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

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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)

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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

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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)

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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)

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546

I. DAVID ABRAHAMS

The existence of all transformed expressions (i.e., integrals of the form (18)) can be

confirmed a posteriori. These equations have general solutions

Ψ1(α,Y ) = A(α)eγ(α)Y+ B(α)e−γ(α)Y,

Ψ2(α,Y ) = C(α)eδ(α)Y+ D(α)e−δ(α)Y,

y ≥ 0,

y ≤ 0,

(21)

(22)

where A(α),...,D(α) are as yet arbitrary functions of α and

γ(α) = (α2− 1)

1

2,δ(α) = (α2− k2)

1

2.

(23)

The branch cut functions are chosen, for α real, to be

γ(α) = |α2− 1|

1

2,

|α| ≥ 1;

γ(α) = −i|1 − α2|

1

2,

|α| < 1,

(24)

and

δ(α) = |α2− k2|

1

2,

|α| ≥ k;

δ(α) = −i|k2− α2|

1

2,

|α| < k.

(25)

Thus, Ψ1(α,Y ) will either grow exponentially or have incoming waves (of the form

e−i|γ(α)|Y −iωt) as Y → ∞ unless A(α) ≡ 0. Similarly, as Y → −∞, it is clear that

Ψ2(α,Y ) is bounded or has outgoing waves if and only if D(α) ≡ 0. The two remaining

unknowns B(α) and C(α) are specified by examining the boundary conditions on

Y = 0.

First, from (12),

?∞

which is given the superscript + to denote that it must be a function analytic for all

α above and on the inverse integration path (which at present is taken as the real

line); see for example Chapter 1 of Noble [29]. Also from (12),

?∞

and, from (12) and (13),

?∞

= −

−∞

= −Ψ−

Ψ1(α,0) =

0

eiαXψ1(Xc1/ω,0)dX ≡ Ψ+

1(α),

(26)

∂Ψ2

∂Y(α,0) =

0

eiαX∂ψ2

∂Y(Xc1/ω,0)dX ≡∂Ψ+

2

∂Y

(α),

(27)

−∞

?0

eiαX

?ρ1

ρ2ψ1(Xc1/ω,0) − ψ2(Xc1/ω,0)

?

dX

eiαXψ2(Xc1/ω,0)dX + 2

?∞

0

eiX(α+k cosθ0)dX

2(α) +

2i

(α + kcosθ0)+,

(28)

where the − superscript denotes functions analytic below and on the contour of the

inverse transform, and the simple pole at α = −kcosθ0 lies just below this path.

Note that the integral of the exponential function converges in a generalized sense

(Lighthill [28]) for real α and k. The remaining condition (13), which is continuity of

normal velocity across y = 0,x ≥ 0, yields

∂Ψ+

1

∂Y∂Y

(α) −∂Ψ+

2

(α) = 0.

(29)

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MATRIX WIENER–HOPF FACTORIZATION

547

Hence, from (21) and (26),

Ψ+

1(α) = B(α),

(30)

and, differentiating (22) and using (27),

∂Ψ+

∂Y

2

(α) = δ(α)C(α).

(31)

Further, differentiating (21) and employing (29) give

∂Ψ−

∂Y

1

(α) +∂Ψ+

2

∂Y

= −γ(α)B(α),

(32)

and finally (21), (22), and (28) yield

ρ1

ρ2B − C = −Ψ−

2(α) +

2i

(α + kcosθ0)+.

(33)

The unknowns B and C can be eliminated from these four equations to give the vector

Wiener–Hopf equation, valid at present only along the line ?(α) = 0:

K(α)Ψ+(α) = Ψ−(α) + F+(α),

(34)

where the kernel is

K(α) =

?

1

µγ(α)

1

−µ/δ(α)

?

(35)

and the unknown column vectors are

Ψ+(α) =

?∂Ψ+

?

2

∂Y

(α),1

µΨ+

1(α)

?T

,

(36)

Ψ−(α) =

−∂Ψ−

∂Y

1

(α),−µΨ−

2(α)

?T

.

(37)

Also,

F+(α) =

2iµ

(α + kcosθ0)+

{0,1}T,

(38)

where µ is the constant

µ =

?ρ2

ρ1.

(39)

The inverse transform path has so far been taken along the real line in the α-

plane. If this contour is to be deformed into the complex plane then it is necessary to

define γ(α) and δ(α) as complex valued functions of the complex variable α. In order

to ensure the behavior defined in (24), (25) it is easy to show that γ(α) and δ(α)

have branch points at ±1,±k, respectively, and the integration path passes above the

branch points at −1,−k, and below those at +1,+k. This path is illustrated as C

in Figure 2, and the branch cuts are selected to run horizontally, as indicated. For

convenience, and ultimately without loss of generality, θ0is confined for the moment

Page 8

548

I. DAVID ABRAHAMS

FIG. 2. The original integration contour C passing just above the left-hand branch cuts and

below the right-hand cuts. The bold lines indicate the finite branch cuts of f(α), and the dashed

lines are the cuts of γ(α). Also shown are the deformed integration path C1employed in expressions

(117), (118) and the regions of regularity D±with their common strip of analyticity D.

as 0 < θ0≤ π/2. All the singularities in K(α) and F+(α) are shown on Figure 2,

and therefore Ψ1(α,Y ),Ψ2(α,Y ) will have these, and only these, cuts and pole. The

inverse integral

?

has the contour C, which can therefore be deformed away from the branch points to

a typical contour C1say, lying within a strip of arbitrary width D (Figure 2). Then

Ψ+(α) and F+(α) are functions analytic in the region above and including D, say, D+,

and D−is the region below and including D. The vector function Ψ−(α) is analytic

in D−and

D ≡ D+∩ D−.

The reasons for choosing the path C1 will be made clear in the next sections, but

it suffices here to state that the path should be bounded (minimum distance unity)

away from the real-line segments between +1 ↔ +k and −1 ↔ −k.

4. Kernel factorization using Pad´ e approximants.

ψj(x,y) =

1

2π

C

e−iαxω/c1Ψj(α,yω/c1)dα,j = 1,2,

(40)

(41)

4.1. Commutative partial decomposition. The boundary value problem de-

fined in section 2 was reduced in the previous section to a two-dimensional matrix

Wiener–Hopf equation (34) valid in the S-shaped strip of analyticity D (see Figure

2). To solve this equation for the unknown vector functions Ψ+(α), Ψ−(α), which are

analytic in D+, D−, respectively, it is necessary to factorize K(α) into the product

form

?

Here again the ± superscripts denote analyticity in D±. Furthermore, det(K±(α))

must be zero free in D±so that [K±(α)]−1are also analytic in their respective half-

planes. The matrix K(α) (42) is the simplest possible nontrivial example of a Wiener–

Hopf kernel with two branch cut pairs. To the author’s knowledge, no procedure has

K(α) =

1

µγ(α)

1

−µ/δ(α)

?

= K−(α)K+(α).

(42)

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MATRIX WIENER–HOPF FACTORIZATION

549

yet been published in the literature for factorization of this class of matrix kernels. In

this section a simple approximate solution scheme is presented which yields an explicit

noncommutative decomposition (42). The accuracy of the procedure can be improved

almost indefinitely, and it will be shown that, without much effort, extremely good

agreement with an exact factorization is possible (< 10−7%). Thus, the solution can,

for all intents and purposes, be treated as an exact matrix factorization.

To proceed, it is useful to first examine the factorization of K(α) when media 1

and 2 have the same propagation speeds, i.e., γ(α) = δ(α) = (1 − α2)1/2. Denoting

this as¯K(α), then

?

which, when µ = 1, is the problem studied previously by various authors, including

Rawlins [31], Hurd [21], and Daniele [14]. This kernel has a commutative factorization,

which is perhaps best written in the Khrapkov form [26],

?

where a±(α),b±(α) are scalar functions with the indicated analyticity properties, I is

the 2 × 2 identity matrix,¯J(α) is the entire matrix

?

and

¯K(α) =

1

µγ(α)

1

−µ/γ(α)

?

=¯K−(α)¯K+(α),

(43)

¯K±(α) = a±(α)cos[∆(α)b±(α)]I +

1

∆(α)sin[∆(α)b±(α)]¯J(α)

?

,

(44)

¯J(α) =

0

γ2

0

−1

?

,

(45)

¯J2= −∆2(α)I.

(46)

The aforementioned constraint on det(¯K±(α)) = a±(α) implies further that a±(α)

are zero as well as singularity free in D±. From (45), ∆(α) = γ(α), and multiplying

¯K−(α) with¯K+(α) and equating with (43) gives

a+(α)a−(α)cos[γ(α)(b+(α) + b−(α))] = 1,

(47)

a+(α)a−(α)sin[γ(α)(b+(α) + b−(α))] = µ.

(48)

By rearrangement, two scalar factorizations are required, namely, the product split

[a+(α)a−(α)]2= 1 + µ2

(49)

and the sum split

tan[γ(α)(b+(α) + b−(α))] = µ,

(50)

or

b+(α) + b−(α) =tan−1(µ)

γ(α)

.

(51)

The first of these is trivial, where for symmetry in a+,a−we choose

a+(α) = a−(α) = (1 + µ2)1/4; (52)

Page 10

550

I. DAVID ABRAHAMS

the second is a standard sum decomposition (see page 21 of Noble [29]),

b±(α) =2tan−1(µ)

πγ(α)

tan−1

?1 − α

1 + α

?±1/2

.

(53)

Not only has a factorization been achieved, but every element of K±(α) has algebraic

behavior as |α| → ∞. This is an essential requirement for a successful solution of the

Wiener–Hopf equation.

Returning now to the factorization of the kernel in (42), it is desirable that the

construction for K−(α), K+(α) reduces to that in (44) as k → 1. Hence, K(α) may

be written as

K(α) = I + µf(α)

γ(α)J(α),

(54)

where

J(α) =

?

0

γ2(α)/f(α)

0

−f(α)

?

?1/4

,

(55)

f(α) = (γ(α)/δ(α))1/2=

?α2− 1

α2− k2

,

(56)

and, crucially, as before,

J2(α) = −γ2(α)I.

(57)

That Riemann surface of f(α) is chosen which has f(0) = 1/√k, and so f(α) ≡ 1

when k = 1 as required. Thus, the Khrapkov factorization forms

?

can be obtained in an identical fashion to that shown previously, and r±(α), s±(α)

will be functions regular in D±. However, Q±(α) are not analytic in D±because

J(α) is not entire. From the choice of cut locations shown in Figure 2, f(α) (and

hence J(α)) has finite branch cuts between 1 ≤ |α| ≤ k. For the moment this fact will

be ignored but will be addressed later in this section. From (58), the commutative

factorization gives

?

+

(59)

Q±(α) = r±(α)cos(γ(α)s±(α))I +

1

γ(α)sin(γ(α)s±(α))J(α)

?

(58)

K(α) = Q−(α)Q+(α) = r+(α)r−(α) cos[γ(α)(s+(α) + s−(α))]I

1

γ(α)sin[γ(α)(s+(α) + s−(α))]J(α)

?

,

which equates with (54) to yield

(r+(α))2(r−(α))2= 1 + µ2f2(α) =µ2γ(α) + δ(α)

δ(α)

,

(60)

s+(α) + s−(α) =

1

γ(α)tan−1(µf(α)).

(61)

Page 11

MATRIX WIENER–HOPF FACTORIZATION

551

It should be stated that, as f(α) has the finite cuts mentioned above and f(0) = 1/√k,

then

?(f(α)) ≥ 0(62)

for all α and hence tan−1(µf(α)) is a single-valued function. Therefore, Cauchy’s

integral theorem can be applied to obtain the sum factors

?

(see Theorem B of Noble [29]) in which α lies above the contour for the + function

and below for the − function. Similarly, it is a simple matter to prove

?±1

which converges as written, and again α lies above C1 for r+(α) and below C1 for

r−(α). Note that this form of r±(α) is zero as well as singularity free in D±as

required. Both (63) and (64) are simple and relatively quick to evaluate numerically,

as will be discussed in the following sections.

s±(α) = ±

1

2πi

C1

tan−1(µf(ζ))

γ(ζ)(ζ − α)dζ

(63)

r±(α) = (1 + µ2)1/4exp

4πi

?

C1

ln[(1 + µ2f2(ζ))/(1 + µ2)]

ζ − α

dζ

?

,

(64)

4.2. Approximate noncommutative factorization. The commutative fac-

tors (58) have now been constructed, but they require modification in order to re-

move the branch cut singularities in J(α). To (approximately) achieve this end in a

remarkably simple fashion, it is convenient to replace f(α) in (55) by its [N/N] Pad´ e

approximant; that is,

f(α) = (γ(α)/δ(α))1/2≈ fN(α) =PN(α)

QN(α),

(65)

where PN(α),QN(α) are the polynomials

PN(α) = a0+ a1α2+ a2α4+ ··· + aNα2N,

(66)

QN(α) = 1 + b1α2+ b2α4+ ··· + bNα2N,

(67)

and N is any positive integer. Note that there is no need to replace the explicit

f(α) term in (54) by fN(α). The notation used above is that employed by Baker [9]

except for the labeling of the coefficients (which is amended due to the absence of

odd powers of α). Because workers in diffraction theory will, in general, be unfamiliar

with Pad´ e approximants, their main properties and derivation are summarized briefly

in Appendix A (see also the relevant work by Bruno and Reitich [10]). It suffices here

to mention that Pad´ e approximants are determined uniquely (for each N) from the

Taylor series expansion of f(α) about the origin (or any other regular point), and

they are also far superior to the Taylor series expansion for obtaining an accurate

approximation to the original function in the whole of the complex plane (away from

singularities). The polynomials in the numerator and denominator are chosen here

to have the same order in α to ensure that fN(α) tends to a constant at infinity.

Note that the zeros of PN(α),QN(α) are all simple, all lie on the real line between

1 < |α| < k (i.e., where the branch cuts are in f(α)), and as N increases so the

poles and zeros of the approximant crowd on these intervals. Choosing PN(α) and

Page 12

552

I. DAVID ABRAHAMS

QN(α) by any other method, for example by a collocation procedure, that is, enforcing

PN(α)/QN(α) = f(α) at a finite number of points on C1, will be a nonunique process

and in general leads to poles and zeros off the real line. As will be shown in the

following sections, excellent agreement is obtained between K(α) and its approximant

counterpart (< 10−7%) for even moderate values of N. This is because the Pad´ e

factorization appears to be an optimal approximate solution of the exact integral

equations defining the decomposition and rapidly tends to the exact solution as N →

∞ (see the complementary article on scalar approximant factorizations by the author

[3]).

The commutative factors (59), with f(α) approximated by (65), are now analytic

in their respective half-planes D+,D−except for simple poles at the zeros of PN(α)

and QN(α) in J(α). The approximation of J(α) is written as

?

and KN(α),Q±

JN(α). It is now a straightforward matter to remove the offending poles in Q±

To do this a new matrix M(α), say, can be constructed which yields

JN(α) =

0

γ2(α)/fN(α)

0

−fN(α)

?

,

(68)

N(α), etc., are as given by (54) and (58) but with J(α) replaced by

N(α).

K+

N(α) = M−1(α)Q+

N(α),

(69)

K−

N(α) = Q−

N(α)M(α),

(70)

where K±

N(α) are analytic in D±as required and because of this form

K−

N(α)K+

N(α) = Q−

N(α)M(α)M−1(α)Q+

N(α) = KN(α).

(71)

Thus, finding M(α) will lead directly to the exact explicit noncommutative factor-

ization elements of KN(α), namely K±

meromorphic elements, and the poles must correspond to the zeros of PN(α),QN(α).

In Appendix B such a matrix is determined directly by enforcing regularity conditions.

However, here it is simpler to construct the elements of M(α) by posing the ansatz

?

n=1

α2−q2

in which pn,qnare the locations of the N positive real zeros of PN(α),QN(α), respec-

tively, ordered as

N(α). Obviously M(α) must be a matrix with

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

?

,

(72)

1 < p1< p2< p3< ··· < pN< k,

The negative real zeros of PN(α),QN(α) are located at −pn, −qn, and An, Bn,¯An,

¯Bn are as yet undetermined constants. Note that Wiener–Hopf product factoriza-

tions are nonunique, and so there is no loss of generality in choosing M(α) as above.

Indeed, Appendix B is presented in order to justify the precise form of M(α). Before

expanding the right-hand sides of (69), (70) it will be necessary to write fN(α) and

its inverse in partial fraction form. That is,

1 < q1< q2< q3< ··· < qN< k.

(73)

fN(α) = C +

N

?

n=1

αn

α2− q2

n

,

(74)

Page 13

MATRIX WIENER–HOPF FACTORIZATION

553

1

fN(α) =1

C+

N

?

n=1

βn

α2− p2

n

,

(75)

where C is the value of fN(α) at infinity,

C = aN/bN≈ 1,

(76)

and the partial sum coefficients are easily shown to be

αn= 2qnPN(qn)/Q?

N(qn),βn= 2pnQN(pn)/P?

N(pn),

(77)

the?denoting differentiation with respect to α.

Now, substituting the approximant form of (58) and (72) into (70) and expanding

give the (1,1) element of K−

?

N(α):

r−(α)cos[γ(α)s−(α)]

1 +

N

?

n=1

Bn

α2− p2

?

n

?

+ r−(α)γ(α)

fN(α)sin[γ(α)s−(α)]

−α

N

?

n=1

An

α2− q2

n

?

.

(78)

This expression does not have poles at α = ±qnbecause they are cancelled by the

zeros of 1/fN(α). It will have poles at −pm,m = 1,...,N, i.e., in D−, unless its

residues are zero. This is achieved by setting

?

r−(−pm)cos[γ(−pm)s−(−pm)]

Bm

(−2pm)

+

βm

(−2pm)γ(−pm)sin[γ(−pm)s−(−pm)]pm

for each m, or, as γ(−pm) = γ(pm), s−(−pm) = s+(pm),

N

?

n=1

An

m− q2

p2

n

?

= 0(79)

Bm= pmβmγ(pm)tan[γ(pm)s+(pm)]

N

?

n=1

An

q2

n− p2

m

,m = 1,...,N.

(80)

Repeating for the (2, 1) element of K−(α) gives, for regularity in D−,

Am=

αm

qmγ(qm)tan[γ(qm)s+(qm)]

?

1 +

N

?

n=1

Bn

q2

m− p2

n

?

,m = 1,...,N.

(81)

One of the unknown column vectors—Bm, say—can be eliminated between these two

expressions, to yield the solution of the system as

A = (I − C)−1c

(82)

in matrix form, where I is the N × N identity matrix, c has components

cm=

qmγ(qm)tan[γ(qm)s+(qm)],

αm

m = 1,...,N,

(83)

Page 14

554

I. DAVID ABRAHAMS

and the square matrix C is given by

Cmn=

N

?

?=l

cmd?

?)(q2

(q2

m− p2

n− p2

?),m = 1,...,N, n = 1,...,N,

(84)

with

dm= pmγ(pm)βmtan[γ(pm)s+(pm)],m = 1,...,N.

(85)

Thus, (82) is a simple and efficient form for calculating the vector of coefficients A

(see section 6). The other vector is determined via the relation

B = DA,

(86)

where the N × N matrix D has elements

Dmn= (87)

dm

q2

n− p2

m

,m = 1,...,N, n = 1,...,N.

The procedure outlined above is repeated to eliminate poles in D−from the second

column of K−

N(α). Omitting details it is found that

¯B = (I −¯C)−1¯d,

(88)

in matrix form, where¯C is the square matrix

¯Cmn=

N

?

?=l

¯dm¯ c?

?)(p2

(p2

m− q2

n− q2

?),m = 1,...,N, n = 1,...,N,

(89)

and ¯ c,¯d are

¯ cm= cmq2

m,

¯dm= dm/p2

m,m = 1,...,N

(90)

in which cm, dmare given in (83), (85). The remaining vector of coefficients,¯A, is

determined via

¯A =¯D¯B,

(91)

where

¯Dmn=

¯ cm

m− p2

q2

n

,m = 1,...,N, n = 1,...,N.

(92)

This completes the construction of M(α) from the enforcement of analyticity of

N(α) in D−, but it does not necessarily enforce regularity of K+

the particular choice of ansatz for M(α), (72), can be shown by direct substitution

into (69) to yield identical values of the coefficients A, B,¯A,¯B (82), (86), (88), (91)

when poles are eliminated in D+. Furthermore, it is easily shown from (81) and (91)

that

?

n=1

?N

n=1

K−

N(α) in D+. However,

det(M(α)) = 1 +

N

?

?

Bn

α2− p2

¯Bn

α2− p2

n

??N

??

1 −

N

?

An

α2− q2

n=1

¯An

α2− q2

?

n

?

+ α2

n

?

n=1

n

(93)

Page 15

MATRIX WIENER–HOPF FACTORIZATION

555

has no singularities at ±pn, ±qn, n = 1,...,N. Thus, it is an entire function which

by inspection tends to the constant value unity as |α| → ∞, and so by Liouville’s

theorem

det(M) ≡ 1.

(94)

Therefore (69), (70) are the noncommutative matrix factors of the approximant kernel

KN(α), (71), and they, together with their inverses, are regular in D+, D−, respec-

tively. We have thus constructed approximate factorizations of the original matrix

kernel (35), and it is clear, although not yet proven (see [3] for proof in the scalar

case), that

K−

N→ K−,

K+

N→ K+

(95)

as N → ∞, where K−K+is the exact noncommutative factorization of K(α). Our

approximate solution should therefore be able to be made indefinitely accurate by

increasing N, and in the numerical evaluation section it will be shown that the ap-

proximate factorization matrices appear to converge very rapidly to the exact result

as N increases.

5. Solution of the boundary value problem. In the previous section an ap-

proximate noncommutative factorization of the Wiener–Hopf kernel into two matrices,

K−(α) and K+(α), regular in D−and D+, respectively, was obtained. These can be

employed in the Wiener–Hopf functional equation (34), which may be arranged into

the form

2iµ

(α + kcosθ0)+[K−(−kcosθ0)]−1

2iµ

(α + kcosθ0)+

K+(α)Ψ+(α) −

?

0

1

?

≡ E(α)

0

1

(96)

≡ [K−(α)]−1Ψ−(α) +

?[K−(α)]−1− [K−(−kcosθ0)]−1??

?

for α ∈ D. The last term on each side of (96) is included to move the simple pole at

α = −kcosθ0, which was defined in section 3 to lie in D−, from the right-hand to the

left-hand side. Thus, the left-hand side of (96) is analytic in D+, and the right-hand

side is analytic in D−, and so by analytic continuation both sides define a vector

function entire in the whole complex α plane D+∪ D−(α), E(α), say. The precise

form of this vector is determined by examining the behavior of both sides of (96) as

|α| → ∞ in their respective half-planes of analyticity. First, by inspection of (72) it

is clear that

M(α) = I + O(α−1),

|α| → ∞

(97)

in D+∪ D−, and by standard techniques we can show that

r±(α) = (1 + µ2)1/4+ O(α−1),

Similarly, asymptotic analysis of (63) reveals

|α| → ∞.

(98)

s±(α) = ±iτ

αlnα + O(α−1),

|α| → ∞

(99)

in the respective half-planes of analyticity, where

τ =1

πtan−1µ.

(100)

Page 16

556

I. DAVID ABRAHAMS

In view of the fact that the density ratio has range 0 < µ < ∞, we find

(101)0 < τ < 1/2.

Thus,

cos(γ(α)s±(α)) ∼ Kατ;

1

γ(α)sin(γ(α)s±(α)) ∼ ±iKατ−1

(102)

as |α| → ∞ in D±; and K is some constant. With these expansions, together with

(74), (76), substituted into (58) and (97), we find

K±(α) ∼ (1 + µ2)1/4K

?

ατ

±iατ+1

ατ

∓iατ−1

?

(103)

and also

[K±(α)]−1∼ (1 + µ2)−1/4K

?

ατ

∓iατ+1

ατ

±iατ−1

?

(104)

for |α| → ∞ in D±.

The next element is to estimate the sizes of Ψ+(α), Ψ−(α) for large |α|. If we

express the dimensionless coordinates in polar form,

X = Rcosθ,Y = Rsinθ,

(105)

then it is straightforward to perform a local analysis around R = 0 which reveals that

ψ1(R,θ) ∼ K1sin(τπ)Rτsinτ(π − θ),

ψ2(R,θ) ∼ K1cos(τπ)Rτcosτ(π + θ) − 2

(106)

(107)

as R → 0 for finite energy density at the edge. Here K1is some unknown constant

and τ is the edge exponent written in (100). Note that in the case of equal densities

in media 1 and 2, µ = 1 and so τ = 1/4 as already found by other authors (see

e.g., Rawlins [31]). There is a direct relationship between the small X expansion of a

function and the large α behavior of its half-range Fourier transform. Referring the

reader to the Abelian theorem quoted in Noble [29, equation (1.74)], and omitting all

detail, we find

Ψ+(α) ∼ K1sin(τπ)cos(τπ)Γ(τ + 1)eiτπ/2

?

−α−τ

iα−τ−1

?

(108)

for |α| → ∞ in D+and

Ψ−(α) ∼

?

0

−2iµα−1

?

+ K1sin(τπ)Γ(τ + 1)e−iτπ/2

?

−α−τ

iα−τ−1

?

(109)

for |α| → ∞ in D−, where Γ(z) is the gamma function.

Both sides of (96) can now be estimated for large |α|. From (103) and (108) the

left-hand side is

?

O

1

α−1

?

,

|α| → ∞,α ∈ D+,

(110)

Page 17

MATRIX WIENER–HOPF FACTORIZATION

557

and the right-hand side is

O

?

1

α−1

?

,

|α| → ∞,α ∈ D−.

(111)

Therefore, by Liouville’s theorem both sides must be equal to

E(α) ≡

?

C

0

?

,

(112)

where C is a constant. This gives the column vector Ψ+(α) as

??

which will have growth

Ψ+(α) = [K+(α)]−1

C

0

?

+

2iµ

(α+k cosθ0)+[K−(−kcosθ0)]−1

?

0

1

??

,

(113)

O

?

ατ

ατ−1

?

(114)

as |α| → ∞ unless C is chosen appropriately. To enforce the behavior written in (108)

it is straightforward to show that C must be

C = −2µ(0,1)[K−(−kcosθ0)]−1

?

0

1

?

,

(115)

and so

Ψ+(α) = 2µ[K+(α)]−1

?

i

(α+k cosθ0)+

0

−1

i

(α+k cosθ0)+

?

[K−(−kcosθ0)]−1

?

0

1

?

.

(116)

This choice of C also ensures the correct growth (109) for Ψ−(α). The scattered

potentials are now known in terms of this column vector via equations (21), (22),

(30), and (31). From (40) we write the solution to the boundary value problem in

dimensional coordinates as

?

ψ2(x,y) =

2π

C1

where C1lies in the strip D as indicated on Figure 2. This contour can be moved back

to C, or indeed to any path as long as it does not cross poles or branch cuts during

the deformation. This shows that the choice of path C1was chosen for convenience,

offering a way of estimating the error of using a Pad´ e approximant solution for K±(α)

(see the next section), and is not germane to the solution method.

By direct substitution of (117) and (118) into conditions (10)–(15) it is clear that

the above solution does indeed satisfy all elements of the boundary value problem.

However, K±(α) are known only approximately through the kernel factorizations

K±

section that K±

regularity as N increases.

ψ1(x,y) =

µ

2π

1

C1

e−iαxω/c1−γ(α)yω/c1(0,1)Ψ+(α) dα, y > 0,

(117)

?

1

δ(α)e−iαxω/c1+δ(α)yω/c1(1,0)Ψ+(α) dα,y < 0,

(118)

N(α) derived in the last section. We show by numerical example in the following

N(α) converge very rapidly to K±(α) in their respective half-planes of

Page 18

558

I. DAVID ABRAHAMS

FIG. 3. The percentage error eN(α(t)), (120), plotted on a Log-scale versus the parameter t,

(121), for N = 2,5,10,13, k = 4.

As a final point to note, if we require K+(α) in D−, and in particular near to the

branch cut between −k < α < −1, then the approximant solution will not be accurate

unless N is large. This is easily overcome, however, by employing the following form

for K+(α):

K+(α) = [K−(α)]−1K(α) ≈ [K−

N(α) is now an excellent approximation to K−(α) for moderate N in the

lower half-plane. The same technique can be employed if we require K−(α) in D+,

and the substitution (119) will be employed in the following section.

N(α)]−1K(α),α ∈ D−,

(119)

where K−

6. Accuracy of approximant solution and far-field evaluation. In this

section we will examine the accuracy and convergence of the approximant factoriza-

tions and illustrate the results by evaluating the far-field diffraction pattern in medium

1 for different Pad´ e numbers. In section 4 an exact factorization of the matrix kernel

KN(α) was obtained, namely K±

kernel K(α), differing only in the replacement of f(α) in J(α) by its Pad´ e approxi-

mant fN(α), (65). If fN(α) is an accurate approximation to f(α) on C1, which is any

infinite line in D, then obviously KN(α) will accurately approximate K(α) on this

line. We define the percentage error as

N(α). But KN(α) is an approximation to the original

eN(α) = 100 × |[fN(α) − f(α)]/f(α)|,

Further, from Abrahams [3] (see also Koiter [25] and Noble [29, Chapter 4.5]), it is

expected that K+

error less than or equal to max{eN(α) : α ∈ D}. The same applies to K−

D−. We therefore need only to calculate the error eN(α) along C1for different Pad´ e

numbers N in order to bound the errors of the approximant factors K±

Numerical experiments for a range of values of the density ratio, i.e., µ, and sound

speed ratio, k, were performed. All gave similar results, and, as expected, the nearer

k approached unity the better the accuracy of the Pad´ e approximants. However, this

convergence property was not found to be very strongly dependent on k. Hence, in

what follows the entirely typical values µ = 1.5, and k = 4 are selected. Figure 3

illustrates the (percentage) error in the Pad´ e approximants on the right half of the

C1path (only half shown because of symmetry) using the parametric representation

α ∈ D.

(120)

N(α) approximates K+(α) in the region of regularity, α ∈ D+, with

N(α) in

N(α) in D±.

Page 19

MATRIX WIENER–HOPF FACTORIZATION

559

FIG. 4. The (1,1) element of K+

µ = 1.5, k = 4.

N(α) plotted on the real line, 0 < α < 5, for N = 2,5,10,13,15,

eN(α(t)), where t is

α =

?

k?1 + eiπ(t/(2k)−1)?,

0 < t < 2k,

t > 2k. t,

(121)

The ordinate is presented on a logarithmic scale because of the dramatic increase

in accuracy with increasing N, and four approximant errors are plotted, namely, for

N = 2,5,10,13. For N = 2 the results reveal that the maximum possible error

in the factorization matrices K±

approximant number of N = 10 (i.e., 10 zeros and poles to replace each finite branch

cut) the upper bound on the error is a mere .00001%. At N = 13 the error is at most

6 × 10−8% and for N ≈ 20 the error is smaller than the rounding error in double

precision on most microcomputers!

Figure 3 indicates the strong convergence of the approximant matrices to the exact

solution. Yet from [3] we can expect the product factorization matrices to converge

even quicker than that shown, and Figure 4 bears this out. The first element of K+

is plotted along the real line segment 0 < α < 5 for N = 2,5,10,13,15 and for k = 4,

µ = 1.5. All these functions are very close in value, and indeed only the N = 2 result

can be discriminated in this figure. It is clear, therefore, that significant convergence

has taken place by N = 5. The plot of the imaginary part of the (1,2) element of

K+

and on examination of the numerical results it is found that K+

differ by more than .00002% for α lying in the first quadrant of the complex plane. As

regards speed of computation, all calculations were carried out using Mathematica on

a 90-Mhz Pentium personal computer. This generated fN(α) and all the coefficients

of M(α) in a few seconds for small systems N = 2,5 and in 160 seconds for N = 13.

The far-field diffraction pattern shown in Figure 6 took approximately 4 minutes to

generate, and this time was not found to alter significantly with Pad´ e number N.

As a final illustration, and to reveal the likely convergence properties (as N in-

creases) for the solutions ψ1, ψ2, we will examine the far-field behavior. Since this pa-

per does not aim to present an exhaustive study of the physical model, we will confine

our attention to y > 0, and for simplicity of analysis we again choose µ = 1.5, k = 4,

and incident wave angle (7) θ0= π/3. This means the pole at α = −kcosθ0= −2

lies to the left of the branch point at −1, and the inverse contour path passes above

N(α) is 2.5%. However, increasing to a still modest

N(α)

N(α) along the line .01 + it, 0 < t < 5, Figure 5, further demonstrates this point,

5(α) and K+

13(α) never

Page 20

560

I. DAVID ABRAHAMS

FIG. 5. The imaginary part of the (1,2) element of K+

α = .01 + it, 0 < t < 5, for N = 2,5,10,13, µ = 1.5, k = 4.

N(α) plotted along the vertical line,

FIG. 6. The modulus of the diffraction coefficient, (125), versus θ (radians) in the upper medium

plotted for different approximant values N = 2,5,10,13 and µ = 1.5, k = 4.

it. We now perform the standard treatment on the integral in (117) to employ the

method of stationary phase: first we change to polar coordinates

x = rcosθ,y = rsinθ,

0 < θ < π,r > 0 (122)

and make the substitution α = −cos(θ + it) after deforming the contour path into a

hyperbola passing through the point α = −cosθ. No poles or other singularities are

crossed in this deformation. The integral may now be rewritten as

?∞

where t lies on the real line. The dominant contribution to this integral comes from

near t = 0, and by usual arguments we find

ψ1=iµ

2π

−∞

eirω

c1cosht(0,1)Ψ+(−cos(θ + it))sin(θ + it) dt,

(123)

ψ1(r,θ) ∼ R(θ)eirω/c1

?rω/c1

,r → ∞,

(124)

Page 21

MATRIX WIENER–HOPF FACTORIZATION

561

where the diffraction coefficient is

R(θ) =µe3iπ/4sinθ

√2π

(0,1)Ψ+(−cosθ),

0 < θ < π,

(125)

and Ψ+(α) is written in (116). Figure 6 is a plot of |R(θ)| over all 0 < θ < π and for

N = 2,5,10,13. As can be seen, it is impossible to distinguish between the different

curves. That is, the approximate solutions, including that for N = 2, give results

which are remarkably close to each other! This indicates that the upper bounds on

the errors of the approximate factorizations are indeed too pessimistic for most results

of physical importance (Abrahams [3]).

As a cautionary note, it must be stated that numerical difficulties are encountered

if the Pad´ e number is too large. This is because, for large N, the poles and zeros crowd

onto the branch cut intervals, thus making an accurate determination of their locations

both essential and rather delicate. Typically the Mathematica routine NSolve started

to introduce inaccuracies at around N ≈ 20, but with a little care this technical

difficulty can be overcome until a much larger Pad´ e number. However, we conclude

that, from the numerical evidence, the Pad´ e factorization matrices K±

very rapidly with N to the exact solutions K±(α) and so this difficulty is entirely

irrelevant. The last figure suggests that using a Pad´ e number of, say, N = 5 or 10 will

yield both an accurate and a computationally efficient solution of the boundary value

problem, and even one as small as N = 2 will probably suffice for most purposes.

N(α) converge

7. Concluding remarks. This paper has presented a new method for obtaining

approximate factorizations of Wiener–Hopf kernels using Pad´ e approximants. As well

as illustrating a technique for matrix kernels, it is also of potentially significant use in

scalar problems (Abrahams [3]). This is because the approximant factorization form

eliminates the necessity of using the Cauchy integral representation and so removes the

problem of singularities of the integrand lying near to or on the integration path. The

scalar Pad´ e approximant representation is the natural extension of Koiter’s method

[25], and it can make significant savings in computer time for calculations involving

complex scalar kernels (e.g., that found in [11]). The present approach can also be

contrasted with an approximate scheme for scalar factorization suggested by Carrier

[12] and used recently on a specific problem by Dahl and Frisk [13]. The simplicity

of the factorization procedure shown in section 4 is due to the fact that quotients of

polynomials (Pad´ e approximants) are trivial to decompose into product terms with

zeros and poles in respective half-planes, and the residues at these points are simple

to evaluate. Further, by increasing the approximant number N the accuracy can be

raised to very high levels, and also there is a comprehensive literature on the full

behavior of approximants. The rapid convergence (with N) of the factorization forms

to the exact solution, shown here for a specific example, is entirely typical. This is

because Pad´ e approximants very accurately imitate the behavior of complex functions

with very general singularity structure away form these singularities.

In the previous section an upper bound on the error of the approximation, K±

α ∈ D±, was given as max{eN(α) : α ∈ D}. The proof that the error is greatest on

the boundary of the half-plane domain is given in [3], and in this article an S-shaped

boundary (Figure 2) was taken as it was bounded away from the finite branch cuts of

f(α) (56). A better (i.e., lower) upper bound on the maximum error may be found by

deforming the strip D in any way as long as it does not cross any singularities of K(α).

Thus, the operation of deformation of D performed in this article is merely employed

to obtain an upper error bound and does not alter the error in approximating K±(α)

N(α),

Page 22

562

I. DAVID ABRAHAMS

itself. The final solution is completely independent of this choice of path. As was

shown in the numerical examples, the error of approximation of K±(α) decreases

rapidly as α moves to an interior point of D±. Fortunately, we almost always require

these kernel factors at interior points and so good accuracy is always expected even

for moderate Pad´ e number N.

A crucial step in the formulation of the approximate factorization is to ensure that

the matrix J(α) (55) has the correct behavior. That is, the elements have polynomial

form as |α| → ∞, i.e., any branch point at infinity has vanishing coefficient, and J2(α)

is proportional to the identity matrix. All problems so far examined by the author

have been able to be cast into this form without difficulty, which is why the method

propounded should have wide applicability. It is interesting to note that Wickham’s

integral equation approach [33] also involves removing terms which are the correct

factors at infinity in the transform space. These rather complex terms as expressed in

his formulation are nothing other than the simple commutative factors Q±(α) in the

present setting! Wickham then shows that the coupled second-kind integral equations

remaining are solvable but require a good deal of effort to evaluate, and this must then

be followed by the nontrivial task of constructing a second eigenvector. The present

approach circumvents the need to tackle the integral equations for the exact elements

of M(α), and this is its key simplifying step. However, in order to understand why the

Pad´ e representation seems to be, in some sense, an optimal approximation, ongoing

research is concerned with the relationship between collocation solutions of the exact

integral equations and the Pad´ e approximants.

To illustrate the important range of physical problems now amenable to solution,

the author has examined elastic wave scattering in an elastic half-space lying in y < 0,

half the surface of which (x < 0, y = 0) is clamped. The remaining half of the surface

x > 0 is traction free, and a Rayleigh wave is incident from infinity along this surface.

The interesting variation in the reflection coefficient of the surface Rayleigh wave as

a function of Poisson’s ratio is calculated in Abrahams [4]. Other work in progress

[3] includes the approximate factorization of scalar kernels arising in fluid-structural

diffraction problems [11], and other models in static and dynamic elasticity.

Appendix A. Derivation of Pad´ e approximants. Given a function f(z),

regular at the origin, with a Taylor series expansion

f(z) =

∞

?

n=0

enzn,

(A.126)

say, then the [L/M] Pad´ e approximant to f(z) is

[L/M] =a0+ a1z + a2z2+ a3z3+ ··· + aLzL

1 + b1z + b2z2+ b3z3+ ··· + bMzM.

The [L/M] approximant has the same Taylor series expansion as f(z) up to and

including order L + M, i.e.,

(A.127)

∞

?

n=0

enzn− [L/M] = O(zL+M+1).

(A.128)

Generally speaking, [L/M] Pad´ e approximants exist and are uniquely determined by

L, M, and the first L+M +1 coefficients of f(z), namely, the en. Multiplying (A.128)

Page 23

MATRIX WIENER–HOPF FACTORIZATION

563

by the denominator polynomial of [L/M] and equating coefficients of zL+1, zL+2, ...,

zL+Mgive an algebraic system of equations:

Eb = −eL,

(A.129)

where

b = (b1,b2,...,bM)T,

ej= (ej+1,ej+2,...,ej+M)T,

(A.130)

(A.131)

and E is the M × M matrix formed from the column vectors

E = (eL−1,eL−2,...,eL−M).

Note that we define en ≡ 0 for n < 0 in the above expressions. Inverting (A.129)

gives all the denominator coefficients, bn, 1 ≤ n ≤ M, and, once determined, the ans

are found by equating coefficients of zn, 0 ≤ n ≤ L. It must be stated that particular

[L/M] approximants of a given function may not exist if the system (A.129) is not

invertible. Details of these cases, and methods for predicting the special [L/M] values,

can be found in Baker [9]. For further information on approximants, see [18, 19].

The essential property of Pad´ e approximants is that they give an approximation

to f(z) far beyond the radius of convergence of its Taylor series expansion. This is

due to the fact that f(w), where

(A.132)

w(z) =

z

1 + cz,

c = constant,

(A.133)

will yield the identical [L/M] approximant to that of f(z) for any c. This invariance

property ensures good agreement between the function and its approximant every-

where in the z-plane except near to the singularities of f(z). Obviously approximants

can only have pole-type singularities, and these may have to represent branch cuts,

essential singularities, etc. In practice, this does not present any difficulty, and the

Pad´ e singularities cluster along cuts or near to actual point singularities of f(z). As a

final point, we could have derived the approximants by equating them with a Taylor

expansion at any regular point of f(z), not just at the origin. This would lead to a

slightly different Pad´ e approximant than that deduced from z = 0, but by a simple

transformation in independent variable, the above procedure can still be followed.

In this article the function f(α), (65), which we approximate, has constant be-

havior as |α| → ∞. Thus, to mimic its behavior everywhere, including the point at

infinity, it is necessary to employ [N/N] Pad´ e approximants. Further, as f(α) is only

a function of α2, in section 4 we use a slightly different notation to that specified in

this appendix. The differences should be transparent to the reader.

Appendix B. Construction of the meromorphic matrix M(α). In section

(4.2) the noncommutative matrix factors K±

phic matrix M(α), (72). In the text M(α) was posed as an ansatz with unknown

coefficients, but for general matrix Wiener–Hopf kernels the form of the regularizing

matrix M(α) will not be easy to spot. This appendix demonstrates a procedure which

will construct M(α) and its coefficients An, Bn,¯An,¯Bnfor the given Wiener–Hopf

kernel (35), and it can be applied in an identical fashion for any kernel.

Our aim is to construct the noncommutative factors K±

mutative split matrices Q±

Nwere constructed using the meromor-

Nonce the partial com-

Nare determined. From (71) we know that

Q+

N[K+

N]−1= [Q−

N]−1K−

N,

(B.134)

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.