Analytical continuation of two-dimensional wave fields
AAnalytical continuation of two-dimensional wave fields
Rapha¨el C. Assier ∗ and Andrey V. Shanin † ∗ Department of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, UK † Department of Physics (Acoustics Division), Moscow State University, Leninskie Gory, 119992, Moscow, Russia
Abstract
Wave fields obeying the 2D Helmholtz equation on branched surfaces (Sommerfeld sur-faces) are studied. Such surfaces appear naturally as a result of applying the reflection methodto diffraction problems with straight scatterers bearing ideal boundary conditions. This is forexample the case for the classical canonical problems of diffraction by a half-line or a segment.In the present work, it is shown that such wave fields admit an analytical continuation intothe domain of two complex coordinates. The branch sets of such continuation are given andstudied in detail. For a generic scattering problem, it is shown that the set of all branchesof the multi-valued analytical continuation of the field has a finite basis. Each basis functionis expressed explicitly as a Green’s integral along so-called double-eight contours. The finitebasis property is important in the context of coordinate equations, introduced and utilisedby the authors previously, as illustrated in this article for the particular case of diffraction bya segment.
In this paper we study 2D diffraction problems for the Helmholtz equation belonging to a specialclass: namely those who can be reformulated as problems of propagation on branched surfaces withfinitely many sheets. At least two classical canonical diffraction problems belong to this class: theSommerfeld problem of diffraction by a half-line with ideal boundary conditions, and the problemof diffraction by an ideal segment. There are also some other important problems belonging tothis class, they are listed in Appendix A. The branched surface for such a problem is referred toas a
Sommerfeld surface and denoted by S . This surface has several sheets over the real Cartesianplane ( x , x ), and these sheets are connected at several branch points.The solution of the corresponding problem is denoted by u ( x , x ) and is assumed to be knownon S . We consider the possibility to continue the solution u into the complex domain of thecoordinates ( x , x ) ∈ C . Namely, we are looking for a function u c ( x , x ) which is analyticalalmost everywhere, obeys the complex Helmholtz equation, and is equal to u ( x , x ) for real ( x , x ).It is shown in the paper that such a continuation can be constructed using Green’s thirdidentity. The integration contours used involve rather complicated loops drawn on S , referredto as double-eight or Pochhammer contours. The integrand comprises the function u on S , acomplexified kernel and their first derivatives.The analytical continuation u c is a branched (i.e. multivalued) function in C . Its branchset T is the union of the complex characteristics of the Helmholtz equation passing through the1 a r X i v : . [ m a t h - ph ] A ug ranch points of S . At each point of C not belonging to the branch set T there exists an infinitenumber of branches of u c , i.e. infinitely many possible values of u c in the neighbourhood of thispoint. However, we prove here that these branches have a finite basis , such that any branch canbe expressed as a linear combination of a finite number of basis functions with integer coefficients.Such property of the analytical continuation is an important property of the initial (real)diffraction problem. Namely, it indicates that one can build the so-called coordinate equations ,which are “multidimensional ordinary differential equations” [1, 2, 3]. Thus, a PDE becomeseffectively solved as a finite set of ODEs.To emphasise the non-triviality of the statements proven in this paper and the importanceof studying the analytical continuation of the field, we can say that we could not generalise theresults to the case of 3D diffraction problems yet. It is possible to build a 3D analog of Sommerfeldsurfaces (for example, it can be done for the ideal quarter-plane diffraction problem), but at themoment it does not seem possible to show that the analytical continuation does possess a finitebasis. Thus, a generalisation of the coordinate equations seems not to be possible for 3D problems.The ideas behind this work were inspired in part by our recent investigations of applicationsof multivariable complex analysis to diffraction problems [4, 5, 6, 7, 8], in part by our work oncoordinate equations [1, 3, 2, 9, 10] and in part by the work of Sternin and his co-authors [11, 12, 13],in which the analytical continuation of wave fields is considered.In [11, 12, 13], the practical problems of finding minimal configurations of sources producingcertain fields, or continuation of fields through complicated boundaries are solved.Different techniques can be used to study the analytical continuation of a wave field. We areusing Green’s theorem as in [14, 11]. Alternatives include the use of Radon transforms [11, 13],series representations or Schwarz’s reflection principle.The rest of the paper is organised as follows. In Section 2 we define the concept of Sommerfeldsurface and introduce the notion of diffraction problems on such object. In Section 3, we specifywhat we mean by the analytical continuation of a wave field u to C , and discuss the notion ofbranching of functions of two complex variables. In Section 4, we give an integral representationthat permits to analytically continue u from a point in S to a point in C . The obtained function u c is multi-valued in C , and, in Section 5, we study in detail its branching structure and specify all itspossible branches by means of Green’s integrals involving double-eight contours. In particular, weshow that there exist a finite basis of functions such that any branch of the analytical continuationcan be expressed as a linear combination, with integer coefficients, of these basis functions. InSection 6, we apply the general theory developed thus far to the specific problem of diffractionby an ideal strip, showing that, in this case, the number of basis functions can be reduced tofour; we describe explicitly and constructively, via some matrix algebra, all possible branches ofthe analytical continuation. Finally, in Section 7, still for the strip problem, we show that ourresults imply the existence of the so-called coordinate equations, effectively reducing the diffractionproblem to a set of two multidimensional ODEs. Let us start by defining more precisely the concept of Sommerfeld surface. Take M samples of theplane R (called sheets of the surface), each equipped with the Cartesian coordinates ( x , x ). Letthere exist N affixes of branch points P , . . . , P N ∈ R with coordinates ( X ( j )1 , X ( j )2 ), j = 1 , . . . , N .Consider a set of non-intersecting cuts on each sheet, connecting the points P j with each other orwith infinity. Finally, let the sides of the cuts be connected (“glued”) to each other according to2n arbitrary scheme. The connection of the sides should obey the following rules: (a) only pointshaving the same coordinates ( x , x ) can be glued to each other; (b) one can glue a single “left”side of a cut to a single “right” side of this cut on another sheet; (c) a side of a cut should be gluedto a side of another cut as a whole.Upon allowing spurious cuts, i.e. cuts glued to themselves, it is possible for the set of cuts tobe the same on all sheets. The result of assembling the sheets is a Sommerfeld surface denotedby S . We assume everywhere that N and M are finite integers. Two examples of Sommerfeldsurfaces are shown in Figure 1.The concept of Sommerfeld surfaces is naturally very close to that of Riemann surfaces of ana-lytic functions of a complex variable. For example, the Sommerfeld surfaces shown in Figure 1 canbe treated as the Riemann surfaces of the functions √ x + ix and (cid:112) ( x + ix ) − a respectively.However, here we prefer to refer to them as Sommerfeld surfaces since the coordinates x and x are real on it, and since we would like to avoid confusion with the complex context that will bedeveloped below. sheet 1 x x sheet 2 III sheet 1 x x (a)(b) sheet 2 III - a a Figure 1: Sommerfeld surfaces for the 2D problems of the Dirichlet half-line (a), and the Dirichletsegment (b). The cuts are shown by thick lines. The sides of the cuts glued to each other bear thesame Roman number. The associated branch points are denoted by thick black dots.Sommerfeld surfaces emerge naturally when the reflection method is applied to a 2D diffractionproblem with straight ideal boundaries. For example, the surfaces shown in Figure 1 help one tosolve the classical Sommerfeld problem of diffraction by a Dirichlet half-line [15] and the problemof diffraction by a Dirichlet segment [1]. A connection between the diffraction problems and theSommerfeld surfaces is discussed in more details in Appendix A, where the class of scatterersleading to finite-sheeted Sommerfeld surfaces is described.There exists a natural projection ψ of a Sommerfeld surface S to R . For any small neigh-bourhood U ⊂ R not including any of the branch points P j the pre-image ψ − ( U ) is a set of M samples of U , as illustrated in Figure 2 . 3igure 2: Diagrammatic illustration of the natural projection ψ of a Sommerfeld surface S with M = 3.Let u be a continuous single-valued function on some Sommerfeld surface S (thus, u is generallyan M -valued function on R ). For any neigbourhood U ⊂ R \{ P , . . . , P N } , let u obey the (real)Helmholtz equation ( ∂ x + ∂ x + (cid:107) ) u ( x , x ) = 0 , (2.1)on each sample of ψ − ( U ). The wavenumber parameter (cid:107) is chosen to have a positive real partand a vanishingly small positive imaginary part mimicking damping of waves.Let u also obey the Meixner condition at the branch points P j . The Meixner condition char-acterises the local finiteness of the energy-type integral (cid:90) (cid:90) ( |∇ u | + | u | ) dx dx , near a branch point. In particular, it guarantees the absence of sources at the branch points of S .A function u obeying the equation (2.1) in the sense explained above and the Meixner conditionwill be referred to as a function obeying the Helmholtz equation on S .Let us now formulate a diffraction problem on a Sommerfeld surface S : find a function u obeying the Helmholtz equation on S that can be represented as a sum u = u in + u sc , where u in is a known incident field, which is equal to a plane wave or to zero, depending on thesheet: u in = (cid:26) exp {− i (cid:107) ( x cos ϕ in + x sin ϕ in ) } , . (2.2)Here ϕ in is the angle of incidence. The incident wave is only non-zero on a single sheet of S , andis zero on the other sheets. Neither u in nor u sc are continuous, but their sum is. The scatteredfield u sc should also obey the limiting absorption principle, i.e. be exponentially decaying as (cid:112) x + x → ∞ .In the rest of the paper, we assume that the existence and uniqueness theorem is proven forthe chosen S and (cid:107) and that the field u is fully known on S . Our aim is to build an analytical continuation u c ( x , x ) of the solution u ( x , x ) of a certaindiffraction problem on a Sommerfeld surface S . The continuation has the following sense.4et x and x be complex variables, i.e. ( x , x ) ∈ C . Naturally, C is a space of real dimen-sion 4. Let T ⊂ C be a singularity set (built below), which is a union of several manifolds of realdimension 2. Let T be such that the intersection of T with the real plane R ⊂ C is the set ofbranch points P , . . . , P N .The continuation u c should obey three conditions. • The continuation u c ( x , x ) should be a multivalued analytical function on C \ T . Eachbranch of u c in any small domain U ⊂ C not intersecting with T should obey the Cauchy–Riemann conditions¯ ∂ x (cid:96) u c = 0 , where ¯ ∂ x (cid:96) ≡ (cid:18) ∂∂ Re[ x (cid:96) ] + i ∂∂ Im[ x (cid:96) ] (cid:19) , (cid:96) ∈ { , } . (3.1) • The continuation u c ( x , x ) should obey the complex Helmholtz equation in C \ T :( ∂ x + ∂ x + (cid:107) ) u c = 0 , where ∂ x (cid:96) ≡ (cid:18) ∂∂ Re[ x (cid:96) ] − i ∂∂ Im[ x (cid:96) ] (cid:19) , (cid:96) ∈ { , } . (3.2)Note that the notation ∂ x (cid:96) is used both in the real (2.1) and in the complex (3.2) context.However one can see that if the Cauchy–Riemann conditions are valid, the complex derivativegives just the same result as the real one. • When considering the restriction of u c ( x , x ) onto R , over each point of R \ { P , . . . P N } ,there should exist M branches of u c equal to the values of u on S .Let us describe, without a proof, the structure of the singularity set T and the branchingstructure of u c . Later on, we shall build u c explicitly, and one will be able to check the correctnessof these statements.The branch points P j are singularities of the field u on the real plane. According to the generaltheory of partial differential equations, the singularities propagate along the characteristics of thePDE (the Helmholtz equation here). Thus, it is natural to expect that T is the union of thecharacteristics passing through the points P j = ( X ( j )1 , X ( j )2 ) ∈ R . Since the Helmholtz equation iselliptic, these characteristics are complex. They are given for j ∈ , . . . , N by L ( j )1 = { ( x , x ) ∈ C , x + ix = X ( j )1 + iX ( j )2 } , (3.3) L ( j )2 = { ( x , x ) ∈ C , x − ix = X ( j )1 − iX ( j )2 } . (3.4)One can see that L ( j )1 , are complex lines having real dimension 2. We will hence refer to them as . Their intersection with R is the set of the points P j , i.e. L ( j )1 ∩ L ( j )2 = P j .We demonstrate below that the 2-lines L ( j )1 , are, generally, branch 2-lines of u c . The branchingof functions of several complex variables is not a well-known matter, thus, we should explain whatit means. Consider for example a small neighbourhood U ⊂ C of a point on L (1)1 , which is not acrossing point of two such lines. Note that the complex variable z = x + ix − ( X (1)1 + iX (1)2 )is a coordinate transversal to L (1)1 . The 2-line L (1)1 corresponds to z = 0. The complex variable z = x − ix
5s then a coordinate tangential to L (1)1 .Let A be some point in U . Consider a path (oriented contour) σ in U starting and ending at A , and having no intersections with L (1)1 . Such a contour, called a bypass of L (1)1 , can be projectedonto the variable z . Denote this projection by σ (cid:48) . One can continue u c along σ and obtain thebranch u c ( A ; σ ). Branches can be indexed by an integer p , which is the winding number of σ (cid:48) about zero. If for some σ (cid:48) having winding number pu c ( A ; σ ) = u c ( A )for any such continuation (here we consider the smallest possible p having this property), then thebranch line L (1)1 has order of branching equal to p . If there is no such p , the branching is said tobe logarithmic.Thus, generally speaking, the branching of a function of several complex variables is similar tothat of a single variable, and it is convenient to study this branching using a transversal complexcoordinate. To provide the existence of such a transversal variable, the branch set should be a set(a complex manifold) of complex codimension 1.For j, k ∈ { , . . . N } , the 2-lines L ( j )1 and L ( k )2 intersect at a single point. For example, if j = k ,this point is P j , while for j (cid:54) = k , this intersection point does not belong to R . The branching of u c near each crossing point has a property that is new comparatively to the 1D complex case: thebypasses about L ( j )1 and L ( k )2 commute.Let us prove this in the case j = k by considering a small neighbourhood U ⊂ C of the point P j = L ( j )1 ∩ L ( j )2 . The case j (cid:54) = k is similar. Introduce the local coordinates z = x + ix − ( X ( j )1 + iX ( j )2 ) , z = x − ix − ( X ( j )1 − iX ( j )2 ) , which are transversal variables to L ( j )1 and L ( j )2 respectively. Take a point A ∈ U \ ( L ( j )1 ∪ L ( j )2 )and a path σ in U \ ( L ( j )1 ∪ L ( j )2 ) starting and ending at A . Consider the projections σ and σ of σ onto the complex planes of z and z respectively.Assume that the path σ is parametrised by a real parameter τ ∈ [0 , σ : ( x ( τ ) , x ( τ )) , or, in the new coordinates, σ : ( z ( τ ) , z ( τ )) . The path σ can be deformed homotopically into a path σ ∗ defined by σ ∗ : ( (cid:15)z ( τ ) / | z ( τ ) | , (cid:15)z ( τ ) / | z ( τ ) | ) , for some small parameter (cid:15) . The projection of σ ∗ onto z (resp. z ) is a small circle σ ∗ (resp. σ ∗ ) ofradius (cid:15) turning (possibly many times) around the origin. Therefore, σ ∗ lies on a torus (product oftwo circles), for which σ ∗ and σ ∗ are strictly longitudinal and latitudinal paths respectively. Thusthese loops commute (this comes from the fact that the fundamental group of the torus is Abelian).Therefore, the path σ can be homotopically deformed into the concatenations σ = σ σ = σ σ , where the path σ occurs for fixed z , and σ occurs for fixed z .6 Integral representation of the analytical continuation
Here we are presenting the technique for analytical continuation of the wave field utilising Green’sthird identity as in e.g. [14, 11].Let U ⊂ ( S \ ∪ j P j ) be a small neighbourhood of a point A ∈ S such that ψ ( A )=( x , x ) ∈ R ,where ψ is the natural projection of S to R . In what follows, in an abuse of notation, we maysometimes identify A and ψ ( A ) when the context permits to do so. Let the contour γ be theboundary of U oriented anticlockwise with unit external normal vector n . Write Green’s thirdidentity for A ∈ U : u ( A ) = (cid:90) γ (cid:20) ∂G∂n (cid:48) ( r , r (cid:48) ) u ( r (cid:48) ) − ∂u∂n (cid:48) ( r (cid:48) ) G ( r , r (cid:48) ) (cid:21) dl (cid:48) , (4.1)where r (cid:48) = ( x (cid:48) , x (cid:48) ) is a position vector along γ , r = ( x , x ) points to A , ∂/∂n (cid:48) corresponds tothe normal derivative associated to the unit external normal vector, and G ( r , r (cid:48) ) = − i H (1)0 ( (cid:107) r ( r , r (cid:48) )) with r ( r , r (cid:48) ) = (cid:112) ( x − x (cid:48) ) + ( x − x (cid:48) ) , (4.2)where H (1)0 is the zeroth-order Hankel function of the first kind. Note that the point A is the onlysingularity of the integrand of (4.1) in the real ( x (cid:48) , x (cid:48) ) plane.The orientation of γ plays no role in (4.1), however we can use graphically the orientation ofcontours to set the direction of the normal vector. Namely, let the normal vector point to the right from an oriented contour.The formula (4.1) can be used to continue u ( x , x ) to u c ( x , x ) in a small domain of C .Namely let A ≡ r = ( x , x ) ∈ C be complex, while ( x (cid:48) , x (cid:48) ) remains real and belongs to γ . If( x , x ) is close to A , the Green’s function G ( r , r (cid:48) ) remains regular for each r (cid:48) ∈ γ . Moreover, beingconsidered as a function of r , the Green’s function G ( r , r (cid:48) ) obeys the Cauchy–Riemann conditions(3.1) and the complex Helmholtz equation (3.2) provided r is a regular point for certain fixed r (cid:48) .Thus, for a small complex neighbourhood of A the formula (4.1) provides a function obeying allrestrictions (listed in Section 3) imposed on u c ( A ).We should note that the continuation u c of u in a small complex neighbourhood of A is uniqueand is provided by letting r become a complex vector in (4.1). The proof is given in Appendix Band its structure is as follows. We start by deriving a complexified Green’s formula obeyed by u c (or by any analytical solution of (3.2)). Then, by Stokes’ theorem, the integration contour forthis formula for A belonging to some small complex neighbourhood of A can be taken to coincidewith γ . In this case, the complexified Green’s formula for u c coincides with (4.1).The procedure of analytical continuation of u using (4.1) fails when G ( r , r (cid:48) ) becomes singularfor some r (cid:48) . Let us develop a simple graphical tool to explore the singularities of G ( r , r (cid:48) ) forcomplex r ≡ A . The function (4.2) is singular when( x − x (cid:48) ) + ( x − x (cid:48) ) = 0 , (4.3)i.e. when A ≡ ( x , x ) and A (cid:48) ≡ ( x (cid:48) , x (cid:48) ) both belong to some characteristic of (3.2). Let us fix thecomplex point A and find the real points A (cid:48) ≡ ( x (cid:48) , x (cid:48) ) such that (4.3) is valid. Obviously, A (cid:48) canhave two values A = (Re[ x ] − Im[ x ] , Im[ x ] + Re[ x ]) , (4.4) A = (Re[ x ] + Im[ x ] , Re[ x ] − Im[ x ]) . (4.5)7hese two points coincide when A is a real point. We will call A and A the first and the secondreal points associated with A and will sometimes use the notation A ( A ) and A ( A ) to emphasisethe link between A , and A . Both points A and A belong to R . Indeed, beside A and A , onecan consider their preimages ψ − ( A ) and ψ − ( A ) on S .The Green’s function G ( r , r (cid:48) ) is singular at some point of the integral contour γ in (4.1) if A ( A ) ∈ γ or A ( A ) ∈ γ , where r points to A .Consider the analytical continuation along a simple path σ as a continuous process. Let σ beparametrised by a real parameter τ ∈ [0 , σ be denoted by A ( τ ). Let A (0) = A ∈ S be the starting real point, and let A (1) be the ending complex point.For each point A ( τ ) find the associated real points A , ( τ ) ≡ A , ( A ( τ )). The position of thesepoints depends continuously on τ .We are now well-equipped to formulate the first theorem of analytical continuation. Theorem 4.1.
Let A ∈ S \ ∪ j P j and let A ∈ C be within a small neighbourhood of A . Let σ bea simple path from A to A parametrised by τ ∈ [0 , as above. Let Γ( τ ) ⊂ S be a continuous setof closed smooth oriented contours (i.e. a homotopical deformation of a contour) such that • Γ(0) = γ ; • for each τ ∈ [0 , A ( τ ) / ∈ ψ (Γ( τ )) , A ( τ ) / ∈ ψ (Γ( τ )) .Then the formula u c ( A ( τ ) + δ r ) = (cid:90) Γ( τ ) (cid:20) ∂G∂n (cid:48) ( r , r (cid:48) ) u ( r (cid:48) ) − ∂u∂n (cid:48) ( r (cid:48) ) G ( r , r (cid:48) ) (cid:21) dl (cid:48) , (4.6) defines an analytical continuation u c of u in a narrow neighbourhood of σ . δ r is an arbitrarysmall-enough complex radius vector, while the radius-vector r points to A ( τ ) + δ r .Proof. We present a sketch of the proof on the “physical level of rigour”. Let the contour Γ( τ ) bechanging incrementally, i.e. consider the contours Γ( τ n ), where 0 = τ < τ < τ · · · < τ K = 1 is adense grid on the segment [0 , τ n ) provides an analytic function u c in a smallneighbourhood of the point A ( τ n ). The grid is dense enough to ensure that such neighbourhoodsare overlapping. Moreover, for any point belonging to an intersection of neighbourhoods of A ( τ n )and A ( τ n +1 ) one can deform the contour Γ( τ n ) into Γ( τ n +1 ) homotopically without changing thevalue of the integral, and hence without changing the value of u c .Note that, formally, the expression (4.6) defines the field u c ambiguously. The values of u and ∂ n (cid:48) u on S are found in a unique way, but the values of G and ∂ n (cid:48) G should be clarified. Namely,according to (4.2), the value depends on the branch of the square root and of the Hankel function(having a logarithmic branch point at zero).For τ = 0, let G be defined on γ = Γ(0) in the “usual” way: the square root is real positive,and the values of H (1)0 ( · ) are belonging to the main branch of this function. Then, as τ changescontinuously from τ = 0 to 1, define the values of G and ∂ n (cid:48) G by continuity. Since the (moving)contour Γ( τ ) does not hit the (moving) singular points A ( τ ), A ( τ ), the branch of G is definedconsistently.The last theorem in this section extends the local result of Theorem 4.1 to a global result.8 heorem 4.2. Let B be a point of C \ ( T ∪ R ) , where T is the union of all the 2-lines L ( j )1 , . Let A be a point belonging to S \ ( ∪ j P j ) and let σ be a smooth path connecting A with B , such that ( σ \ A ) ∩ ( T ∪ R ) = ∅ . Then there exists a family of contours Γ( τ ) associated with σ and obeying the conditions of The-orem 4.1. We omit the proof of this theorem. It is almost obvious, but not easy to be formalised. Oneshould consider the process of changing τ from 0 to 1, and “pushing” the already built contourΓ( τ ), which is considered to be movable, by the moving points A ( τ ) and A ( τ ) (or, to be moreprecise, by small disks centred at A ( τ ) and A ( τ )).Some issues may occur with such deformation. For example, the contour may become pinchedbetween A , ( τ ) and P j for some j , or between A ( τ ) and A ( τ ). The condition( σ \ A ) ∩ T = ∅ guarantees that the points A ( τ ), A ( τ ) do not pass through P j , and thus the contour Γ( τ ) cannotbe pinched between A , and P j . The condition( σ \ A ) ∩ R = ∅ guarantees that the contour Γ( τ ) cannot be pinched between A ( τ ) and A ( τ ).Theorems 4.1 and 4.2 demonstrate that u c can be expressed almost everywhere as an integral(4.6) containing u ( x , x ) defined on the real surface S , and some known kernel G . The continuation u c has a more complicated structure than u on S . This is explained by the fact that the set ofclosed contours on S has a more complicated structure than S itself. This will be investigated inthe next section. We will now study the branching of u c as follows. Let us consider a starting point A ∈ ( S \ ∪ j P j ),an ending point B located near a set on which we study the branching (i.e. near R or L ( j )1 , ), anda simple path σ connecting A to B satisfying( σ \ A ) ∩ ( T ∪ R ) = ∅ , i.e. σ is such that the condition of Theorem 4.2 is valid. According to the Theorems 4.1 and 4.2,one can continue u c from A to B along σ . Let the result of this continuation be denoted by u c ( B ). Let σ (cid:48) be a local path starting and ending at B and encircling a corresponding fragmentof the branch set. For example, one can introduce a local transversal variable near the branch setand build a contour σ (cid:48) in the plane of this variable encircling zero for a single time in the positivedirection. One can consider a continuation of u c ( B ) along σ (cid:48) and obtain a branch of u c that isdenoted by u c ( B ; σ (cid:48) ). This is a continuation of u c from A along the concatenation σσ (cid:48) of thecontours σ and σ (cid:48) . Let the parameter τ parametrise the contour σσ (cid:48) ; τ = 0 correspond to A ; τ = 1 / σ ; τ = 1 correspond to the end of σσ (cid:48) . Consistently with thisparametrisation, the contour of integration used to obtain u c ( B ) and u c ( B ; σ (cid:48) ) will be denoted byΓ(1 /
2) and Γ(1) respectively. 9f u c ( B ; σ (cid:48) ) ≡ u c ( B ), then the contour σ (cid:48) yields no branching. If u c ( B ; ( σ (cid:48) ) p ) ≡ u c ( B ), forsome integer p >
1, then the branching has order p (the smallest strictly positive integer with suchproperty). Here u c ( B ; ( σ (cid:48) ) p ) is the result of continuation along the concatenation σσ (cid:48) . . . σ (cid:48) , where σ (cid:48) is taken p times. The two following theorems establish the type of branching of u c . Theorem 5.1.
The points of the real plane R other than P j do not belong to the branch set of u c .Proof. Note that R is not an analytical set, so the concept of a transversal variable is not fullyapplicable to it (and of course this is the reason of non-branching at it). Anyway, the schemesketched above can be applied to this case. Let B be close to R but not close to any of the P j .Consider a local path σ (cid:48) starting and ending at B and encircling R . Consider the motion of A ( A )and A ( A ) as the point A travels along σ (cid:48) . To be consistent with the parametrisation, we will referto these points as A , ( τ ) for τ ∈ [1 / , B since B is close to R . Depending on the particular contour σ (cid:48) , it may happen that during this motion A encircles A a single time, several times or not at all. Note that A (1 /
2) = A (1) = A ( B ) and A (1 /
2) = A (1) = A ( B ).One can show that if A does not encircle A then the contour Γ(1) is homotopic to Γ(1 / u c ( B ; σ (cid:48) ) = u c ( B ).Let A encircle A a single time. The case of several times follows from this case in a clearway. To consider this case, we follow the principles of computation of ramified integrals that canbe found for example in [16, 11, 13]. Namely, it is known that the integral changes locally duringa local bypass. The fragments of the contour Γ(1 /
2) that are located far from the points A (1 / A (1 /
2) are not affected by the bypass σ (cid:48) . The fragments that are close to these points butdo not pass between these points are not affected either. The only parts that are affected are theparts of Γ(1 /
2) passing between the points A (1 /
2) and A (1 / /
2) (Figure 3a into a corres-ponding fragment of Γ(1) (Figure 3b). One can see from Figure 3c that the fragment of Γ(1) can bewritten as the sum of the initial fragment of Γ(1 /
2) and an additional contour δ Γ. The contour δ Γis a “double-eight” contour having the following property: it bypasses each of the points A (1 / A (1 /
2) zero times totally (one time in the positive and one time in the negative direction).We say that this double-eight contour is based on the points A (1 /
2) and A (1 / G(1/2) A (1/2) A (1/2) G(1/2) A A dG G(1) A (1) A (1) (a) (b)(c) A A dG (d) dG ' Figure 3: Transformation of a fragment of the integration contour Γ as A encircles A /
2) passing between A (1 /
2) and A (1 / /
2) and several double-eight contours based on the points A (1 /
2) and A (1 / S .Let us show that for a double-eight contour δ Γ based on A (1 /
2) and A (1 / (cid:90) δ Γ (cid:20) ∂G∂n (cid:48) ( r , r (cid:48) ) u ( r (cid:48) ) − ∂u∂n (cid:48) ( r (cid:48) ) G ( r , r (cid:48) ) (cid:21) dl (cid:48) = 0 . (5.1)This will yield that u c ( B ; σ (cid:48) ) = u c ( B ) and that there is no branching on R . In order to do so,connect the points A (1 /
2) and A (1 /
2) with an oriented contour δ Γ (cid:48) (see Figure 3d, δ Γ (cid:48) is shownby a black line). Squeeze the contour δ Γ in such a way that it goes 4 times along δ Γ (cid:48) .We now claim that it is not necessary to account for the integral in the close vicinity of A (1 /
2) or A (1 / G hastwo logarithmic branch points at A (1 /
2) and A (1 / δ Γneeds to be considered on the Sommerfeld surface of G . Such surface has infinitely many sheets,and can be constructed by considering a straight cut between A (1 /
2) and A (1 /
2) and suitable“gluing”. What is important is that locally, around A (1 /
2) and A (1 / G behaves like a complexlogarithm. Hence, locally, the difference in G between two adjacent sheets is constant. Therefore,when considering the part of δ Γ in the vicinity of A (1 /
2) and A (1 / A , (1 / u being single valued on δ Γ, the logarithmic singularities of G cancelout. Thus, a consideration of the integral near A (1 /
2) and A (1 /
2) is not necessary, and it ispossible to reduce δ Γ to four copies of δ Γ (cid:48) .As explained in detail in Appendix C, according to the formulae (4.4)-(4.5) linking the complexvariables ( x , x ) with the real coordinates of the points A , A , and to formulae (4.2), a bypassabout A in the positive direction (in the real plane) leads to the following change of the argumentof H (1)0 ( z ): z → e iπ z . Similarly, a bypass about A in the positive direction changes the argumentof the Hankel function as z → e − iπ z .As a result, one can rewrite the integral of (5.1) as (cid:90) δ Γ (cid:20) ∂G∂n (cid:48) ( r , r (cid:48) ) u ( r (cid:48) ) − ∂u∂n (cid:48) ( r (cid:48) ) G ( r , r (cid:48) ) (cid:21) dl (cid:48) = (cid:90) δ Γ (cid:48) (cid:20) ∂G (cid:48) ∂n (cid:48) ( r , r (cid:48) ) u ( r (cid:48) ) − ∂u∂n (cid:48) ( r (cid:48) ) G (cid:48) ( r , r (cid:48) ) (cid:21) dl (cid:48) , (5.2)where G (cid:48) ( r , r (cid:48) ) = − i (cid:16) − H (1)0 ( e − iπ (cid:107) r ( r , r (cid:48) )) + 2 H (1)0 ( (cid:107) r ( r , r (cid:48) )) − H (1)0 ( e iπ (cid:107) r ( r , r (cid:48) )) (cid:17) . (5.3)The notation H (1)0 ( e iπ z ) means the value of H (1)0 ( · ) obtained as the result of continuous rotationof the argument z about the origin for the angle π in the positive direction.Apply the formula (see e.g. [17], sec. 1.33, eqs (202)–(203)) well-known from the theory ofBessel functions : − H (1)0 ( e − iπ z ) + 2 H (1)0 ( z ) − H (1)0 ( e iπ z ) = 0 , (5.4)to conclude that the expression on the right-hand side of (5.2) is equal to zero and conclude theproof. Here and below, the formula (5.4) seems to play a fundamental role in the process ofanalytical continuation.Note also that every branch of u c is continuous at any point A ∈ R \ ∪ j P j , thus these pointsare regular points of any branch of u c . This can be seen using the complexified Green’s theorem of11ppendix B. Indeed, using the same idea of variable translation, it can be shown that for a point A ∈ C \ ( T ∪ R ) in a close complex neighbourhood of A and a given branch u c ( A ), we can write u c ( A ) = (cid:82) ˜ γ ( u c ∇ C G − G ∇ C u c ) for some small contour ˜ γ surrounding A in C . Since the differentialform u c ∇ C G − G ∇ C u c is closed (see Appendix B), Stokes’ theorem allows us to deform ˜ γ to acontour γ in R surrounding A (see Figure 12) without changing the value of the integral. A ischosen close enough to A so that we can let A → A along a simple small straight path withoutany singularities of the integrand hitting the contour γ , showing, in doing so, the continuity of u c at A . Theorem 5.2.
Let P j ∈ R be a branch point of order p (the definition of the order of the branchpoint is clarified below). Then both L ( j )1 and L ( j )2 are branch 2-lines of u c of order p .Proof. First, let us define what it exactly means for P j ∈ R to be a branch point of S of order p .The situation is simple if ψ − ( P j ) is a single point. Then p is the order of branching of S at ψ − ( P j ). If ψ − ( P j ) is a set of several points, then p is the least common multiple of orders of allpoints ψ − ( P j ). A bypass encircling P j p times in the positive direction returns each point locatedclosely to P j to itself.Consider a point B ∈ C \ T that is close to L ( j )1 . Let the loop σ (cid:48) bypass L ( j )1 once in thepositive direction (i.e. its projection onto the local transversal coordinate z = x + ix − ( X ( j )1 + iX ( j )2 )bypasses zero once in the positive direction). Let σ (cid:48) be also such that it does not bypass anyother 2-line of T . As before, parametrise σ (cid:48) by τ ∈ [1 / ,
1] and denote by Γ( τ ) the associatedintegration contours used for the analytical continuation along σ (cid:48) . Then A ( τ ) bypasses P j oncein the positive direction, and A ( τ ) does not bypass any branch point.The fragments of Γ(1 /
2) that are far from A (1 /
2) or not passing between A (1 /
2) and P j arenot affected by σ (cid:48) . For each fragment of Γ(1 /
2) passing between A (1 /
2) and P j , the double-eightcontour δ Γ is added to obtain a fragment of Γ(1). In this case, the double-eight contour is basedon the points A (1 /
2) and P j . The graphical proof is similar to that in Figure 3.Let us now show that the order of branching at L ( j )1 is equal to p . Consider a fragment ofΓ(1 /
2) passing between A and P j . A single bypass along σ (cid:48) changes (locally)Γ(1 / σ (cid:48) −→ Γ(1 /
2) + δ Γ , where δ Γ is illustrated in Figure 4. Upon performing a second bypass σ (cid:48) , we hence getΓ(1 /
2) + δ Γ σ (cid:48) −→ Γ(1 /
2) + δ Γ + δ Γ (1) , where δ Γ (1) is the double-eight contour based on the points A and P j , and obtained from δ Γ byrotating A about P j once in the positive direction.The projection ψ ( δ Γ (1) ) coincides with ψ ( δ Γ), however, the new contour passes along othersheets of S , and the branch of H (1)0 is different on it. Finally, after p rotations one gets theintegration contour Γ(1 /
2) + δ Γ + δ Γ (1) + · · · + δ Γ ( p − . In order to show that the branching of L ( j )1 is of order p , we need to show that that (cid:90) δ Γ+ δ Γ (1) + ··· + δ Γ ( p − (cid:20) ∂G∂n (cid:48) ( r , r (cid:48) ) u ( r (cid:48) ) − ∂u∂n (cid:48) ( r (cid:48) ) G ( r , r (cid:48) ) (cid:21) dl (cid:48) = 0 , (5.5)12 dG j PC C C C Figure 4: Illustration of δ Γ and the points of application of formula (5.4) to δ Γimplying that u c ( B ; ( σ (cid:48) ) p ) = u c ( B ), and that the branching has order p . Algebraically, in terms ofhomology classes, this can be written as δ Γ + δ Γ (1) + · · · + δ Γ ( p − = 0 . (5.6)We will show this by decomposing δ Γ and all its subsequent “copies” into three main parts: (i)the two circles around P j ; (ii) the two circles around A and (iii) the four straight lines between A and P j , and showing that the contribution of each of these three parts to the integral (5.5) isindeed zero.(i) By Meixner conditions, u and ∂ (cid:48) n u are integrable in the close vicinity of P j , where G ( r , · ) iswell behaved. Hence we can “shrink” the circles to P j , and the integrals over the two circlesbecome zero and do not contribute to (5.5).(ii) For the circles around A , the argument is slightly more subtle. Let ( s , · · · , s p ) be the sheetsof S accessible by turning around P j . Whatever sheet u is on the initial outer circle, after p rotation, it will have “visited” all sheets s , · · · , s p once. The same is true for the innercircle. Hence, the overall contribution of the circles to (5.5) can be written as (cid:90) A (cid:26)(cid:20) ∂G∂n (cid:48) ( r , r (cid:48) ; h ) u ( r (cid:48) ; s ) − ∂u∂n (cid:48) ( r (cid:48) ; s ) G ( r , r (cid:48) ; h ) (cid:21) + · · · + (cid:20) ∂G∂n (cid:48) ( r , r (cid:48) ; h p ) u ( r (cid:48) ; s p ) − ∂u∂n (cid:48) ( r (cid:48) ; s p ) G ( r , r (cid:48) ; h p ) (cid:21)(cid:27) dl (cid:48) − (cid:90) A (cid:26)(cid:20) ∂G∂n (cid:48) ( r , r (cid:48) ; h (cid:48) ) u ( r (cid:48) ; s ) − ∂u∂n (cid:48) ( r (cid:48) ; s ) G ( r , r (cid:48) ; h (cid:48) ) (cid:21) + · · · + (cid:20) ∂G∂n (cid:48) ( r , r (cid:48) ; h (cid:48) p ) u ( r (cid:48) ; s p ) − ∂u∂n (cid:48) ( r (cid:48) ; s p ) G ( r , r (cid:48) ; h (cid:48) p ) (cid:21)(cid:27) dl (cid:48) , (5.7)for some ( h , · · · , h p ) and ( h (cid:48) , · · · , h (cid:48) p ) corresponding to given sheets of G ( r , · ), where the ;notation in the argument of a functions specifies which sheet it is on, and where (cid:82) A specifiesintegration along a small circle encircling A . Now we can group the terms for which u lieson the same sheet. For each such pair, we can use the reasoning used below (5.1) in theproof of Theorem 5.1 to show that even if G may be on a different sheet for each element ofthe pair, the logarithmic singularities cancel out. Hence one can safely “shrink” the circlesto A without leading to any contribution to (5.5).13iii) Finally, for the straight lines, one can check that due to the identity (5.4), the branch of H (1)0 (and hence of G ) does not matter. Namely, let C , C , C , C be some points of δ Γ projectedonto a single point C ∈ R . These points are shown in Figure 4, but for clarity they areshown close to each other, not above one another.Consider the points C and C . They belong to the same sheet of S , but the branches of H (1)0 are different at these points. Namely, if the argument of H (1)0 is equal to z at C , then itis equal to ze − iπ at C (see again Appendix C). Since the contours have different directionsat C and C , the contribution of the two outer lines to the integral over δ Γ is of the form (cid:90) δ Γ (cid:48) (cid:20) ∂G (cid:48) ∂n (cid:48) ( r , r (cid:48) ) u ( r (cid:48) ; s ) − ∂u∂n (cid:48) ( r (cid:48) ; s ) G (cid:48) ( r , r (cid:48) ) (cid:21) dl (cid:48) , (5.8)for some s ∈ ( s , · · · , s p ), where G (cid:48) ( r , r (cid:48) ) = − i ( H (1)0 ( (cid:107) r ( r , r (cid:48) )) − H (1)0 ( e − iπ (cid:107) r ( r , r (cid:48) )) andwhere δ Γ (cid:48) is a straight line between A and P j .Perform a bypass σ (cid:48) . In the course of this bypass, the points C and C encircle A inthe positive direction. Thus, the corresponding part of the integral along δ Γ (1) contains H (1)0 ( ze iπ ) − H (1)0 ( z ) and its derivative. However, due to (5.4), H (1)0 ( ze iπ ) − H (1)0 ( z ) = H (1)0 ( z ) − H (1)0 ( ze − iπ ) , (5.9)and hence the value of G (cid:48) is unaffected by such bypass. As before, after p bypasses, u wouldhave “visited” all the sheets ( s , · · · , s p ) and the overall contribution of the two outer linesto (5.5) can be written (cid:90) δ Γ (cid:48) (cid:20) ∂G (cid:48) ∂n (cid:48) ( r , r (cid:48) ) u (cid:48) ( r (cid:48) ) − ∂u (cid:48) ∂n (cid:48) ( r (cid:48) ) G (cid:48) ( r , r (cid:48) ) (cid:21) dl (cid:48) , (5.10)where u (cid:48) ( · ) = u ( · , s ) + · · · + u ( · , s p ).The same consideration can be conducted for the points C and C on the inner lines to showan overall contribution equal to minus that of (5.10). Hence the overall contribution of thestraight lines to (5.5) is also zero.We have therefore proved that the equality (5.5) is correct, and hence that L ( j )1 is a branch2-line of order p . The case of σ (cid:48) bypassing L ( j )2 can be considered in a similar way. Note thatthis time, if σ (cid:48) bypasses L ( j )2 once in the positive direction (and no other branch 2-line), then thecorresponding A ( τ ) bypasses P j once in the negative direction, and A ( τ ) does not bypass anybranch point. Theorem 5.3.
Consider two points
A, B ∈ ( C \ T ) , and let σ and σ be piecewise-smooth pathsin ( C \ T ) connecting A with B . Assume that it is possible to continue homotopically σ to σ in ( C \ T ) . Let u c ( A ) be some branch of u c in some neighbourhood of A . Then the branches of u c ( B ) obtained by continuation of u c ( A ) along σ and along σ coincide: u c ( B ; σ ) = u c ( B ; σ ) . This theorem follows naturally from the principle of analytical continuation, which remainsvalid in 2D complex analysis. In the following theorem, we show that any closed contour in C \ T can be deformed to a concatenation of canonical building blocks .14 heorem 5.4. Let A be a point in C \ T , and let σ be a closed path in C \ T starting and endingat A . Then, within C \ T , σ can be homotopically transformed into a concatenation contour σ (cid:48) ofthe form σ (cid:48) = σ m α σ m α . . . σ m H α H , (5.11) for some positive integer H . For h ∈ { , . . . , H } , the powers m h belong to Z and the elementarycontours σ α h are fixed for each α h . The indices α h are pairs of the type ( (cid:96), j ) , where (cid:96) ∈ { , } and j ∈ { , . . . , N } . These contours can themselves be represented as concatenations σ α h = γ α h σ ∗ α h γ − α h , (5.12) where each contour γ α h goes from A to some point near L ( j ) (cid:96) , and σ ∗ α h is a local contour encircling L ( j ) (cid:96) once. On each contour σ ,j the value x − ix is constant. On each contour σ ,j the value x + ix is constant. Examples of such contours can be found in Figure 5.
L LLL
P P g L (2) s * A s g - Figure 5: Illustration of some contours σ α , γ α and σ ∗ α Proof.
Consider a small neighbourhood U ⊂ C \ T of some point A . By the principles of analyticcontinuation, all possible branches u c ( A ) are obtained by continuation along loops ending andstarting at A that cover all possible combinations of the homotopy classes of C \ T . Each σ α represents one of the homotopy classes, and hence by allowing σ (cid:48) to take the form (5.11), allpossible combinations of such classes are covered. Finally since any closed contour σ can bewritten as a combination of homotopy classes, then we can in principle deform any closed contour σ to one akin to σ (cid:48) .Note that the elementary paths σ α have the following property. Since either x − ix or x + ix is constant on such a path, either A or A remains constant as the path σ α is passed.In the next theorem, we formulate the main general result of the paper, which is that thereexists a finite basis of elementary functions such that any branch of the analytical continuation u c is a linear combination, with integer coefficients, of such functions. Theorem 5.5.
For a neighbourhood U ⊂ C \ T of a given point A , one can find a finite set ofbasis functions g ( A ) , . . . , g Q ( A ) , A ∈ U , which are analytical solutions of the complex Helmholtzequation (3.2) in U , and such that any branch u c ( A ; σ ) , can be written as a linear combination u c ( A ; σ ) = Q (cid:88) q =1 b q ( σ ) g q ( A ) , (5.13)15 here b q ( σ ) are integer coefficients that are constant with respect to A . The dimension of the basis, Q , can be defined from the topology of S .Proof. According to Theorem 4.2, Theorem 5.1, Theorem 5.2 and the basic principles of ana-lytical continuation, it is enough to prove the statement of the theorem for an arbitrary smallneighbourhood U .Indeed, if the theorem is true for such a neighbourhood, the elements of the basis, then, can beexpressed as linear combinations of Q linearly independent branches of u c . The coefficients of thecombination are constant, therefore the elements of the basis have singularities only at T , and thetype of branching is the same as that of u c . Any other neighbourhood U (cid:48) can be connected with U with a path in C \ T , and the formula (5.13) can be continued along this path.Hence, below, we prove the theorem for a point A , for which the mutual location of the points A , (the real points associated to A ) and P j (the branch points on S ) is convenient in some sense.Choose the “convenient” neighbourhood U as follows. Let us assume that the point A ≡ ( x , x )is far enough from the branch points P j , is close to R but does not belong to R . This results ina configuration akin to that illustrated in Figure 6. G AA dG P PP P dG Figure 6: Illustration of some integration contours used to define the basis functionsDefine the basis functions g q as follows. Let g be defined by the integral (4.6) with the contourΓ shown in Figure 6. Such a function g is equal to u c obtained by analytical continuation froma small neighbourhood of a point belonging to S as per Theorem 4.1.All the other basis functions are constructed as follows. Let δ Γ (cid:96),j be a double-eight contourbased on the points A (cid:96) , (cid:96) ∈ { , } , and P j , j ∈ { , . . . , N } , as illustrated in Figure 6. Considerall preimages ψ − ( δ Γ (cid:96),j ) on S . Denote them by δ Γ (1) (cid:96),j , . . . , δ Γ ( M ) (cid:96),j , where we remind the reader that M is the finite number of sheets of S . Some of them are linearly independent. The functions g , . . . , g Q are the integrals of the form (4.6) taken with all linearly independent contours from theset { δ Γ (1) (cid:96),j , . . . , δ Γ ( M ) (cid:96),j } .To see this, let us continue the function u c from A along some closed path σ . Deform the path σ into a path σ (cid:48) as in Theorem 5.4. Each building block of σ (cid:48) , σ (cid:96),j , can be analysed by the proceduredescribed in the proof of Theorem 5.2. A local contour σ ∗ (cid:96),j produces several local double-eightloops, and the path γ − (cid:96),j stretches the loops into those shown in Figure 6.As mentioned in the statement of Theorem 5.5, the exact value of the dimension Q of the basisdepends on the topology of S and hence on the specific diffraction problem considered. In the16ext section we show (among other results) that in the case of the Dirichlet strip problem, we have Q = 4. Everything so far has been done for a generic scattering problem described in introduction. Wewill here deal with the specific problem of diffraction by a finite strip ( − a < x < a, x = 0). Thecanonical problem of diffraction by a strip has attracted a lot of attention since the beginning of the20th century, and various innovative mathematical methods have been designed and implementedto solve it: Schwarzschild’s series [18], Mathieu functions expansion [19], modified Wiener-Hopftechnique [20, 21], embedding and reduction to ODEs [22, 23]. It has important applications,including in aero- and hydro-acoustics, see [24] for example.The problem can be formulated as follows: find the total field u satisfying the Helmholtzequation and Dirichlet boundary conditions ( u = 0) on the strip, resulting from an incident planewave. The scattered field (the difference between the total and the incident field) should satisfythe Sommerfeld radiation condition, and the total field should satisfy the Meixner conditions atthe edges of the strip.Here we assume that this physical field u is known and we consider the associated Sommerfeldsurface S . It has two branch points P = ( a,
0) and P = ( − a, N = 2. They are eachof order 2, and the Sommerfeld surface S has two sheets ( M = 2). The surface S is shown inFigure 1b. Let us assume that sheet 1 is the physical sheet, while sheet 2 is its mirror reflection.We will now apply the general theory developed in the paper in order to unveil the analyticalcontinuation u c .Let A ≡ ( x , x ) ∈ C \ ( T ∪ R ) be some point near R . Consider all possible continuationsof u c ( A ) along closed paths. According to Theorem 5.4, any such path can be represented as aconcatenation of elementary paths σ (cid:96),j for (cid:96) ∈ { , } and j ∈ { , } . The elementary paths can bechosen in a such a way that corresponding trajectories of the points A and A are as shown inFigure 7. A A s sss
Re[ ] x Re[ ] x a - a Figure 7: Trajectories of A , corresponding to the elementary paths σ (cid:96),j As it follows from the considerations in previous sections, each path σ (cid:96),j is an elementarybypass about the branch 2-line L ( j ) (cid:96) . The 2-lines L ( j )1 are bypassed in the positive direction, whilethe 2-lines L ( j )2 are bypassed in the negative direction.Now let us choose the basis functions g j of Theorem 5.5. For this, consider the contours Γ and the double-eight loops δ Γ (cid:96),j for (cid:96), j ∈ { , } shown in Figure 8. For definiteness, we assume17hat the points marked by a small black circle on the contours belong to the physical sheet of S . A A
Re[ ] x Re[ ] x a - a dG dG A A
Re[ ] x Re[ ] x dG dG G A A
Re[ ] x Re[ ] x P P PP PP
Figure 8: Basis contours for diffraction by a segmentFormally, since the branch points P j are of order 2, there should exist two double-eight loopsfor each pair of indexes (cid:96), j : δ Γ (cid:96),j and δ Γ (1) (cid:96),j . However, due to the identity (5.6), the loops δ Γ (1) (cid:96),j arenot needed to be included into the basis, reducing the number of candidates for the basis functionsfrom 9 to 5.Moreover, using contour deformation and cancellation, one can check the following contouridentity δ Γ , + δ Γ , = δ Γ , + δ Γ , , (6.1)by showing that each side of (6.1) is equal to 2Γ . Therefore, one of the double-eight loops, forexample δ Γ , , does not need to be included into the basis since it depends linearly on the other 3contours. This effectively reduces the number of basis functions to 4, as required and these basisfunctions can be explicitly written as g ( x , x ) = (cid:90) Γ [ . . . ] dl (cid:48) , g ( x , x ) = (cid:90) δ Γ , [ . . . ] dl (cid:48) , (6.2) g ( x , x ) = (cid:90) δ Γ , [ . . . ] dl (cid:48) , g ( x , x ) = (cid:90) δ Γ , [ . . . ] dl (cid:48) , (6.3)where the [ . . . ] integrand is the same as in (4.6).Upon introducing the vector of functionsW = ( g , g , g , g ) T , (6.4)the analytic continuation of u c is fully described by the following theorem. Theorem 6.1.
The analytical continuation along each closed fundamental path σ (cid:96),j affects thevector of basis functions as follows: W σ (cid:96),j −→ M (cid:96),j W , (6.5) where the × constant matrices M (cid:96),j are given by M , = − − − , M , = − − − , (6.6)M , = −
10 1 0 00 0 1 00 0 0 − , M , = − −
10 1 0 00 0 1 00 − − . (6.7)18he statement of the theorem can be checked directly by studying how the double-eight contoursare transformed under the action of each σ (cid:96),j , though it is omitted here for brevity.Nevertheless, the following theorem and its proof can be considered as a check that the matricesgiven in Theorem 6.1 are indeed correct. Theorem 6.2.
Let j, k, (cid:96) ∈ { , } . The following statements are correct:a) Each branch 2-line L ( j ) (cid:96) has order 2, i.e. : u c ( A ; ( σ (cid:96),j ) ) = u c ( A ) . (6.8) b) The bypasses σ ,j and σ ,k commute for any pair j, k : u c ( A ; σ ,j σ ,k ) = u c ( A ; σ ,k σ ,j ) . (6.9) c) The intersecting branch 2-lines L ( j )1 and L ( k )2 have the additive crossing property: u c ( A ) + u c ( A ; σ ,j σ ,k ) = u c ( A ; σ ,j ) + u c ( A ; σ ,k ) . (6.10) Proof.
To prove a) it is enough to show that for all j, (cid:96) ∈ { , } , we haveM (cid:96),j = I , where I is the 4 × j, k ∈ { , } ,we have M ,j M ,k = M ,k M ,j . Finally, to prove c), it is enough to show that for all j, k ∈ { , } , we haveI + M ,j M ,k = M ,j + M ,k . All this can be shown to be true directly for the matrices (6.6) and (6.7).Let us now discuss the proven relations. The relation (6.8) can be considered as an alternativecheck of Theorem 5.2. Indeed, since P and P are branch points of order 2 on S , the branch2-lines L ( j ) (cid:96) have order 2.The second relation (6.9) follows from a fundamental property of multidimensional complexanalysis discussed at the end of Section 3: the bypasses about L ( j )1 and L ( k )2 commute.Finally, the relation (6.10) is discussed in details in [4]. This relation means that the function u c can be represented locally as a sum of two functions having branch 2-lines, separately, at L ( j )1 and at L ( k )2 . As shown in [4] and [5] the additive crossing property plays a fundamental role in theprocess of integration of functions of several complex variables. Here we continue to study the Dirichlet strip problem from the previous section.The property (6.5) of the vector W reminds of the behavior of a Fuchsian ordinary differentialequation on a plane of a single complex variable. Namely, the poles of the coefficients of a FuchsianODE are branch points of its solution, and the vector composed of linearly independent solutions19s multiplied by a constant monodromy matrix as the argument bypasses a branch point. Indeed,here, the M (cid:96),j play the role of such monodromy matrices. It is also well known that, conversely, if avector of linearly independent functions of a single variable have this behaviour, then there existsa Fuchsian equation obeyed by them (This is Hilbert’s twenty-first problem). This can be shownusing the concept of fundamental matrices and their determinants (the Wronskian), see [25].Here the situation is more complicated. There are two independent complex variables insteadof one. Moreover, the behavior of the components of W at infinity are not explored. However, wecan still prove some important statements.Throughout the paper, it is implicitly assumed that the field u , and, thus, its continuationbasis W, depend on the angle of incidence ϕ in . Let us now consider four different incidence angles ϕ inI , ϕ inII , ϕ inIII , and ϕ inIV . It hence leads to four different wave fields, and to four basis vectors, alldefined by (6.4): W I , W II , W III , W IV . Let us construct the 4 × ≡ (W I , W II , W III , W IV , ) . (7.1)We claim here (without proof) that the matrix V is non-singular almost everywhere (in C minus a set having complex codimension 1), so that we can freely write V − .Note that the whole matrix is only branching at T , and that the equations (6.5) are valid forthe matrix V as a whole: V σ (cid:96),j −→ M (cid:96),j V . (7.2)This allows us to formulate the following theorem, linking the theory of differential equationsto the strip diffraction problem: Theorem 7.1.
There exist two × matrix functions Z ( x , x ) and Z ( x , x ) , meromorphic in C , such thata) the matrix function V satisfies the following differential equations ∂ x V = V Z and ∂ x V = V Z ; (7.3) b) these matrix functions obey the consistency relation: Z Z − Z Z = ∂ x Z − ∂ x Z , (7.4) where ∂x (cid:96) for (cid:96) ∈ { , } are the complex derivatives defined in (3.2).Proof. For a) assume that V is known and that V − exists almost everywhere (in C minus a setof complex codimension 1). In this case, the coefficients Z and Z are simply given byZ = V − ∂ x V and Z = V − ∂ x V . (7.5)Let us show that the matrices Z and Z are single-valued in C . The only sets at which one canexpect branching are the branch 2-lines of V, i.e. L ( j ) (cid:96) . Make a bypass σ (cid:96),j about a 2-line L ( j ) (cid:96) andstudy the change of Z and Z as the result of this bypass:Z k σ (cid:96),j −→ V − M − (cid:96),j ∂ x k (M (cid:96),j V) = Z k , k ∈ { , } , (7.6)because M (cid:96),j are constant matrices. Thus, the coefficients Z and Z are not changing at the branch2-lines of V, and, therefore, they are single-valued in C . A detailed study shows that they have20imple polar sets at the lines L ( j ) (cid:96) , and, possibly, polar sets at the zeros of det(V) though we omitthis discussion here for brevity.To prove b), differentiate the first equation of (7.3) with respect to x , and the second equationwith respect to x . The expressions in the left are equal, and we get( ∂ x V) Z + V ∂ x Z = ( ∂ x V) Z + V ∂ x Z . Applying (7.3) and multiplying by V − , obtain (7.4).As it follows from Frobenius theorem, this relation guarantees the solvability of the system(7.3).A detailed form of the coefficients Z and Z can be found in [1, 2, 3]. Here our aim was justto demonstrate that the existence of the coordinate equations is connected with the structure ofanalytical continuation of the solution.Finally, before concluding the paper, it is interesting to note that the idea of considering a setof different incident angles and trying to link the solutions to each other by means of differentialequations is somewhat reminiscent of Biggs’ interpretation of embedding formulae [26]. We have provided an explicit method to analytically continue two-dimensional wave fields eman-ating from a broad range of diffraction problems and described the singular sets (in C ) of theiranalytical continuation. We have shown that, even though the analytical continuation may havepotentially infinitely many branches, each branch can be expressed as a linear combination offinitely many basis functions. Such basis functions are expressed as Green’s integrals over a realdouble-eight contour. The effectiveness of the general theory was illustrated via the example ofdiffraction by an ideal strip, for which we proved that only 4 basis functions were needed. Usingthese, we were able to completely describe the analytical continuation and study its branching be-haviour. Finally, we have shown that this finite basis property was directly related to the existenceof the so-called coordinate equation for the strip problem. Acknowledgement
R.C. Assier would like to acknowledge the support by UK EPSRC (EP/N013719/1). Both au-thors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, forsupport and hospitality during the programme “Bringing pure and applied analysis together viathe Wiener–Hopf technique, its generalisations and applications” where some work on this paperwas undertaken. This work was supported by EPSRC (EP/R014604/1) and, in the case of A.V.Shanin, the Simons foundation. Both authors are also grateful to the Manchester Institute forMathematical Sciences for its financial support.21 eferences [1] Shanin AV. 2003 A generalization of the separation of variables method for some 2D diffractionproblems.
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Introduction to complex analysis Part II. Functions of several variables .American Mathematical Society.
A Diffraction problems on a plane and on Sommerfeld sur-faces
In this appendix, we aim to describe the wide class of 2D diffraction problems for which the theorydeveloped in the paper is valid.Consider an incident plane wave u in impinging on a set of scatterers. The aim is to find theresulting total field u satisfying the Helmholtz equation and subject to some specified boundaryconditions on the scatterers. For the problem to be well-posed, we also require that u have boundedenergy at the edges of the scatterers (Meixner condition) and that the scattered field u − u in beoutgoing (radiation condition).For the theory of the paper to be applicable to such problem, it should obey the four followingsimple rules: 231 all scatterers faces should be straight segments, possibly intersecting, possibly of infinitelength;R2 the boundary conditions should be of Neumann or Dirichlet type, they are allowed to bedifferent on each side of the segment;R3 the segments should either have the same supporting line or, failing that, their supportinglines should all intersect in one common point;R4 the angles between the supporting lines of each segment should be rational multiples of π .Provided that R1–R4 are satisfied, the diffraction problem can be reformulated as a propagationproblem on a Sommerfeld surface, without scatterers per se , but with a finite number N of branchpoints and a finite number M of sheets.One can formalise the procedure of building the Sommerfeld surface S and the solution u on it provided that the solution u in the physical domain is known. Consider the origin of thepolar coordinates ( ρ, ϕ ) to be the intersection point mentioned in the rule R3 (or anywhere on thesupport line if there is only one). The solution u can be continued past the faces of the scatterersby following the reflection equations across each face of the scatterers:(Dirichlet): u ( ρ, Φ + ϕ ) = − u ( ρ, Φ − ϕ ) , (A.1)(Neumann): u ( ρ, Φ + ϕ ) = + u ( ρ, Φ − ϕ ) , (A.2)where ( ρ, Φ) is a point belonging to a scatterer’s face.In other words, one should take the physical domain with the solution u on it, make cutsalong each scatterers face, make enough copies of the physical plane by reflection across some lines(the supporting lines of the scatterers and of their successive reflections), and attach the obtainedreflected planes to each other according to these reflection equations. Examples.
As illustrated in Figure 9, examples satisfying the conditions R1–R4 with N = 1branch point include: the Dirichlet-Dirichlet (Dir-Dir) or Neumann-Neumann (Neu-Neu) half-plane ( M = 2), the Dirichlet-Neumann (Dir-Neu) half-plane ( M = 4) and any Dir-Dir or Neu-Neu wedge with internal angle pπ/q with ( p, q ) ∈ N > and q > p , for which M is equal to thedenominator of the irreducible form of the fraction q/ (2 q − p ). All these can be solved by theWiener-Hopf technique, see e.g. [20] for Figure 9a , [27] for Figure 9d and [28] for Figure 9b and9d .Figure 9: Example of scatterers (thick blue lines) with N = 1 branch point (thick red dot).Examples with higher-numbers of branch points, illustrated in Figure 10, include the Dir-Diror Neu-Neu strip ( N = 2, M = 2), the Dir-Neu strip ( N = 2 , M = 4), or more complicated24igure 10: Example of scatterers (thick blue lines) with N > π/ π/
4, for whichbranch points not belonging to the physical scatterers start to occur ( N = 9 , M = 8), etc.Finally, configurations with multiple non-intersecting scatterers are also possible, as illustratedin Figure 11. The case of multiple aligned strips (Figure 11a) is a diffraction problem that haspreviously been investigated in e.g. [1] (via coordinate equations) or [29] (via an iterative Wiener–Hopf method).Figure 11: Example of multiple non-intersecting scatterers (thick blue lines), the supporting linesare illustrated in thin black lines B Complex Green’s theorem
In this appendix we aim to formulate a complex version of Green’s theorem. In order to do sowe need to introduce the notion of complex gradient of a function of several complex variables.Because of the topic of the paper, it is enough to focus on C for which it can be defined as follows.For a function v ( x , x ) of two complex variables x and x , the complex gradient of v , denoted ∇ C v , is defined as the complex 1-form ∇ C v ≡ ∂ x v dx − ∂ x v dx . The complex Green’s theorem in C can hence be formulated as follows. Theorem B.1 (Complex Green’s theorem) . If two functions v ( x , x ) and w ( x , x ) both obey thecomplex Helmholtz equation (3.2) in some neighbourhood (included in C ) where they are analytic,i.e. where they satisfy the Cauchy-Riemann conditions (3.1), then, in this neighbourhood, d ( v ∇ C w − w ∇ C v ) = 0 , where d is the usual exterior derivative operator for differential forms. roof. Following [30], it is convenient to decompose the exterior derivative d as d = ∂ + ¯ ∂ , where ∂ and ¯ ∂ are the so-called Dolbeault operators. For any function (0-form) f ( x , x ), they are definedby ∂f = ∂ x f dx + ∂ x f dx and ¯ ∂f = ∂ ¯ x f d ¯ x + ∂ ¯ x f d ¯ x , while for any complex 1-form ω = f ( x , x ) dx + f ( x , x ) dx + g ( x , x ) d ¯ x + g ( x , x ) d ¯ x , theyare given by ∂ω = ∂f ∧ dx + ∂f ∧ dx + ∂g ∧ d ¯ x + ∂g ∧ d ¯ x , ¯ ∂ω = ¯ ∂f ∧ dx + ¯ ∂f ∧ dx + ¯ ∂g ∧ d ¯ x + ¯ ∂g ∧ d ¯ x . Hence, using the fact that v , w and their derivatives are analytic we can show that¯ ∂ ( v ∇ C w − w ∇ C v ) = 0, leading to d ( v ∇ C w − w ∇ C v ) = ∂ ( v ∇ C w − w ∇ C v )= ( ∂ x ( v∂ x w ) + ∂ x ( v∂ x w ) − ∂ x ( w∂ x v ) − ∂ x ( w∂ x v )) dx ∧ dx , where we have used that dx (cid:96) ∧ dx (cid:96) = 0 for (cid:96) ∈ { , } and that dx ∧ dx = − dx ∧ dx .Upon expanding out the derivatives, we obtain d ( v ∇ C w − w ∇ C v ) = ( v ( ∂ x w + ∂ x w ) − w ( ∂ x v + ∂ x v )) dx ∧ dx = 0 , as required, since both v and w satisfy the complex Helmholtz equation.Note that the same result can be proven in C . In order to do so, one has to define the complexgradient of a function v ( x , x , x ) as ∇ C v = ∂ x v dx ∧ dx + ∂ x v dx ∧ dx + ∂ x v dx ∧ dx . These complex gradients definitions are closely related to and can be expressed in terms of theHodge star operator.The complex Green’s theorem, combined with Stokes’ theorem for complex differential forms,makes it possible to show that for any two functions v and w satisfying the hypotheses of TheoremB.1, and two contours γ and γ that can be deformed homotopically to each other within theregion of analyticity of v and w , we have (cid:90) γ ( v ∇ C w − w ∇ C v ) = (cid:90) γ ( v ∇ C w − w ∇ C v ) . (B.1)Hence, the value of the integral is not changed when the contour is deformed, even within C .Moreover, note that when restricted to a contour γ in R , we have v ∇ C w − w ∇ C v = (cid:20) v ∂w∂n − w ∂v∂n (cid:21) dl. The link with what we have done in the paper becomes clear by choosing v ≡ u c and w ≡ G .To prove that u c ( A ) is indeed defined uniquely by (4.1) in a small complex neighbourhood of A , proceed as follows . Let A ≡ ( x A , x A ) ∈ C [ x , x ] \ T in a small neighbourhood of A ∈ S , Below, square brackets following either C or R are used to specify the coordinate system under consideration. u c ( A ) be the value obtained by letting r become a complex vector, and choosing A closeenough to A such that G ( r , r (cid:48) ) remains regular for r (cid:48) ∈ γ .Consider now the change of variable ( ξ , ξ ) = ( x − x A , x − x A ) so that in C [ ξ , ξ ], we have A ≡ (0 , A ∈ C [ ξ , ξ ] ∩ R [ ξ , ξ ], and u c can be studied as a function obeying theHelmholtz equation on the real plane R [ ξ , ξ ]. This can be done by considering the function˜ u c ( ξ , ξ ) = u c ( ξ + x A , ξ + x A ), and, as such, it is given by the Green’s formula u c ( A ) = ˜ u c (0 ,
0) = (cid:90) ˜ γ (cid:20) ˜ u c (˜ r (cid:48) ) ∂G∂n (cid:48) ( , ˜ r (cid:48) ) − G ( , ˜ r (cid:48) ) ∂ ˜ u c ∂n (cid:48) (˜ r (cid:48) ) (cid:21) d ˜ l (cid:48) = (cid:90) ˜ γ ( u c ∇ C G − G ∇ C u c ) , where ˜ γ is a real contour encircling the origin of R [ ξ , ξ ], ˜ r (cid:48) is a real vector pointing to a pointin ˜ γ and is the zero vector. The last equality comes from the fact that a form is independentof the coordinate system in which it is expressed, and from the translational invariance of G thatsatisfies G ( x − x A , x − x A ; x (cid:48) − x A , x (cid:48) − x A ) = G ( x , x ; x (cid:48) , x (cid:48) ).Now when viewed in C [ x , x ], ˜ γ is not a contour in R [ x , x ], but by (B.1) we can deformthis contour to the contour γ ⊂ R [ x , x ] used in (4.1) without changing the value of the integralto get u c ( A ) = (cid:90) γ ( u c ∇ C G − G ∇ C u c ) = (cid:90) γ (cid:20) u c ( r (cid:48) ) ∂G∂n (cid:48) ( r , r (cid:48) ) − G ( r , r (cid:48) ) ∂u c ∂n (cid:48) ( r (cid:48) ) (cid:21) dl (cid:48) , as expected, where r (cid:48) is a real vector pointing to γ and r is a complex vector pointing to A , whichis exactly what we would have obtained by letting r become complex in (4.1). The contours usedand the subsequent deformation are illustrated in Figure 12.Figure 12: Diagrammatic Illustration of the contours ˜ γ and γ and how one is deformed to theother C Bypasses and argument of the Hankel function
Let A = ( x , x ) ∈ C \ ( T ∪ R ). We are interested in the behaviour of the function G ( x (cid:48) , x (cid:48) ) = H (1)0 (cid:16) (cid:107) (cid:112) ( x − x (cid:48) ) + ( x − x (cid:48) ) (cid:17) ,