Appl. Math. Mech., 35 (1971), pp. 1009–1016.
[28] M. J. LIGHTHILL, Introduction to Fourier Analysis and Generalised Functions, Cambridge
University Press, Cambridge, 1958.
[29] B. NOBLE, Methods Based on the Wiener–Hopf Technique, 2nd ed., Chelsea Press, New York,
1988.
[30] A. N. NORRIS AND J. D. ACHENBACH, Elastic wave diffraction by a semi-infinite crack in a
transversely isotropic material, Quart. J. Mech. Appl. Math., 37 (1984), pp. 565–580.
[31] A. D. RAWLINS, The solution of a mixed boundary value problem in the theory of diffraction
by a semi-infinite plane, Proc. Roy. Soc. London Ser. A, 346 (1975), pp. 469–484.
[32] A. D. RAWLINS, A note on Wiener–Hopf matrix factorization, Quart. J. Mech. Appl. Math.,
38 (1985), pp. 433–437.
[33] G. R. WICKHAM, Mode conversion, corner singularities and matrix Wiener–Hopf factorization
in diffraction theory, Proc. Roy. Soc. London Ser. A, 451 (1995), pp. 399-423.
[34] N. WIENER AND E. HOPF,¨Uber eine Klasse singul¨ arer Integralgleichungen, S. B. Preuss. Akad.
Wiss., (1931), pp. 696–706.