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Emb edding formulae for scattering by three-dimensional structures
E. A. Skelton1, R. V. Craster2, A. V. Shanin3 & V. Valyaev
1

3

3

Department of Mathematics, Imperial Col lege of Science, Technology and Medicine, London SW7 2BZ, U.K. 2 Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada Department of Physics (Acoustics Division), Moscow State University, 119992, Leninskie Gory, Moscow, Russia

Abstract The far field diffraction b ehaviour for canonical scattering problems involving corners or sectors, in three-dimensions, are considered. The far-field results are obtained using ideas based up on emb edding formulae and therefore complement and extend existing results. Sp ecific geometries such as the flat cone and a corner formed by a solid octant are considered in detail. The formulae derived for the diffraction b ehaviour are also computed in sp ecial cases and compared with known results. Key words: Diffraction by cones; Emb edding formulae

1

Introduction

The diffraction of waves by a sharp edge is a fundamental problem in many areas: fracture mechanics, radar cross-section measurements, non-destructive evaluation of structures and acoustic wave scattering, and much work has taken place on characterising the far field scattered by canonical ob jects: a half-plane, a wedge or a cone for use in conjunction with Keller's geometrical ray theory of diffraction [1]. Two-dimensional structures such as wedges or half-planes are typically approached using the Sommerfeld integral [2] or the Weiner-Hopf technique [3] and there is a vast literature extracting the farfields for various canonical scattering geometries. However, in three-dimensions much less has been done, although naturally cones of circular cross-section [4,5] do allow for some simplification and can be approached through KantorovichLebedev transforms; other shapes such as elliptical cross-section cones can also be tackled [6,7] and this includes the degenerate case of a flat-cone. General
Preprint submitted to Elsevier Science 28 August 2009


three-dimensional corners and the flat-cone (a sector of a plane) can be tackled using an approach pioneered by Smyshlyaev and co-workers [8­10]. Their approach is to extract the far-field using properties of the Laplace-Beltrami operator on the unit sphere and the far field is represented as a contour integral involving the spherical Green's function of the Laplace-Beltrami operator. Numerical evaluation of the far-field is not easy, [10,11], and can be timeconsuming. As noted by one of us, [12], for a flat quarter plane computations are optimised using embedding formulae. In this case one has to evaluate the far-field created by an edge Green's function and from it construct the far field for any plane wave incidence, thus after solving a single "master" canonical problem the others are then just a manipulation of this. In the current article we build upon [12] showing that similar ideas can be used for a planar sector of any angle, and we also extend the ideas to fully three-dimensional shapes such as the edge of a cube (a solid o ctant). The latter extension relies upon the higher order operators intro duced in [13] for wedge geometries and illustrate how the two-dimensional ideas used there translate into three dimensions. Embedding formulae are a relatively new addition to the techniques utilised in diffraction theory: the fundamental idea is that instead of solving, and re-solving, the physical problem of interest for each different incoming plane wave angle of incidence, instead one solves a single (or much reduced number of ) canonical problem(s). Then one constructs the far field of the physical problem just in terms of the canonical far field. Thus, numerically one need only solve the canonical problems, the subsequent manipulations are then a trivial numerical exercise. The metho d emerged for scattering by planar strips (cracks) in two-dimensional acoustics and initially utilised integral equation techniques. Williams [14] showed that, for a finite straight rigid strip, the directivity for all incident angles is obtained from just that found for a plane wave incident at the grazing angle. It is a rather remarkable result that hints at some deeper result buried within the governing equation and asso ciated boundary conditions. Building on this framework several authors pursued this integral equation approach for a variety of scattering problems [15­21]. The approach taken in the current article follows a slightly different route. In [22] it was shown that embedding formulae emerge directly from the governing equations independent of the solution pro cedure and three dimensional objects could also be embedded (see also [12]). The approach utilises recipro city, the intro duction of a differential operator that generates an eigensolution that has edge conditions that are more singular than usual, and uniqueness. The canonical problem requires an edge Green's function with, in two (three) dimensions, a line (point) source placed on the sharp edges. For straight and parallel cracks or strips the number of canonical solutions is equal, in the absence of any symmetries, to the number of edges in two-dimensions. More recently [13,23] extended embedding formulae to wedge and angular geometries (of rational angle) or to cracks inclined at angles to each other, although this is still in two-dimensions. The overly singular edge Green's function ap2


proach is arguably a bit awkward to use with existing numerical schemes and [24] demonstrates how linear superposition can be used to create embedding formulae using the far fields from problems involving incoming plane waves. The current article takes the ideas utilised in [13], namely the higher order operators required for wedge geometries, and shows how they can be used to generate embedding formulae for some three dimensional structures namely planar sectors and an o ctant; the planar sectors are a generalisation of the quarter plane result [12] to any angle. To illustrate the utility of the formulae derived we evaluate the far-field numerically for the sector and compare it with that derived in [8­10]. The structure of this article is as follows: To illustrate the ideas behind embedding formulae and the three-dimensional extension we begin in section 2 with the flat cone. This is an extension of the quarter-plane treated in [12] to a sector of any angle (< ). Crucial to the embedding metho dology are edge Green's functions, described in section 2.1.1 and a differential operator that transforms the standard problem into an overly singular eigensolution. The differential operator takes different forms and can be of either first or second order. Sections 2.1.2-2.1.4 illustrate this and generate the appropriate embedding formulae. A brief numerical verification and comparison of the embedding formulae versus the standard diffraction results is undertaken in section 2.2. The flat cone is quasi-two-dimensional, and closely related to the quarter-plane, and so we move on to consider fully three-dimensional corner structures. In section 3 a corner formed by a solid o ctant is considered and operators utilised in [13] for right-angled wedges (in two-dimensions) are used. Edge Green functions (section 3.1.1) again play a vital role and embedding formulae are again deduced. We briefly discuss extensions to yet more general geometries showing that the embedding idea is not restricted to simple geometries; this and some closing remarks are in section 4.

2

The Flat Cone
i t

For time-harmonic motion, where the variables are proportional to e- wave equation becomes the Helmholtz equation
2 ( 2 + k 0 ) u = 0 ,

, the

(2.1)

with k0 = /c, and c the wave speed. In the first problem considered here the Helmholtz equation is satisfied everywhere in 3-dimensional space, described by Cartesian co ordinates (x, y , z ). A flat cone scatterer, shown in figure 1, is present in the form of a sector of a plane, x > 0, 0 < y < x tan , z = 0, upon which the Dirichlet condition u=0 (2.2) 3


z Incident plane wave

y 1 edge 1
1



2 edge 2 2

x

Fig. 1. The planar sector showing the notation used for angles.

is satisfied. The lines x > 0, x > 0, y = 0, y = x t a n , z = 0, and z = 0, (2.3) (2.4)

are the edges of the sector, and are denoted by 1 and 2 , respectively. The angle between the edges is . Hence, when = /2 this reduces to the problem addressed by Shanin [12]. The incident field is the plane wave uin = exp[-i(kx x + ky y + kz z )] (2.5)

2 2 2 2 where k0 = kx + ky + kz , and, as in [12], the edge conditions from the theory of diffraction by an ideal half-plane are
1 2 u 1,2 sin

1, 2

2

(2.6)

where 1,2 and 1,2 are lo cal cylindrical radial and angular co ordinates measured from the edges 1 and 2 , respectively. The vertex conditions, [12], are u = O (1), u = o ( r
-
1 2

) as r 0,

(2.7)

and these ensure that the energy remains lo cally finite. The total field u consists of the incident plane wave and a scattered field. Overall the scattered field has complicated structure, due to plane wave scat4


tering at the surface, and diffraction around the edges and the vertex. In the far-field the leading order terms of the scattered field are identified as a plane wave present in the geometric scattered region, together with a diffracted term decaying as 1/r as r where r is the distance from the vertex, together with higher order terms. Thus the diffracted term has the form u ( , r ) = 2 eik0 r f ( ) + O (ei k0 r
k0 r

(k 0 r )- 2 ),

(2.8)

where represents the angular co ordinates of the observation point (e.g. spherical polar co ordinates). The diffraction co efficient f ( ) also depends on the angle of incidence of the plane wave, 0 , and to make that explicit it is convenient to write f = f ( ; 0 ). (2.9) The ob jective of this paper is to find expressions for the diffraction co efficient f ( ; 0) in terms of the edge Green's functions for the problem. Hence, if the far-field edge Green's functions are known everywhere, the solution to the scattering problem is constructed for any combination of angle of incidence and observation, without the need to re-solve the problem each time.

2.1 Flat Cone Embedding Formulae

We begin with the flat cone (planar sector) and demonstrate that different embedding formulae are found for various operators and provide a brief numerical comparison with results from the standard approach.

2.1.1 Edge Green's functions Edge Green's functions, Gx (x, y , z ; X ) and Gs (x, y , z ; S ), are intro duced via a limit pro cess. For Gx (x, y , z ; X ) this is achieved by placing a point source of strength / in the plane of the scattering cone, at a small distance away from the edge 1 , with X the distance from the vertex to the nearest point on the edge to the source, and taking the limit as 0: 2 + k
2 0

^ Gx (x, y , z ; X, ) = on

(x - X ) (y + ) (z ),

(2.10) (2.11) (2.12) (2.13)

^ Gx (x, y , z ; X, ) = 0

x > 0 , 0 < y < x t a n , z = 0 , 1 1,2 2 ^ on 1,2 , Gx (x, y , z ; X, ) 1,2 sin 2 1 ^ ^ Gx (x, y , z ; X, ) = O (1), Gx = o(r - 2 ) as r 0, 5


and ^ Gx (x, y , z ; X ) = lim Gx (x, y , z ; X, ).
0

(2.14)

In the following analysis, the lo cal behaviour of Gx (x, y , z ; X ) near the edge 1 is required, and in particular the behaviour there of integrals of the type


I (x, y , z ) =

0

h(X )Gx (x, y , z ; X )dX.

(2.15)

This lo cal result is given in [12], as
1 1 1 h(x) - 1 2 I (x, 1 , 1 ) = - 1 2 sin + O 1 sin 2 2

, near 1 .

(2.16)

It is long (the and

a special case of the result (A.9), for a point source near an infinitely wedge, with r0 = , 0 = and source strength / rather than -4 , leading order term), together with a term `source-free' on the edge 1 represents the field scattered by the edge 2 , (the O ( 1 ) term).

The Green's function corresponding to the other edge, 2 , Gs (x, y , z ; S ), is defined in a similar way, with s being the co-ordinate measured from the origin along the edge 2 , and the argument S is the distance from the origin along that edge of the source. It is convenient to define some further notation here. Following Shanin [12], the directivities fx and fs of the edge Green's functions are defined from their far-field asymptotic expansions as eik0 r fx ( ; X ) + O ei k0 r eik0 r fs ( ; S ) + O ei k0 r

Gx (x, y , z ; X ) = 2 Gs (x, y , z ; S ) = 2 and by symmetry

k0 r

(k 0 r )

-2

, ,

(2.17) (2.18)

k0 r

(k 0 r )

-2

Gx (x, y , z ; X ) = Gs (x cos + y sin , x sin - y cos , z ; X ), fx ( , ; X ) = fs ( cos + sin , sin - cos ; X ), in which ( , ) are "Cartesian" co-ordinates = sin cos , = sin sin ,

(2.19) (2.20)

(2.21)

representing the pro jection of the point on the x-y plane. Additionally, the asymptotic behaviour of the edge Green's functions, Gs and Gx , near the opposite edges, 1 and 2 , respectively, are 6


2CG (x; S ) 1 2 1 sin 2 C G (s ; X ) 1 2 Gx (s, 2 , 2 ; X ) = 2 sin Gs (x, 1 , 1 ; S ) =

3 1 2 + O 1 , near 1 , 2 3 2 2 + O 2 , near 2 , 2

(2.22) (2.23)

where (1 , 1 , x), (2 , 2 , s) are cylindrical polar co-ordinate systems with 1 ,2 respectively being the axis. Recipro city allows the position of source and observer to be interchanged, hence CG (x; s) = CG (s; x). (2.24)

2.1.2 Embedding formula using the operator Hx = / x + ik

x

The embedding technique developed in [22]--[23] is applied here. Briefly, this entails finding an operator that, when applied to the total field of the original problem, pro duces a new field that satisfies the original equations, the boundary conditions, the radiation condition and which has the appropriate asymptotic behaviour near to any edges to ensure that it is `source-free' there. When these properties are satisfied, uniqueness arguments apply to show that the new field is identically zero everywhere. Far-field asymptotic analysis of the formal definition of the new field then allows an expression for the directivity of the original scattered field to be extracted. In this section the operator Hx = + ikx , x (2.25)

which has been used successfully in the past, [22], is applied to the total field u. The resulting field Hx [u] has the following properties: (1) Hx [u] satisfies the Helmholtz equation (2) Hx [u] satisfies the Dirichlet boundary condition on the surfaces of the scatterer (3) Hx [uin ] 0 and hence Hx [u] = Hx [us ], in which us is the outgoing scattered field of the original problem. Thus Hx [u] also satisfies the radiation condition. However, the remaining requirement that the new field should have the correct 1 2 `source-free' behaviour at the edges ( 1,2 as in equation (2.6)), although met on 1 , is not met on 2 , as shown below: From equation (2.6) the field near 1 is expressed in the form u(x, 1 , 1 ) =
1 2Cx (x) 1 2 2 1 sin 1 + o(1 ) near 1 , 2

(2.26)

7


in which Cx (x) is an as yet unknown function of x. Hence Hx [u] =
1 2 (Cx (x) + ikx Cx (x)) 1 1 2 2 1 sin + o(1 ) near 1 , 2
1 2 which has the required 1 behaviour. Similarly, near to 2 the field is 1 2 C s (s ) 1 2 2 2 2 sin + o(2 ). 2

(2.27)

u(s, 2 , 2 ) =

(2.28)

Differentiation with respect to x is accomplished by considering a rotation of the x and y axes through angle , so they correspond with the s axis and ~ perpendicular to it a t axis, such that ~ 2 = t2 + z 2 , 2 and ~ s = x cos + y sin , t = -x sin + y cos . Performing the differentiation gives, near 2 Hx [u] =
1 2 - sin Cs (s) - 1 2 2 2 sin + O 2 2

~ tan 2 = -z /t,

(2.29) (2.30)

(2.31)

and hence Hx [u] do es not have the required behaviour there and so, in this particular example, appears to fail the required criteria for generating embedding formulae. However, from equations (2.15) and (2.16) another expression can be constructed with this leading order behaviour near to 2 :


sin

0

Cs (S )Gs (x, y , z ; S )dS =

1 - sin Cs (s) - 1 2 2 2 2 sin + O 2 , (2.32) 2

and hence, by defining,


u = Hx [u] - sin

0

Cs (S )Gs (x, y , z ; S )dS ,

(2.33)

a function is constructed which has the required asymptotic behaviour near the edge 2 . The second term above is a superposition of Green's functions with sources along the edge 2 , each of which satisfies the Helmholtz equation, the boundary conditions on the scatterer faces, and has the required edge behaviour near to 1 . Importantly, after a detailed study one can prove that the vertex condition is also satisfied. Hence the combination of terms u satisfies all the requirements needed for the embedding metho d. Then, from uniqueness arguments u 0 (2.34) and Cs (S )Gs (x, y , z ; S )dS. (2.35) Hx [u(x, y , z )] = sin
0

8


This is the weak form of the embedding formula, and corresponds to Shanin's [12] equation (18) for a cone with angle /2, but now generalised to a cone of general angle . The, at present, unknown function Cs is determined using a recipro city argu^ ment, between a point source of strength / used in the definition of Gs (and hence in the limiting case as 0 of Gs ) at distance S along edge 2 and a point at angular lo cation 0 at distance r in the far-field. The leading order term of the far-field due to the source with this amplitude lo cated near to 2 , G21 say, is obtained directly from equation (2.18) G
21

2

eik0 r fs (0 ; S ). k0 r

(2.36)

If the source is lo cated in the far-field then, near the vertex of the scatterer the incident field from the source is approximately that of a plane wave of amplitude - /eik0 r /4 r . Hence the leading order term of the total field near the point S on 2 , u12 say, is obtained, by comparison with equation (2.28) which is for a plane wave of unit amplitude, as u
12

2 eik0 r 2Cs (s) 1 2 2 sin - 4 r 2

near 2 .

(2.37)

At the point under discussion, s = S , 2 = and 2 = , hence G
12

-

eik0 r 2Cs (S ) -eik0 r Cs (S ) . = 4 r 2 r

(2.38)

Applying the recipro city principle, that the same result is obtained if the source and observer positions are interchanged, shows that G
12

=G

21

(2.39)

and hence an expression for the unknown function Cs (S ) is obtained as C s (S ) = - ( 2 )2 fs (0 ; S ), k0 (2.40)

allowing equation (2.35) to be expressed as Hx [u(x, y , z )] = -(2 )2 sin k0
0

fs (0 ; S )Gs (x, y , z ; S )dS.

(2.41)

After noting that in the far-field / x corresponds to ik0 , the far-field approximation of this equation results in the embedding formula 4 2 i sin f ( ; 0 ) = 2 k 0 ( + 0 )
0

fs (0 ; S )fs ( ; S )dS.

(2.42)

9


corresponding to Shanin's [12] equation (12) in the limit as the sector angle is /2, but now generalised to a flat cone of general angle .

2 .1 .3

Using the operator Hy = / y + ik

y

In [12] Shanin also presents a first order embedding formula based on the operator Hy for the problem of scattering by a quarter plane. This is particularly appropriate to that geometry since the y -axis is the second edge of the scatterer and the result follows immediately by symmetry. This is not the case for a scatterer of arbitrary angle, considered here. It is clear that the field defined by Hy [u] satisfies the requirements listed as 1­3 of the previous subsection, but in this case it fails the requirement of the edge behaviour, not only on 2 but also on 1 . Fortunately a slight mo dification allows us to subtract the singular behaviour along both of the edges, setting


(2.43) pro duces a field satisfying all the requirements for the embedding metho d. Thus u 0, and


u = Hy [u] -

0

Cx (X )Gx (x, y , z ; X )dX + cos

0

Cs (S )Gs (x, y , z ; S )dS ,

(2.44)

Hy [u(x, y , z )] =

(2.45) as the weak form of the embedding formula for this operator. Equation (2.40) relating Cs (S ) and fs (0 ; S ), derived previously, holds in this case to o, together with the corresponding equation relating Cx (X ) and fx (0 ; X ). Substituting these into equation (2.45) and applying the far-field approximation results in the embedding formula f ( ; 0 ) = 4 2 i 2 k0 ( + 0 )
0

0

Cx (X )Gx (x, y , z ; X )dX -cos

0

Cs (S )Gs (x, y , z ; S )dS ,

fx (0 ; X )fx ( ; X )dX - cos

0

fs (0 ; S )fs ( ; S )dS ,

(2.46) corresponding to Shanin's [12] equation (13) for a cone with angle /2, but now generalised to a cone of general angle . Similarly we can use the operator Hs = / s + ik0 (0 cos + 0 sin ) as a rotation of axes gives


Hs [u] = cos Hx [u] + sin Hy [u] = sin

0

Cx (X )Gx (x, y , z ; X )dX , (2.47)

10


to get f ( ; 0 ) = 4 2 i sin 2 k0 (cos ( + 0 ) + sin ( + 0 ))
0

fx (0 ; X )fx ( ; X )dX ,

(2.48) which could also be considered to correspond to Shanin's [12] equation (13) for a cone with angle /2, but generalised to a cone of general angle .

2.1.4 Using a "second order" operator Embedding formulae are also obtained using suitable higher order operators, provided the requirements outlined above are met. In [12] the second order operator Hxy [u] = Hx [Hy [u]] is used to obtain a different embedding formula for the right angled flat cone. Here, for a flat cone with angle a corresponding embedding formula is obtained using the operator H
xs

=

+ ik0 x

0

+ ik0 (0 cos + 0 sin ) . s

(2.49)

The asymptotic behaviour near the 2 edge is found, from equations (2.31) and (2.32), to be Hxs [u] -


sin Cs (s) - 1 2 + ik0 (0 cos + 0 sin ) 2 2 sin s 2

Cs (S ) + ik0 (0 cos + 0 sin )Cs (S ) Gs (x, y , z ; S )dS. S 0 (2.50) Similarly, near 1 the asymptotic behaviour is sin 1 sin Cx (x) - 1 + ik0 0 1 2 sin x 2 Cx (X ) + ik0 0 Cx (X ) Gx (x, y , z ; X )dX. sin X 0 (2.51) Thus a field u satisfying all the properties required is constructed:


Hxs [u] -

u = Hxs [u] - sin


0

Cs (S ) + ik0 (0 cos + 0 sin )Cs (S ) Gs (x, y , z ; S )dS S (2.52)

- sin

0

Cx (X ) + ik0 0 Cx (X ) Gx (x, y , z ; X )dX. X 11


Hence, again u 0, and thus the weak form of embedding from this operator is


Hxs [u(x, y , z )] = sin + sin

0 0

Cs (S ) + ik0 (0 cos + 0 sin ) Cs (S ) Gs (x, y , z ; S )dS S Cx (X ) + ik0 0 Cx (X ) Gx (x, y , z ; X )dX. (2.53) X

After noting that equation (2.40) for Cs , and the corresponding relation for Cx remain valid, and applying the far-field asymptotic expansions the embedding formula obtained using this second order operator is 4 2 sin в 3 k0 ( + 0 ) (cos ( + 0 ) + sin ( + 0 )) fs (0 ; S ) + ik0 (0 cos + 0 sin ) fs (0 ; S ) fs ( ; S )dS S 0 + fx (0 ; X ) + ik0 0 fx (0 ; X ) fx ( ; X )dX . (2.54) X 0

f ( ; 0 ) =

This formula requires derivatives with respect to the source position of the directivities of the edge Green's functions, whose calculation requires more computational effort than the Green's functions themselves. It is therefore desirable to rearrange the formula to avoid the necessity of calculating them. This is achieved by first considering Gx (x, y , z ; X )/ X , which satisfies the Helmholtz equation everywhere except at the source position. However, the combination Gx / X + Gx / x has no singularity as x X and therefore satisfies the Helmholtz equation everywhere. Additionally it satisfies the boundary conditions on the faces of the flat cone, the radiation condition, and the edge condition on 1 . It do es however still have singular behaviour near 2 , since, following the metho d outlined in (2.28)­(2.32), but with u replaced now with Gx and Cs (s) replaced now with CG (s; X ),
1 Gx - sin CG (s; X ) - 1 2 2 2 2 sin + O 2 x 2

near
1

2



sin Hence,

0

2 CG (S ; X )Gs (x, y , z ; S )dS + O 2

near 2 .

(2.55)

Gx Gx + - sin X x

0

1 2 CG (S ; X )Gs (x, y , z ; S )dS O 2

near 2 , (2.56)

12


and the left hand side of this equation now satisfies the Helmholtz equation, the boundary conditions on the faces, the edge conditions on both edges and the radiation condition. Therefore, by a further application of the uniqueness argument it is identically zero everywhere. Thus Gx Gx =- + sin X x and in the far-field fx (0 ; X ) = -ik0 0 fx (0 ; X ) + sin X
0 0

CG (S ; X )Gs (x, y , z ; S )dS ,

(2.57)

CG (S ; X )fs (0 ; S )dS.

(2.58)

Similarly, by considering the behaviour of Gs / S + Gs / s Gs Gs =- + sin S s which has the far-field form fs (0 ; S ) = -ik0 (0 cos + 0 sin ) fs (0 ; S )+sin S
0 0

CG (X ; S )Gx (x, y , z ; X )dX ,

(2.59)

CG (X ; S )fx (0 ; X )dX.

(2.60) Thus, by substituting (2.58) and (2.60) into the embedding formula (2.54) the embedding formula is obtained in the form of a double integral, corresponding to Shanin's [12] equation (14) as 4 2 sin в 3 k0 ( + 0 ) (cos ( + 0 ) + sin ( + 0 ))
0 0

f ( ; 0 ) =

[fx ( ; X )fs (0 ; S ) + fx (0 ; X )fs ( ; S )] CG (X ; S )dX dS. (2.61)

2.2 Numerical Results For the embedding formulae to be of value it is important to demonstrate that they repro duce the results found using the standard formula for the directivity: f ( ; 0 ) = i e-
i



g r ( , 0 , ) d

(2.62)

where gr is the "reflected" part of the Green's function on the sphere with a cut. The details of this formula are to be found in [10]. We choose to compare this with the embedding formula (2.42) and require the edge Green's functions. The embedding formula is written in terms of the spherical edge 13


Green's function v 1 ( , ) derived in [25] as f ( ; 0 ) = 1 4 i( ( ) + (0 ))
i-1/2 -i-1/2

(v 1 ( , )v 1 (0 , + 1) + v 1 ( , + 1)v 1 (0 , ))d

(2.63) where ( ) = sin ( ) sin(( )). To allow a brief and simple comparison across several sector angles we just consider the incidence direction to be at the axis of symmetry of the cone, i.e. it has spherical co ordinates = /2, = + /2. The scattering directions are taken in the sagittal plane, i.e. they have spherical co ordinates = sc , = + /2 for various sc . In this case the diffraction co efficient is purely imaginary and it is plotted versus sc in figure 2. Numerically we find that the real part of the diffraction co efficients calculated either way is extremely small. In figure 2 the embedding result and the standard result are compared and the results are visually indistinguishable the maximal difference is 3 · 10-4. It is naturally reassuring that the numerical comparison is easily performed and accurate.
0 -0.05 Imag(f) -0.1 -0.15 -0.2 0 0.5 1 =/4 =/2 =3/4 1.5

Fig. 2. Numerical results: the directivity function along the sagittal plane for three sectoral angles. The crosses are from the emb edding formulae and the lines from the standard Smyshlyaev result.

3

The Solid Cone

In the second problem considered here a 3-D solid cone o ccupies the 1/8 space, o ctant, defined by x > 0, y > 0, z > 0, as shown in figure 3. The cone surfaces are the three quarter planes, subtending angle /2: (i) x = 0, y > 0, z > 0, S1 say, (ii) y = 0, x > 0, z > 0, S2 say, and (iii) z = 0, x > 0, y > 0, S3 say, which are mutually at right angles. The edges are denoted 1 , the positive x-axis, y = 0, z = 0, 2 , the positive y -axis, x = 0, z = 0 and 3 , the positive z -axis, x = 0, y = 0. An acoustic medium o ccupies everywhere except for the o ctant x > 0, y > 0, z > 0. The incident field is a plane wave described by equation (2.5). 14


z

3 edge surface S Incident plane wave
2

surface S1 2 edge
2

2 y


1

1

edge

1

surface S x

3

Fig. 3. Octant geometry formed by three p erp endicular quarter planes.

Dirichlet boundary conditions, (2.2), are taken on the three planar surfaces of the cone and the edge conditions are obtained from the theory of diffraction by an ideal infinitely long wedge subtending an angle of /2, as
2 3 u 1,

2,3

sin

21, 3

2,3

(3.1)

where 1,2,3 and 1,2,3 are lo cal cylindrical radial and angular co ordinates measured from the edges 1 , 2 and 3 , respectively. As in the previous example the total field consists a scattered field, which can itself be considered as the geometric scattered region together with a diffr of the diffracted term is again characterised by the directivity, f ( ; 0), defined as in equation (2.8). of the incident field and a plane wave present in acted term. The far-field diffraction co efficient, or

3.1 Solid Cone Embedding Formulae

3.1.1 Edge Green's functions Edge Green's functions, G1x (x, y , z ; X ), G1y (x, y , z ; Y ) and G1z (x, y , z ; Z ), are now intro duced via a limit pro cess as described in detail in Appendix A. For this geometry the angle between the planes is /2, and thus p = 1 and q = 2 in the notation used in Appendix A. For G1z (x, y , z ; Z ), for example, 2 this is achieved by placing a point source of strength 1/ 3 in the plane of the scattering cone, at a small distance away from the edge 3 , with Z the distance from the vertex to the nearest point on the edge to the source, and 15


taking the limit as 0:
2^ ( 2 + k 0 ) G 1z

2 = 4 (x - / 2) (y - / 2) (z - z0 )/ 3 =
2

4 (3 - ) (3 - 0 ) (z - z0 )/ 3 3

(3.1)

where 3 , 3 and z are lo cal cylindrical polar co ordinates and the fluid o ccupies 0 < 3 < 3 /2, 0 = 3 /4 and 0. The different power of on the right hand side here compared to equation (2.10) reflects the different geometry here. These edge Green's functions must also satisfy the boundary, edge and radiation conditions required of the original problem: ^ G1z (x, y , z ; Z, ) = 0
3 ^ G1z (x, y , z ; Z, ) 1, 2

on 21, 3
2,3

S

1,2,3

, ,

(3.2) (3.3) (3.4)

2,3

sin

on

1,2,3

and ^ G1z (x, y , z ; Z ) = lim G1z (x, y , z ; Z, ).
0

The important lo cal result for this Green's function, obtained from Appendix A, is that near 3
0

(3.5) together with corresponding results for the integrals of G1x (x, y , z ; X ) and G1y (x, y , z ; Y ). For this more complicated geometry we need to intro duce the `dipole edge Green's functions', G2x (x, y , z ; X ), G2y (x, y , z ; Y ) and G2z (x, y , z ; Z ), satisfying the boundary, edge and radiation conditions. and, for G2z (x, y , z ; Z ) for example,
2 ( 2 + k 0 ) G 2z

h(Z )G1z (x, y , z ; Z )dZ = -4h(z )3 3 sin

-

2

2 23 23 3 + O 3 sin 3 3

,

= 4 (3 - ) (3 - 0 ) (z - z0 )/ 3 3

4

(3.6)

in which a shorthand notation has been used, where the limit as 0 is now assumed. The details of the derivation and properties of these `dipole edge Green's functions' is presented in Appendix A. The important lo cal result for this dipole Green's function, corresponding to equation (3.5), is obtained from Appendix A as
2 8 43 -4 3 + O 3 , near 3 , h(Z )G2z (x, y , z ; Z )dZ = - h(z )3 3 sin 3 3 0 (3.7) together with corresponding results for the integrals of G2x (x, y , z ; X ) and G2y (x, y , z ; Y ).



16


3.1.2 Using the operator H

2x

= 2 / x2 + k

2 x

In [13] and [23] it was demonstrated that for wedge shaped geometries higher order operators were needed to construct embedding formulae than those needed for line geometries. In [13] the higher order operators Hpx were defined as i kx Hpx = (ik0 )p Tp - Tp (3.8) k0 x k0 where Tp is the Tchebychev polynomial of order p, and used for wedge geometries of more general angle. In particular for 2­D wedges with angle /2 oper2 ators of a least second order are necessary and the operator H2x = 2 / x2 + kx was used. Hence, for this problem in which the cone surfaces are all mutually perpendicular the simple second order operator H2x will be used as the basis for constructing an embedding formula. The resulting field H2x [u] has the following properties: (1) H2x [u] satisfies the (2) H2x [u] satisfies the of the cone. On S2 2 u/ x2 0 there H2x [u] is rewritten Helmholtz equation. Dirichlet boundary condition on each of the surfaces and S3 u 0 there and hence clearly u/ x 0 and also. On S1 , since u satisfies the Helmholtz equation, as 2 2 - y2 z
2

2 2 H2x [u] = kx - k0 -

u,

(3.9)

and since u 0 on S1 clearly 2 u/ y 2 0 and 2 u/ z 2 0 there to o. (3) H2x [uin ] 0, hence H2x [u] satisfies the radiation condition. However, although this new field satisfies the edge conditions on 1 , it do es not have `source-free' behaviour near to the edges 2 and 3 . For example, near to 2 ,
2 3 u(y , 2, 2 ) = Cy (y )2 sin 4 4 22 42 3 3 + Dy (y )2 sin + o(2 ), 3 3

(3.10)

where the unknown functions Cy (y ) and Dy (y ) are still to be determined. Hence, after differentiating with respect to x as in §2.1, it is found that
2 42 4 22 2 -4 -2 3 + Dy (y )2 3 sin +O (2 ) near 2 , (3.11) H2x [u] = - Cy (y )2 3 sin 9 39 3

and hence do es not have the required behaviour near 2 . These singular combinations of 2 and 2 are precisely the terms which o ccur as the leading order terms of integrals similar to those of equations (3.7) and (3.5), allowing the construction of an expression which is non-singular near 2 : H2x [u]- 1 12
0

Cy (Y )G2y (x, y , z ; Y )dY +

1 9

0

2 3 Dy (Y )G1y (x, y , z ; Y )dY = O (2 ).

(3.12)

17


Similarly, near

3

2 2 43 4 23 -4 -2 3 H2x [u] = - Cz (z )3 3 sin + Dz (z )3 3 sin + O (3 ), 9 3 9 3

(3.13)

and H2x [u]- 1 12
0

Cz (Z )G2z (x, y , z ; Z )dZ +

(3.14) Hence, by combining results (3.12) and (3.14) a function u can be constructed for this problem as


1 9

0

2 3 Dz (Z )G1z (x, y , z ; Z )dZ = O (3 ).

u =H2x [u] - - 1 12
0

1 12

0

Cy (Y )G2y (x, y , z ; Y )dY + 1 9
0

1 9

0

Dy (Y )G1y (x, y , z ; Y )dY (3.15)

Cz (Z )G2z (x, y , z ; Z )dZ +

Dz (Z )G1z (x, y , z ; Z )dZ ,

a combination which has the required asymptotic behaviour near all three edges 1 , 2 and 3 . The additional terms added to H2x [u] above are all combinations of Green's functions which, away from the edges, satisfy the Helmholtz equation, the Dirichlet boundary conditions on the cone surfaces and the radiation condition. Then, applying the uniqueness argument again the function u is found to be identically zero everywhere and 1 9 1 - 9
0 0

H2x [u(x, y , z )] = -

1 12 1 Dz (Z )G1z (x, y , z ; Z )dZ + 12 Dy (Y )G1y (x, y , z ; Y )dY +

0 0

Cy (Y )G2y (x, y , z ; Y )dY Cz (Z )G2z (x, y , z ; Z )dZ , (3.16)

which is the weak form of the embedding formula for this problem. The unknown functions Cy (Y ), Cz (Z ), Dy (Y ) and Dz (Z ) are determined using similar recipro city arguments to those used in the previous section. Thus, for 2 example consider a monopole point source of strength 4 / 3 , used in the ^ definition of G1y (and hence in the limiting case as 0 of G1y ) at distance Y along edge 2 and the same source at angular lo cation 0 at distance r in the far-field. The leading order term of the far-field due to the source near 2 , GY F say, is obtained from its directivity G
YF

2

eik0 r f1y (0 ; Y ). k0 r

(3.17)

Correspondingly, if this source is lo cated in the far-field then, near the vertex of the cone the incident field from the source is approximately that of a plane 18


wave of amplitude -eik0 r /( 3 r ) and hence the leading order terms of the total field near the point Y on 2 , uF Y say, is obtained, by comparison with equation (3.10) as u
FY

2

-

ei

k0 r
2 3

2 3 Cy (y )2 sin

r

4 22 42 3 + Dy (y )2 sin 3 3

near 2 .

(3.18)

At the point under discussion, y = Y , 2 = and 2 = 3 /4, hence G
FY

-

ei

k0 r

C y (Y ) . r

(3.19)

Applying the recipro city principle, that the same result is obtained if the source and observer positions are interchanged, shows that G
YF

=G

FY

(3.20)

and hence an expression for the unknown function Cy (Y ) is obtained as C y (Y ) = - 2 f1y (0 ; Y ), k0 (3.21)

Cz (Z ) follows by replacing y , Y with z , Z in (3.21). In order to obtain an 4 expression for Dy (Y ), consider first the dipole source of strength 4 / 3 , used ^ in the definition of G2y (and hence in the limiting case as 0 of G2y ) at distance Y along edge 2 . The leading order term of the far-field due to this source near 2 , DY F say, is obtained from the directivity of the dipole edge Green's function, as eik0r DY F 2 f2y (0 ; Y ). (3.22) k0 r A second expression for this term is obtained by noting that the source term 2 ^ ^ in the definition of G2y is - 3 / 2 of the source term in the definition of G1y . 2 ^ Formally, this is -- 3 / 0 of the G1y source term, where 0 is the angular co ordinate of the source lo cation. From the recipro city discussed above, the ^ far-field due to the G1y source at 0 is the same as the field at 0 near to 2 due to the source in the far-field. Hence, the far-field of the dipole source is expressed as
4 2 eik0 r 1 20 40 3 3 -2 + Dy (y )2 sin Cy (y )2 sin 2 3 3 3 0 3r

D

YF

- =

0 =

3 4

,y =Y ,2 =

1e
4 3

i k0 r

r

2 4 20 4 40 2 3 3 Cy (y )2 cos + Dy (y )2 cos 3 3 3 3

.
0 =
3 4

,y =Y ,2 =

(3.23) 19


Hence, 4 eik0 r Dy ( Y ) . (3.24) 3r Thus, by equating equations (3.22) and (3.24) the unknown function Dy (Y ) is obtained in terms of the dipole directivity as D
YF

=-

Dy ( Y ) = - and Dz (Z ) follows by replacing y , is then obtained for this problem weak embedding formula (3.16), Dz (Z ), using equations (3.21), (3 2 -k0 2 in the far-field, as f ( ; 0 ) = - 32 2 6 k 0 ( - 0 ) +
0

3 f2y (0 ; Y ), 2 k0

(3.25)

Y with z , Z in (3.25). An embedding formula by taking the far-field approximation of the substituting for Cy (Y ), Cz (Z ), Dy (Y ) and .25), and noting that 2 / x2 corresponds to

{f2z (0 ; Z )f1z ( ; Z ) - f1z (0 ; Z )f2z ( ; Z )} dZ
0

{f2y (0 ; Y )f1y ( ; Y ) - f1y (0 ; Y )f2y ( ; Y )} dY . (3.26)

This embedding formula corresponds to Shanin's [12] equation (12) generalised from a flat cone with perpendicular edges to the case of a solid cone with perpendicular faces. It is useful for practical purposes, that is, for computations such as those in section 2.2, to be able to write this embedding formula in terms of the edge Green's functions on a sphere so they are in a similar form to (2.63). We briefly derive this form of the formulae, to do so we require the spherical co ordinates, x,y,z , x,y,z , shown in figure 4, and then we intro duce edge Green's functions on a sphere v1z ( , ) and v2z ( , ) as the results of limiting pro cedures: v1z ( , ) = lim v1z ( , , ), ^
0

v2z ( , ) = lim v2z ( , , ), ^
0

(3.27)

^ where v1z ( , , ) and v2z ( , , ) are solutions of the following problems: ^ ~^ v1z ( , , ) = -
2 3

1 ( z - ) ( z - ) o n S, z sin

(3.28) (3.29) (3.30) (3.31)

v1z ( , , ) = 0 on S, ^ 4 1 ~^ v2z ( , , ) = - 3 ( z - ) ( z - ) o n S, sin z v2z ( , , ) = 0 on S ^

where S is the surface of the unit sphere without the piece excised by cone, and S is the boundary of S , i.e. the cuts corresponding to cross-sections of 20


z



z

z

x
x

x y

y

y

Fig. 4. Spherical coordinates for the edge Green's functions.

the cone. The differential op angular part of the Laplacian functions v1z ( , , ) and v2z ^ ^ point sources placed near the

~ erator is operator, ( , , ) are pro jection o

~ ~ ~ = + 2 - 1 and is the 4 = and = 34 . The edge Green's the fields of monopole and dipole f the z -edge of the cone.

Equation (3.26) consists of four terms that are obtained from each other by interchanging variables as 0 and z y , so we need only consider the first term ~ f ( ; 0 ) = -
3 2 6 k 0 ( 2 - 0 ) 0

f2z (0 , Z )f1z ( , Z )dZ .

(3.32)

After considerable algebra, and using the properties of the edge Green's functions, in both physical space and in their form on a sphere, one can eventually deduce the first term of the embedding formula in the mo dified Smyshlyaev f o rm i 1 ~ f ( ; 0 ) = 2 - 2) 1 2 ( 0 e-
i



v1z ( , )v2z (0 , - 2) + v1z ( , - 2)v2z (0 , ) -1 (3.33)

-

2 v1z ( , )v2z (0 , ) d -1
2

21




2 - 2 1 1 2

2 - 1 2 3 4

Fig. 5. The contour of integration required for the integral in (3.33).

where the contour is shown in figure 5, and thus one can build an expression for f ( ; 0). Notably the contour is the usual large lo op enclosing the positive real axis for , but with some poles excluded. The points j and j + 2, where j are the points of the spectrum of the Laplace-Beltrami Dirichlet problem, are all possible poles of the integrand. Potential problems are pro duced by points 1 and 2 - j for j < 2, and so these points are explicitly excluded from the contour; this is analogous to the subtraction of poles in [12].

3.1.3 Other second order operators Further embedding formulae are obtained using operators corresponding to differentiation in the other co ordinate directions. Thus, making use of the operator H the embedding formula
2y

=

2 +k y2

2 y

(3.34)

f ( ; 0 ) =

- 2 6k ( 2 - 0 )
3 0

0

{f2x (0 ; X )f1x ( ; X ) - f1x (0 ; X )f2x ( ; X )} dX
0

+

{f2z (0 ; Z )f1z ( ; Z ) - f1z (0 ; Z )f2z ( ; Z )} dZ (3.35) 22


z

z=z Incident plane wave Surface S
1 1
2

0



5

4

Surface S

4

3

Surface S


1

2

2 y

2



1

Surface S

3

x

Fig. 6. Additional parallel face geometry.

is obtained, and using the operator H the embedding formula 6 k ( + - -
3 0 2 2 2 0 2 0 2z

=

2 2 2 2 + k0 (1 - 0 - 0 ) z2

(3.36)

f ( ; 0 ) =

) +

0

{f2x (0 ; X )f1x ( ; X ) - f1x (0 ; X )f2x ( ; X )} dX
0

{f2y (0 ; Y )f1y ( ; Y ) - f1y (0 ; Y )f2y ( ; Y )} dY (3.37)

is obtained.

4

Extensions and closing remarks

It becomes clear that the entire embedding pro cedure can be applied to scattering by quite general shapes. The key being the existence of an operator that kills the incoming field and preserves the boundary conditions. The results of the previous section can be used with only minor alterations for some related geometries. In particular the mo dification is straightforward if an additional face is present, parallel to one of the existing faces, as shown in figure 6. Suppose that an additional face S4 is present, parallel to S3 with x > 0, y > 0, z = z0 , pro ducing edges 4 , x > 0, y = 0, z = z0 , and 5 , x = 0, y > 0, z = z0 , and the scattering cone o ccupying x > 0, y > 0, 0 < z < z0 . Because 23


the extra faces and edges intro duced here are parallel to existing faces and edges the same operators can be used, as they will still preserve the Dirichlet boundary condition on all the faces, and have similar effects on the edges. Any integrations then take place along the length of all relevant edges. Hence, with slightly mo dified notation for the edge Green's functions and using the operator H2x , the embedding formula

- f ( ; 0)= 3 2 2 6 k 0 ( - 0 ) +

0

{f2y (0 ; Y , z0 )f1y ( ; Y , z0) - f1y (0 ; Y , z0 )f2y ( ; Y , z0)} dY
0 z 0
0

{f2y (0 ; Y , 0)f1y ( ; Y , 0) - f1y (0 ; Y , 0)f2y ( ; Y , 0)} dY {f2z (0 ; Z )f1z ( ; Z ) - f1z (0 ; Z )f2z ( ; Z )} dZ (4.1)

+

readily emerges and similar formulae are found if one applies the operator H or H2z .

2y

Finally, we can insert more parallel sides (at x = x0 and y = y0 ) to create a cuboid o ccupying 0 < x < x0 , 0 < y < y0 , z < z0 . This results in six faces on which the second order differential operators previously defined still preserve the Dirichlet boundary conditions. There are 12 edges, and for each operator extra terms, as described previously, are required to be included for each operator in order to preserve the edge conditions. The subsequent integrations then take place only along the lengths of the relevant edges. The subsequent result is quite lengthy to repro duce, so only that based on the operator H2x is repro duced here. The results for the operators H2y and H2z are obtained in a completely analogous manner and are omitted here. The embedding formula obtained from the H2x operator is 24


z Incident plane wave



3

y 1
1


1


2

2

2

x

Fig. 7. Cone formed by 2 quarter planes p erp endicular to a sector of angle .

- в f ( ; 0)= 3 2 2 6 k 0 ( - 0 )
y
0

0

+ + + + + + +

y y y z z z

{f2y (0 ; 0, Y , z0 )f1y ( ; 0, Y , z0) - f1y (0 ; 0, Y , z0 )f2y ( ; 0, Y , z0)} dY
0

0 0 0 0 0 0

{f2y (0 ; x0 , Y , z0 )f1y ( ; x0 , Y , z0 ) - f1y (0 ; x0 , Y , z0 )f2y ( ; x0 , Y , z0 )} dY
0

{f2y (0 ; 0, Y , 0)f1y ( ; 0, Y , 0) - f1y (0 ; 0, Y , 0)f2y ( ; 0, Y , 0)} dY
0

{f2y (0 ; x0 , Y , 0)f1y ( ; x0 , Y , 0) - f1y (0 ; x0 , Y , 0)f2y ( ; x0 , Y , 0)} dY {f2z (0 ; 0, y0, Z )f1z ( ; 0, y0, Z ) - f1z (0 ; 0, y0, Z )f2z ( ; 0, y0, Z )} dZ

0

0

{f2z (0 ; 0, 0, Z )f1z ( ; 0, 0, Z ) - f1z (0 ; 0, 0, Z )f2z ( ; 0, 0, Z )} dZ
0

z 0

{f2z (0 ; x0 , 0, Z )f1z ( ; x0, 0, Z ) - f1z (0 ; x0 , 0, Z )f2z ( ; x0 , 0, Z )} dZ
0

{f2z (0 ; x0 , y0 , Z )f1z ( ; x0 , y0, Z ) - f1z (0 ; x0 , y0 , Z )f2z ( ; x0 , y0, Z )} dZ . (4.2)

Thus, the integrations take place along the edges where the variable is y or z . We now relax the requirement that all edges are mutually perpendicular and move the edge 2 in the x­y plane away from the y -axis to make angle with the x-axis, as shown in figure 7. Thus the surface S3 is the same surface as in the flat cone example of a previous section. It is now necessary to find a suitable differential operator preserving the Dirichlet boundary condition on the faces S1 , S2 and S3 , and then to determine a suitable combination of monopoles, dipoles . . . to assemble along the edges to satisfy the edge conditions. 25


It is simplest to cho ose an operator based on / z and to pro ceed as before. This preserves the Dirichlet boundary condition on surfaces S1 and S2 , and since the angle between each of these surfaces and S3 is still /2, the operator H2z based on T2 ( / z ) preserves the Dirichlet boundary condition on all three surfaces. On the edge 3 the operator H2z preserves the edge condition. On the edge 1 the behaviour is that described previously for the o ctant case. Similarly, since the angle between faces S1 and S3 is also /2 it can be deduced that the behaviour near the edge 2 has similar characteristics to that near 1 . Thus, the embedding formula is obtained in this case as f ( ; 0 ) =
3 2 2 6k0 ( 2 + 2 - 0 - 0 ) 0

{f2x (0 ; X )f1x ( ; X ) - f1x (0 ; X )f2x ( ; X )} dX
0

+

{f2s (0 ; S )f1s ( ; S ) - f1s (0 ; S )f2s ( ; S )} dS . (4.3)

This is of the same form as (3.37), but with the variable Y for the integration replaced by S , the variable measured along 2 , and the Green's functions f1x , f2x , f1s and f2s are those for this cone shape. Hence, we see that a combination of the ideas used in this article can be utilised to consider scattering by quite general geometries. Moreover, it is not unusual for there to be several embedding formulae and these can be used to cross-validate the results. However, we sound a note of warning, unlike the o ctant or cuboid discussed previously, for the cone in figure 7 other embedding formulae cannot be obtained merely by interchanging the co ordinate for the differentiation in the operator, i.e. H2x and H2y or H2s are not in general suitable for obtaining an embedding formula. To see this consider the operator H2x , it do es not in general preserve the Dirichlet boundary condition on the face S1 . However, as 1 and 2 are both perpendicular to 3 , the face S2 can be rotated around 3 onto the face S1 . An operator based on this angle may preserve the Dirichlet boundary condition on S1 . However, for example, if = /3 then even the operator H3x do es not in general preserve the Dirichlet boundary condition on S1 and is not a suitable operator. In addition to preserving the Dirichlet boundary conditions on the faces a suitable operator for obtaining an embedding formula is also required to pro duce edge behaviour which can be expressed in terms of edge Green's functions (monopoles and higher orders). In this example the direction of the differentiation is neither parallel to nor perpendicular to the edge 2 and is also not in the plane of one of the faces there, hence considerably more analysis is needed to investigate the edge behaviour there and to determine suitable operators for this geometry. In conclusion it is therefore clear that scattering from many three-dimensional geometries can be considered using the embedding formulae philosophy es26


poused in [22] and subsequent articles. The embedding is far easier for planar structures, but can be generalised to right-angled geometries reasonably straightforwardly. Even more general angled structures can be contemplated, but the identification of the appropriate operator becomes more arduous. None the less the fact that embedding can be utilised suggests that this route could dramatically reduce the effort required to generate far-field directivity functions for these geometries.

5

Acknowledgements

The authors gratefully acknowledge the support of the UK EPSRC through grant number EP/D045576/1. AVS thanks the Russian Foundation for Basic Research for their support within grant number 07-02-00803. RVC thanks NSERC Canada for their support through the Discovery Grant scheme.

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[11] B. D. Bonner, I. G. Graham, V. P. Smyshlyaev, The computation of conical diffraction coefficients in high-frequency acoustic wave scattering, SIAM J. Num. Anal. 43 (2005) 1202­1230. [12] A. V. Shanin, Modified Smyshlyaev's formulae for the problem of diffraction of a plane wave by an ideal quarter plane, Wave Motion 41 (2005) 79­93. [13] R. V. Craster, A. N. Shanin, Emb edding formulae for diffraction by rational wedge and angular geometries, Proc. R. Soc. Lond. A 461 (2005) 2227­2242. [14] M. H. Williams, Diffraction by a finite strip, Q. Jl. Mech. Appl. Math. 35 (1982) 103­124. [15] A. K. Gautesen, On the Green's function for acoustical diffraction by a strip, J. Acoust. Soc. Am. 74 (1983) 600­604. [16] P. A. Martin, G. R. Wickham, Diffraction of elastic waves by a p enny-shap ed crack: analytical and numerical results, Proc. R. Soc. Lond. A 390 (1983) 91­ 129. [17] C. M. Linton, P. McIver, Handb ook of mathematical techniques for wave/structure interactions, Chapman-Hall CRC Press, 2001. [18] N. R. T. Biggs, D. Porter, D. S. G. Stirling, Wave diffraction through a p erforated breakwater, Q. Jl. Mech. appl. Math. 53 (2000) 375­391. [19] N. R. T. Biggs, D. Porter, Wave diffraction through a p erforated barrier of non-zero thickness, Q. Jl. Mech. appl. Math. 54 (2001) 523­547. [20] N. R. T. Biggs, D. Porter, Wave scattering by a p erforated duct, Q. Jl. Mech. Appl. Math. 55 (2002) 249­272. [21] N. R. T. Biggs, D. Porter, Wave scattering by an array of p erforated breakwaters, IMA J. Appl. Math. 70 (2005) 908­936. [22] R. V. Craster, A. V. Shanin, E. M. Doubravsky, Emb edding formulae in diffraction theory, Proc. R. Soc. Lond. A 459 (2003) 2475­2496. [23] E. A. Skelton, R. V. Craster, A. V. Shanin, Emb edding formulae for diffraction by non-parallel slits, Q. Jl. Mech. Appl. Math. 61 (2008) 93­116. [24] N. R. T. Biggs, A new family of emb edding formulae for diffraction by wedges and p olygons, Wave Motion 43 (2006) 517­528. [25] A. V. Shanin, Coordinate equations for a problem on a sphere with a cut associated with diffraction by an ideal quarter plane, Q. J. Mech. Appl. Math. 58 (2005) 289­308.

28


A

Appendix: Local behaviour of a monopole point or dipole source near the edge of a wedge of rational angle

In the text we require edge Green's functions and their behaviour in the near field, to obtain this it is necessary to consider some source and wedge interaction problems and the purpose of this appendix is to sketch the derivation of the near field. We treat the monopole source is detail. The Helmholtz wave equation is satisfied by fluid outside an infinitely long wedge whose angle is a rational multiple of , p /q , as shown in figure A.1. Cylindrical polar co ordinates (r, , z ), where the z -axis coincides with the vertex edge, r is measured from the vertex, and is measured from one of the wedge faces are used. A point source is lo cated in the fluid midway between the wedge faces, at radial distance r0 from the vertex line. Thus, the governing equations for u are
2 (2 + k0 )u1 (R; R0 ) = -4 (r - r0 ) ( - 0 ) (z - z0 )/r0 , 0 < < 20 , (A.1)

in which together with the Dirichlet boundary conditions on the wedge faces u1 (R; R0 ) = 0 at = 0 and = 20 . (A.3) 0 = (1 - p/2q ), (A.2)

Utilising a Fourier series expansion in , a Hankel transform in radius, r , and a Fourier transform in the axial direction z the following solution, for r r0 > 0, emerges u 1 (R ; R 0 ) = - i 0


sin n=1 n o dd

n n sin 20 2



J
-

n 20

( r0 ) H

n 20

( r )ei

(z -z0 )

d,

(A.4) in which = k - . As the source approaches the edge of the wedge r0 0 and the leading order term of u1 (R; R0 ) results from the n = 1 term of the
2 0 2

Y

POINT SOURd E C ``` ``` (r0 , 0 ) ``` ``

p q

``` `

Fig. A.1. Wedge geometry

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summation: u 1 (R ; R 0 ) 0 - i
20

+1

r0 2

20

sin

20




-

20

H

20

( r )ei

(z -z0 )

d.

(A.5) 0 It is frequently useful to consider a source whose strength varies as r and then to use the limit as r0 0. This limiting case is obtained by omitting the /2 r0 0 factor above and replacing the `' with `=':
- /2 0

u

1lim

(R; z0 ) =

0

- i
n 20

+1
0

1 2

20

sin

20




-

20

H

20

( r )ei

(z -z0 )

d.

(A.6) In this exposition the r factor will be retained, but with the understanding that the limit pro cedure will be subsequently applied. Next, the near field behaviour of equation (A.6) as r 0 is examined, using known properties of Hankel functions, and
/2 0

u 1 (R ; R 0 )

- 0

20

20

+1

r0 r

20

sin

20

-

1 + A 2 r 2 + . . . ei

(z -z0 )

d.

(A.7) as r 0. Inverting the Fourier transforms using delta functions and their derivatives yields the near field behaviour of u1 (R; R0) as r0 u1 (R; R0 ) -4 r
20

sin

20

2 1 + Ak0 r 2 + . . . (z - z0 ) ,

+ Ar 2 + . . . (z - z0 ) + . . . . (A.8) This form of the solution is particularly useful for evaluating integrals with respect to z0 , for example, as r 0
0

r0 h(Z )u1(r, , z ; r0 , 0 , Z )dZ 4 r

20

sin

20 (A.9)

2 1 + Ak0 r 2 + . . . h(z ) + Ar 2 + . . . h (z ) + . . . .

For the dipole, the inhomogeneous Helmholtz equation (A.1) is replaced by
2 (2 +k0 )u2 (R; R0) = -4 (r -r0 ) (-0 ) (z -z0 )/r0 , 0 < < 20 , (A.10)

and following through the analysis above gives the near field for the dipole as -2 u 2 (R ; R 0 ) 0 r0 r
0

sin

0 (A.11)

2 1 + B k0 r 2 + . . . (z - z0 ) + B r 2 + . . . (z - z0 ) + . . . .

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Hence the near field behaviour of the required integrals with respect to z0 are obtained as
0

2 h(Z )u2 (r, , z ; r0 , 0 , Z )dZ 0

r0 r

0

sin

0 (A.12)

2 1 + B k0 r 2 + . . . h(z ) + B r 2 + . . . h (z ) + . . . , as r 0.

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