In this paper we derive an expression for the point source Green's function for the
reduced wave equation, valid in an angular sector whose angle is equal to a rational
multiple of 77. This Green's function can be used to find new expressions for the field
produced by the diffraction of a spherical wave by a wedge whose angle can be
expressed as a rational multiple of n. The expressions obtained will be in the form of
source terms and real integrals representing the diffracted field. The general result
obtained is used to derive a new representation for the solution of the problem of diffraction by a mixed hard-soft half plane

Rawlins, AD

English

Brunel University Research Archive

Math. Proc. Camb. Phil. Soc. (1989), 105, 185 185Printed in Great BritainA Green's function for diffraction by a rational wedgeBY A. D. RAWLINSDepartment of Mathematics and Statistics, Brunei University,Uxbridge, Middlesex UBS 3PH(Received 14 October 1987; revised 18 March 1988)AbstractIn this paper we derive an expression for the point source Green's function for thereduced wave equation, valid in an angular sector whose angle is equal to a rationalmultiple of 77. This Green's function can be used to find new expressions for the fieldproduced by the diffraction of a spherical wave by a wedge whose angle can beexpressed as a rational multiple of n. The expressions obtained will be in the form ofsource terms and real integrals representing the diffracted field. The general resultobtained is used to derive a new representation for the solution of the problem ofdiffraction by a mixed hard-soft half plane.1. IntroductionIn this work we shall give new solutions to various problems of diffraction of aspherical acoustic wave by a wedge whose angle can be expressed as a rationalmultiple of n. The solutions, corresponding to various soft (Dirichlet) and/or hard(Neumann) boundary conditions on the wedge faces, is given in terms of geometricalacoustic source terms, and real integrals representing the diffracted field, for allpositions of source and receiver.In acoustics such a field representation is useful for physical interpretation andanalytical calculations. Besides the problems of diffraction by a soft wedge or a hardwedge, we include in our consideration the field produced by diffraction of thespherical source wave by a hard-soft wedge, that is, a wedge, one face of which isrigid, the other soft. This type of boundary value problem has applications in noisebarrier design (see Rawlins[9]), and in underwater acoustic problems connected withsound propagation in coastal channels (see Jacobson et al. [4, 5]).In 1896 A. Sommerfeld obtained an exact closed form solution to the problem ofplane wave diffraction by a half plane. He used the method of images on Riemannsurfaces corresponding to multivalued solutions of the reduced wave equation, andindicated how this method could be used to obtain exact closed form solution for theproblem of diffraction by a wedge. The method of constructing the required many-valued solution of the wave equation was simplified by Sommerfeld in a subsequentpaper [13]. Subsequently Carslaw[2] replaced the image method by a directconstruction of the solution which yields simpler formulae for various types of wavesources. These solutions had been obtained independently by Macdonald [6, 7] bysumming the Fourier series representation of the Green's function. Other methods of186 A. D. RAWLINSobtaining some or all of the principal results have since been put forward by variousauthors, including Bromwich[l], Whipple[14], Garnir[3] and Oberhettinger[8]. Thelast two authors give extensive bibliographies.Recently Rawlins[10, 11] derived some new representations for the Green'sfunction for plane and cylindrical wave diffraction by a rational wedge. The resultfor the cylindrical wave Green's function was based on the plane wave Green'sfunction. Here we derive in a different, more direct, and simpler manner the Green'sfunction for a spherical point source. The method used here can be applied withoutdifficulty to deal with more general types of sources, in particular the plane wave andcylindrical wave source already considered.In §2 we shall give the geometry of the physical diffraction problems. In §3 weshall define a periodic Green's function as the solution of a periodic boundary valueproblem for a wedge of open angle 2a. By means of this periodic Green's function weshow in §4 that one can obtain expressions for the Green's function for Dirichlet,Neumann or mixed Dirichlet—Neumann boundary value problems for a wedge ofopen angle a. In §5 we give a general complex integral representation for the periodicGreen's function. This complex integral representation for the periodic Green'sfunction is not new. It has appeared in various guises in the papers already cited. In§ 6 we show how this Green's function can be reduced to source terms and real integralsrepresenting the diffracted waves for the special case of a rational wedge a = pn/q (pand q positive integers). Finally in §7 we give an application in the form of a newsolution for the problem of acoustic diffraction of a spherical wave by a hard—softhalf plane.2. Geometry of diffraction problemsWe shall be interested in finding solution to diffraction problems in wedge shapedregions. To be specific, we shall consider acoustic diffraction by time harmonic waves(with harmonic time variation eiwt assumed, but not shown explicitly, in the rest ofthe paper).We shall work in Euclidean space of three dimensions with cylindrical polarcoordinates (r, 8, z). In this space we shall assume there is a wedge of open angle awith faces defined by the planes 0 = 0 and 8 = ex.. For a = 0 the wedge disappears;for a = v it becomes a half space. For a =2n the wedge becomes a semi-infinite planewhose upper and lower faces are considered as distinct.In physical problems a is necessarily not greater than 2n. However, in themathematical analysis that follows we shall be dealing with functions that makemathematical sense for all real a. Therefore it is of interest to consider wedge anglesa which are virtual with opening a > 2n; their existence is purely mathematical.We shall consider two points P = (r, 8, z) and Po = (r0,80, z0) which both lie withinthe open angle region 0 < 8 < a, and letR(P, Po) = (r» + r2 + (z - z0)2 - 2rr0 cos (8 - 0O))»,R(P,P0) is the shortest distance (in the Euclidean plane) between Po and P; andD(P,P0) is the shortest distance of the broken line from Po to the edge of the wedgeand thence to P.We shall assume that a time harmonic point wave source is situated at Po.A Green's function for diffraction by a rational wedge 1873. Periodic boundary value problem defining a Green's functionWe shall define a periodic Green's function for the reduced wave equation operatorV2 + k2 (where k = w/c, c being the velocity of sound) by means of a periodic boundaryvalue problem posed on a wedge, real or virtual, of open angle 2a.We shall denote this periodic Green's function by Ga(r,6,z; ro,8o,z0; k) orGa{P,P0; k) for an angular opening 2a. Ga(P,P0; k), which is a function of P and Po,is defined uniquely by the following periodic boundary value problem.(a) It is a solution ofo) (0 < r <oo,0 < 6 < 2a, -oo < z < o o ) .(b) I t satisfies the periodic boundary conditionsGa(r,0,z; r0,60,z0; k) = Ga(r,2a,z; r0,60,zo;k),-^(r,O,z;ro,6o,zo;k) = -^(r,2a,,z;ro,6o,zo;k).(c) It satisfies the Sommerfeld radiation conditionlim R(P, Po) (8G;£pP°' *> + ikGa(P, Po; *)) = 0.(d) It satisfies the edge conditionGJP,P0; k) = 0(1) and |gradGa(P,Po; k)\ = o(l/r) as r ^ O .We remark that it follows from the above properties that the Green's function willbe symmetric, i.e.Once the above periodic Green's function has been obtained, it is a simple matter toobtain Green's functions for acoustically hard, soft or mixed hard-soft boundaryvalue problems for wedges of half this open angle.4. Green's functions for hard and soft boundaries for a wedge of open angle a.To simplify notation in this section we shall let the periodic Green's functionGa(r,6,z; ro,6Q,zo; k) be denoted by Ga(6) or equivalently Ga(60). Then it is notdifficult to show, using the fact that Ga(2noc + 6) = Ga(d) where n is any integer,that: the Green's function Gsa(P,P0\ k) for a soft wedge of open angle a is given by/~*8 / D £> . ln\ /^ / /3 \ t~1 / t ) - . ZJ\ .\jr \x , ±n , IC) — Ur \U I — VT_I ^ a — (71 ,the Green's function G*(P,P0; k) for a hard wedge of open angle a is given byIn the same way it can be shown thatG«-ft(P,P0; k) = G2a(d) + G2a(2a-d)-Gis the appropriate Green's function for the wedge which is soft on the face 6 = 0 andhard on the face 6 = a. The corresponding Green's function for a hard boundary onthe face 6 = 0 and a soft boundary on the face 6 = a is given by^•8(P,P0; k) = Gta(0)-Gta{2*-6)-G188 A. D. RAWLINSFor n < a < 2n and n/2 < a ^  2n the functions Ga(6) and G^Jd) respectively,mathematically relate to a virtual wedge angle, which explains the necessity toconsider such wedges.5. A complex integral representation of the periodic Green's functionA complex integral representation for the periodic Green's function defined in §3for S(P,PQ) = — 4:nS(\P—PO\) has been shown by Carslaw and others [3] to be givenGct(r,6,z;ro,do,zo;k) = 7±-f 9(0 , T ^ ^ a i / w "W2^aJ^ cos(£7r/a)-cos((0-0o)7r/a)where 9(0 = ^ p r , «(g) = (r2 + r\ + (z -zof - 2rr0 cos £)*. (2)The square root is defined uniquely by ^(0) =Z)(P,P0), so that &(Q is analyticeverywhere except along branch cuts that extend vertically up and down frombranch points above and below the real axis, respectively. These branch points arelocated at 2nl±ifl, where ft = cosh"1 {(r2 + rl + (z — z0)2)/(2rr0)}, and I is any integer:see Figure 1. In the chosen cut plane — n/2 ^ arg^2(£) ^ n/2. The contour ofintegration ^ in the expression (1) is such that the starting point is given by c1+icoand the termination point is given by c2 + ioo, where — n ^ Cj ^  0, n ^  c2 ^ 2TT; andthe contour lies between the branch points £ = + if} and does not intersect any of theother branch cuts which run parallel to the imaginary axis: see Figure 1.We shall now use the complex integral representation (1) to obtain a differentrepresentation for the special case of a being a rational multiple of n, i.e. a = pn/q,where p and q are positive integers. This new representation will be in terms of pointsource terms and real integrals. From the results we have already given in §4 thismeans we can represent the acoustic field scattered by a soft, hard, or mixedsofi^hard wedge with an open angle a = pn/q in terms of point sources and realintegrals representing the diffracted field.6. Green's function for a rational wedgeIf the angle is a = pn/q, where p and q are positive integers, the expression (1)becomesQ (P p. ft- 2ipn J / ( € ) cos (fy/p) - cos ((d-0o) q/p)' (6)By using the identityg sin (£p/q) = ^ sin (g/p)cos(&/q)-cos((d-d0)p/q) Jl0cosa/p)cos((ff6)/p + 2nm/q) ( '(see Rawlins [10]) we can rewrite (3) in the formpn/g(P,P0;k)= S Iv(d-d0 + 2nmp/q), (5)W-^L™~£X*,fyA Green's function for diffraction by a rational wedge 189f-planeFig. 1. Diagram of the paths of integration in the f-plane. The shaded regions showwhere the integral (6) is uniformly convergent. The wavy lines are the branch cuts forthe functionWe now distort the path of integration %> in the integral (6), so that it takes up thenew path <<?', as shown in Figure 1. The new path of integration # ' lies along regionswhere the integral is uniformly convergent. In distorting the contour <& to take upthe contour # ' the line 0 < Re £ ^  n, Im £ = 0, is crossed, and therefore if any zerosof cos(£/p} — cos (l/r/p), (viz £ = ±(ifr + 2npN) where N is an integer) are capturedthen they will give rise to pole contributions. ThusIPW) = S H[TJ- \rjf + 2npN\] <$(i/r + 2npN)(n + iy)(x-ip)cosh (y/p) — cossin ((n + iy)/p)idycos ((iy + n)/p) — cos (f/p)sin ((x — ip)/p)dxcos ((x — ip)/p—cos190 A. D. RAWLINSwhere C0ioTZ<0H[x] = hUfora;> 1is the Heaviside step function, and where the summation is performed for all integervalues of N which satisfy the inequality — n ^ \jr 4- 2npN ^ n.In the above expression the first integral has had its limits of integration brokenup, because the path traverses branch cuts. It can be shown, using the appropriatelydefined values of @(iy) on the branch cuts, that this whole term vanishes because theintegrand is odd in y. The last integral in the above expression also gives zerocontribution because lim \'&{x — ip)\ ->0 for 0 ^  x ^ rr. Thusp->00NBy using the identity (see Rawlins [10])+_._, 9{n^ r(,( i r r ; y ^ # t . (7)cos(C,/p) — cos (i/r/p)_ cos £— cos xjrwhere an(ifr) = sin((n+l)i/r/p)/{p sin(ijr/p)}, in the expression (7) we haveAT1 f°° ^(n+iy) sin ((n + iy)/p) dy2n p~x J_00 coshn - 0 _„„ cosh y + cos xjrNow by using the fact that &(n + iy) = ^{n — iy) we can write the above expressionasN1 , • (A f°° ${n-iy) cosh- —ap_1(^) s in | —' ' —7T- 1 5 (a Or) sinf^=^) T ^ ~ ^ c o s h ( ^ -71 n-li \ P I Jo COsh^ + COS„ , . / ( P - 1 - » A fTO ^(?r - iy) cosh (y(p - 1 - n)/y) dy\— t*«_iIuTj S i l l I 7T I I \ ; f . I o )V P J Jo cosh^ + cos^- J v 7A Green's function for diffraction by a rational wedge 191Thus finally on substituting (8) into (5) we getGv.lq(p>po'k)= S 2H[n-\6-60 + 2nmp/q + 2npN\]<<0(d-do + 2nmp/q + 2npN)1 "^ sin (d-60 + 2nmp/q) sin (n/p)pn m.o sin((0-<cosh (y/p)^(n-iy)dyi0 coshy + cos (6 —60 + 2ninp/q)"-1 p~2 [sin ((n +1) (6 - 60 + 2nmp/q)/p) sin (nn/p)sin ((6- 60 + 2nmp/q)/p)cosh ((p — n) y/p) ^(n — iy) dyJo cosh ?/ + cos (# — ^ 0 + 2nmp/q)sin (n(0 — 60 + 27Tmp/q)/p) sin (sin ((d-60 + 2nmp/q)/p)cosh ((j>— 1 —ri)y/p)e§{Ti — iy)dy\cosh y + cos (0 — 80 + 2nmp/q) JrJo, (9)where the summation over N is for all integer values of N which can make theargument of the Heaviside step function non-negative, and where $(£,) is given byequation (2).As a specific application we shall give the solution to the problem of acousticdiffraction of a spherical sound wave by a hard-soft half plane.7. Diffraction by a hard-soft half planeIn terms of the Green's function the solution Uh s for the problem of diffraction ofthe spherical wave &(d — d0) by a hard-soft half plane is given byUh,s = Gin(r,0,z,r0,60,zo;k) + Gin(r,6,z,r0, -d0,z0; k)-Gi7,(r, 6, z, r0, An - 60, zo;k)- Gin(r, d, z, ro,4n + 60, zo;k).By using the expression (9) with p = 4, q = 1, it is not difficult to show that-o.) fJo4V o;Jo cosh< + cos (0-6o)dt\/2n Jo cosh< +cos (^  + ^ 0)cosh lt&(n-it)-dt192 A. D. RAWLINSThe first term of the above expression represents the incident field which will onlyexist in the region n ^ \6 — 00\. The second term represents the reflected field from therigid upper face of the half plane, which will only be present for n ^\8 + 60\. The thirdterm represents the reflected wave from the soft lower face of the half plane, whichwill only be present for n ^ \0 + 6o — in\. The remaining integrals represent the fielddiffracted by the edge of the half plane.In conclusion we remark tha t if we replace &(£) in the expression (9) by eifcrcos? or//Q2 ) (&??(£)) we get the appropriate Green's function for a plane or cylindrical wavesource, respectively. If we replace ^(£) by l/!M(Q, we obtain the Green's function forLaplace's equation, which has applications in incompressible fluid mechanics andelectrostatics. Inversion of this problem will give the Green's function for a lensproblem as well. The author hopes to pursue this aspect in a later paper.REFERENCES[1] T. J. L'A BROMWICH. Diffraction of waves by a wedge. Proc. London Math. Soc. (2) 14 (1915),450-463.[2] H. S. CARSLAW. Diffraction of waves by a wedge of any angle. Proc. London Math. Soc. (2) 18(1919), 291-307.[3] H. G. GARNIR. Fonctions de Green de l'operateur metaharmonique pour les problemes deDirichlet et de Neumann poses dans un angle ou un diedre. Bull. Roc. Roy. Sci. Liege 21(1952), 119-140, 207-231.[4] M. I. JACOBSON and J. G. CLARK. Refraction/reflected ray transmission in a divergentchannel. J. Acoust. Soc. Amer. 41 (1967), 167-176.[5] M. I. JACOBSON and J. T. WARFIELD. Surface reflected/bottom-reflected ray transmission ina divergent channel. J. Acoust. Soc. Amer. 43 (1968), 15-24.[6] H. M. MACDONALD. Electric Waves (Cambridge University Press, 1902).[7] H. M. MACDONALD. A class of diffraction problems. Proc. London Math Soc. (2) 14 (1915),410-427.[8] F. OBERBBTTINOER. On the diffraction and reflection of waves and pulses by wedges andcorners. J. Res. Nat. Bur. Standards 61 (1958), 343-365.[9] A. D. RAWLINS. The solution of a mixed boundary value problem in the theory of diffractionby a semi-infinite plane. Proc. Roy. Soc. London Ser. A 346 (1975), 469-484.[10] A. D. RAWLINS. Plane wave diffraction by a rational wedge. Proc. Roy. Soc. London Ser. A 411(1987), 265-283.[11] A. D. RAWLINS. Cylindrical wave diffraction by a rational wedge. Proc. Roy. Soc. London Ser.A 411 (1987), 285-303.[12] A. SOMMERFELD. Mathematische Theorie der Diffraktion. Math. Ann. 47 (1896), 317-374.[13] A. SOMMERFELD. Uber verzweigte Potentiale in Raum. Proc. London Math. Soc. (1) 28 (1897),395-429.[14] F. J. WHIPPLE. Diffraction by a wedge and kindred problems. Proc. London Math. Soc. (2) 16(1917), 94-111.

A class of diffraction problems.

Cylindrical wave diffraction by a rational wedge.

Diffraction by a wedge and kindred problems.

Diffraction of waves by a wedge of any angle.

Diffraction of waves by a wedge.

Electric Waves

Fonctions de Green de l'operateur metaharmonique pour les problemes de Dirichlet et de Neumann poses dans un angle ou un diedre.

Mathematische Theorie der Diffraktion.

On the diffraction and reflection of waves and pulses by wedges and corners.

Plane wave diffraction by a rational wedge.

Refraction/reflected ray transmission in a divergent channel.

Surface reflected/bottom-reflected ray transmission in a divergent channel.

The solution of a mixed boundary value problem in the theory of diffraction by a semi-infinite plane.

Uber verzweigte Potentiale in Raum.

A Green's function for diffraction by a rational wedge

A Green's function for diffraction by a rational wedge

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