Rawlins, A D

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             T R / 1 4 / 8 6     O c t o b e r  1 9 8 6  A  G r e e n ' s  f u n c t i o n  f o r  d i f f r a c t i o n  b y  a  r a t i o n a l  w e d g e .  by  A .  D .  Rawl in s                          -1-  1.  IntroductionIn 1896 A. Sommerfeld obtained an exact closed form solution to the problem of plane wave diffraction by a half plane. He used the method of images on Riemann surfaces corresponding to multivalued solutions of the reduced wave equation, and indicated how this method could be used to obtain exact closed form solution for the problem of diffraction by a wedge. The method of constructing the required many- valued solution of the wave equation was simplified in a subsequent paper, Sommerfeld (1897). Subsequently Carslaw (1919) replaced the image method by a direct con- struction of the solution which yields simpler formulae for various type of wave sources.  These solutions had been obtained independently by Macdonald (1902), (1915) by summing the Fourier series representation of the Green's function. Other methods of obtaining some or all  of the principal results have since been put forward by various authors, including Bromwich (1915), Whipple (1917), Garnir (1952) and Oberhettinger (1958).  The last two authors give extensive bibliographies. Recently Rawlins (1986a,b,c) derived some new representations for the Green's function for plane,  cylindrical  and spherical  wave diffraction by a rational wedge. The results  for the cylindrical  and spherical  wave Green's function were based on the plane wave Green's function.  Here we derive in a different,  more direct ,  and simpler manner the Green's function for a spherical  point source.  The method used here can be applied without difficulty to deal with more general  types of sources,  in particular the plane wave and cylindrical  wave source already considered.  In section 2 we shall  give the geometry of the physical diffraction problems. I n  s e c t i o n  3  w e  s h a l l  d e f i n e  a  p e r i o d i c  G r e e n ' s  f u n c t i o n  a s  t h e  s o l u t i o n  o f  a  per iodic  boundary  va lue  problem for  a  wedge  of  open  angle  .  By  means  of  th i s  α2periodic  Green 's  funct ion we show in sect ion 4 that  one can obtain expressions for  the Green's function for Dirichlet ,  Neumann or mixed Dirichlet/Neumann boundary value problems for a wedge of open angle α .  In section 5 we give a general  complex integral  representation for the periodic Green's function.  This complex integral  representation for the periodic Green's function is  not new.   I t  has appeared in various guises in the papers already cited.   In section 6 we show how this Green's function can be reduced to source terms and real  integrals representing the diffracted waves for the special  case of a rational wedge q/pπ=α  (p and q posit ive integers).   2. Geometry of diffraction problems.We shall be interested in finding solution to diffraction problems in wedge shaped regions. To be specific we shall consider acoustic diffraction by time harmonic waves (with harmonic time variation iwte  assumed, but not shown explicitly, in the rest of the paper).   -2-  We shal l  work in  Eucl idean space of  three dimensions with cyl indrical  polar  coordinates (r , ,z) .  In this space we shall  assume there is  a wedge of open angle α  θwith faces defined by the planes θ  = 0 and  = α .  For α  = 0 the wedge disappears;  θfor  i t  becomes a half  space.  For π=α π=α 2  the wedge becomes a semi-infinite plane whose upper and lower faces are considered as dist inct .  In  physical  problems α  i s  necessar i ly  not  greater  than 2 .   However ,  in  the πmathematical  analysis  that  fol lows we shal l  be  deal ing with funct ions that  make mathematical  sense for  a l l  real  α .  Therefore  i t  i s  of  interest  to  consider  wedge angle  α  which are  vir tual  with opening π>α 2 ,  their  exis tence is  purely mathematical .  We shal l  consider  two points  )z,,r(P θ≡  and )z,,r(P 0000 θ≡  which both l ie  within the open angle  region 0 < θ  <  α ,  and le t   ),(cosrr2)zz(rr)P,P(R 00202200 θ−θ−−++=   .)zz()rr)P,P(D 20200 −++=   R(P,P0)  is  the  shortes t  dis tance ( in  the Eucl idean plane)  between P0  and P;  and D(P,P0) is  the shortest  distance of the broken l ine from P0 to the edge of the wedge and thence to  P .  We shal l  assume that  a  t ime harmonic point  wave source is  s i tuated at  P0 .   3 . Periodic  boundary value problem defining a  Green 's  funct ion.We shal l  def ine a  per iodic  Green 's  funct ion for  the reduced wave equat ion operator  ,  where c  is  the veloci ty  of  sound)  by means of  a  per iodic  c/wk(k22 =+∇boundary value problem posed on a  wedge,  real  or  vir tual ,  of  open angle  2α .  We shal l  denote  this  per iodic  Green 's  funct ion by G  or  )k;z,,r;z,,r( 000 θθα)k;P,P(G 0α  for an angular opening 2α .   which is a function of P and P)k;P,P(G 0α 0 ,  i s  def ined uniquely by the fol lowing per iodic  boundary value problem. (a)  I t  i s  a  solut ion of  ,)P,P()k;P,P(G)k( 0022 δ=α+∇       .z,20,r0 ∞<<∞−α<θ<∞<<  (b)  I t  sa t isf ies  the per iodic  boundary condi t ions    ,)k;z,,r;z,2,r(G)k;z,,r;z,0,r(G 000000 θαα=θα  (c)  I t  sa t isf ies  the Sommerfeld radiat ion condi t ion    .0)k;P,P(ikG)P,P(R)k;P,P(G)P,P(R)P,P(Riml 00000=⎟⎟⎠⎞⎜⎜⎝⎛ α+∂α∂∞→   (d)  I t  sa t isf ies  the edge condi t ion  .0ras)r/1(o|)k;P,P(Ggrad|and),1(0)k;P,P(G 00 →=α=α  -3-  We remark  tha t ,  i f  fo l lows  f rom the  above  p roper t i e s  tha t  the  Green ' s  func t ion  wi l l  be  symmet r ic  i . e .      .k)P;,(PαGk);P(P,αG 00 =  Once  the  above  per iodic  Green ' s  func t ion  has  been  obta ined ,  i t  i s  a  s imple  mat te r  to  obtain Green 's  funct ions for  acoust ical ly  hard,  sof t  or  mixed hard/sof t  boundary value problems for  wedges of  half  this  open angle .   4 .  Green 's  funct ions for  hard and sof t  bounderies  for  a  wedge of  open angle  α .  To s implify  notat ion in  this  sect ion we shal l  le t  the  per iodic  Green 's  funct ion k);z,θ,rz;θ,(r,αG 000 be denoted by  or  equivalent ly  .  Then i t  i s  not  )(θαG )(θαG 0diff icul t  to  show, (using the fact  that  )(θαGθ)α(2nαG =+  where n  is  any integer)   that :  The  Green ' s  func t ion   fo r  a  so f t  wedge  o f  open  ang le  α  i s  g iven  by  ;k)(P,PsαG 0;θ)α2(αG)(θαGk);P,(PsαG 0 −+=  t he  Green ' s  func t ion for  a  ha rd  wedge  o f  open  ang le  α  i s  g iven  by  ;k)(P,PhαG 0.θ)(2ααG)(θαG)k;P,(PhαG 0 −+=  In  the  same  way  i t  can  be  shown tha t  θ)α4(α2Gθ)α2(α2Gθ)α2(α2G)(θα2G;k)P,(Ps/hαG −−+−−+=  i s  the  appropr ia te  Green ' s  func t ion  for  the  wedge  which  i s  sof t  on  the  face  θ  =  0 ,  and hard on the face θ  = α .  The corresponding Green's function for a hard boundary on the face θ  = 0 and a soft  boundary on the face θ  = α  is  given by .θ)(4ααGθ)(2ααGθ)(2ααG(θθαGk);P(P,h/sαG 2222 −++−−−=  For    and    the  func t ions   and   r espec t ive ly ,  2παπ ≤< 2παπ/2 ≤< )(θαG )(θ2αGmathemat ica l ly  re la te  to  a  v i r tua l  wedge  ang le ,  which  exp la ins  the  necess i ty  to  cons ider  such  wedges .   5 .  A  complex  in tegra l  r epresen ta t ion  o f  the  pe r iod ic  Green ' s  func t ion ,A  complex  in tegra l  r epresenta t ion  for  the  per iodic  Green ' s  func t ion  def ined     in section 3 for |),PP(|δ4π)Pδ(P, 00 −−=  has been shown by Carslaw and others (1919) to be given by  ∫ −−ζα= ,π/α)θ)cos(( θcos( ζos(ζ dζα)sin( ζin)(GCi21)k;z,θ,r;zθ,,r(αG 000  (1 )  where                          .)cosζrr2z)(zr(r)R(ζ,)R(ζζ)ikR(e)G(ζ 2102202 −−++=−=  (2 )  -4-  The square root is defined uniquely by R(0)=D(P,P0),  so that R( ζ ) is analytic everywhere except along branch cuts that extend vertically up and down from branch points above and below the real axis,  respectively. These branch points are located  at  ,  where β  = ,  and  is any integer,  see iβπ2 ±l )}/(2rr)z(zr{(r 0202021 −++−cosh lf ig .1 .  In  the chosen cut  plane 2π/)R(ζarg2π/ ≤≤− .  The contour  of  integrat ion C in the expression (1) is such that the starting point is given by , and the ∞+c1 it e rmina t ion  poin t  i s  g iven  by  ∞+ ic2 ,  where  ;π2cπ,0cπ 21 ≤≤≤≤−  and  thecontour lies between the branch points iβζ ±=  and does not intersect any of the   other branch cuts which run parallel to the imaginary axis,  see fig.1. We shall  now use the complex integral representation (1) to obtain a different r epresen ta t ion  fo r  the  spec ia l  case  o f  α  be ing  a  ra t iona l  mul t ip le  o f ,πi .e.   (p and q positive integers).  This new representation will  be inq/pπα =terms of point source terms and real integrals.  From the results we have already given in section 4 this means we can represent the acoustic field scattered by a soft,  hard, or mixed soft/hard wedge with an open angle q/pπ=α  in terms of point sources and real integrals representing the diffracted field. 6. Green's function for a rational wedge.If the angle q/pπ=α ,  where p and q are positive integers the expression(1) becomes   ))q/pθ((θcosq/p)(cosdζq/p)(sinq)G(ζipπ21;k)(P,PqpπG00 −−= ∫   (3) By using the identity  ⎟⎟⎠⎞⎜⎜⎝⎛ +−−−==−−ζ ∑qπm2pθθcos/p)(ζcosp)/(ζsin1q0m)p/q)θ((θcos/q)(ζcosp/q)(sinq00  (4) see Rawlins (1986), we can rewrite (3) in the form     πmp/q),2θ(θI1q0m;k)(P,PqpπG 0p0 +−−== ∑     (5) where   .p)/(ψcosp)/(ζcosdζp)/(ζsinp1)G(ζcπi22)(ψIp −= ∫    (6)  We now distort the path of integration C, in the integral (6),  so that i ttakes  up  the  new path  C',  as  shown in fig.1.  The new path of  integration C1 lies -5- along regions where the integral is uniformly convergent.  In distorting thecontour C to take up the contour C' the line ,0ζIm,πζRe0 =≤≤  is crossed,and therefore if any zeros of cos pN)2(ψζψ/pζ/p π+±=− viz(),cos()(  where N isan integer) are captured then they will give rise to pole contributions.  Diagram of the paths of integration in the ζ -plane. The shaded regionsshow where the integral (6) is uniformly convergent. The squiggly linesare the branch cuts for the function G( ζ ). fig.1.  Thus   pN)π2G(ψ]pNπ2ψ[πHNI(ψ( ++−= ∑     /p)(ψcos(y/p)coshdyi(y/p)sinhpiG(iy)ββββiπ21−⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧ −−∞+∞++ ∫∫∫     ⎟⎟⎠⎞⎜⎜⎝⎛−⎟⎟⎠⎞⎜⎜⎝⎛ +++∞−∞+ ∫pψcospπyicosdyiy)/p)i((πsinp1y)iG(πiπ21     ⎟⎟⎠⎞⎜⎜⎝⎛−⎟⎟⎠⎞⎜⎜⎝⎛ −−−+ ∫∞→pψcospρixcosdx)ρ)/pi((xsinp1ρ)iG(xπ0iπ21limρ where ⎪⎪⎭⎪⎪⎬⎫⎪⎪⎩⎪⎪⎨⎧>=<=1xfor11xfor0xfor0H[x]21  is  the  Heavis ide s tep funct ion,  and where the symbol   means summation is performed for all integer values of N ∑Nwhich satisfy the inequality πNpπ2ψπ <+<− . -6-  In the above expression the first integral has had its limits of integrationbroken up, because the path traverses branch cuts. It can be shown, using the appropriately defined values of G (iy) on the branch cuts that this whole termvanishes because the integrand is odd in y. The last integral in the above expression also gives zero contribution because πx0,0ρ)iG(xlimρ≤≤→−∞→ . Thus    N)pπ2G(ψ]Npπ2ψ[πHNp(ψI ++−= ∑     .pψcospπiycosdyiiy)/p)((πsinp1iy)G(πiπ22⎟⎟⎠⎞⎜⎜⎝⎛−⎟⎟⎠⎞⎜⎜⎝⎛ +++∞−∞+ ∫    (7) By using the identity (see Rawlins 1986)   ψcosζcosn)/p)}2(pζ(sinn)/p)(pζ(sin){(ψna2p0np)/(ζsin)(ψ1pa/p)(ψcos/p)cps(ζ/p)(ζsinp1−−−−−−=+−=−∑ where in the expression (7) we have /p)},(ψsinp{//p))ψ1((nsin)(ψna += πpN)2]G(ψπpN2ψ[πHN)(ψpI ++−= ∑    ψcosycoshdyiy)/p)π((siniy)πG()ψ(aπ211p +++∞−∞− ∫−   .dypn)2(piy)(πsinpn)(piy)(πsinψcosycoshiy)πG((ψnapnπ21 20 ⎭⎬⎫⎩⎨⎧⎟⎟⎠⎞⎜⎜⎝⎛ −−+−⎟⎟⎠⎞⎜⎜⎝⎛ −+++∞−∞−=− ∫∑  Now by using the fact that iy)πG(iy)πG( −=+  we can write the above expression as  πpN)2]G(ψπpN2ψH[πN)ψ(pI ++−= ∑   ψcosycosh(y/p)dycoshiy)πG(0pπsin)ψ(paπ11 +−∞⎟⎟⎠⎞⎜⎜⎝⎛−− ∫   ⎪⎩⎪⎨⎧+−−∞⎟⎟⎠⎞⎜⎜⎝⎛ −−=− ∫∑ ψcoscoshy n)/p)dypiy)cosh(y(πG(0p n)(pπ)sinψ(a2p 1nπ1 n   ⎪⎭⎪⎬⎫+−−−∞⎟⎟⎠⎞⎜⎜⎝⎛ −−−− ∫ ψcosycosh n)/p)dy1(y(pcoshiy)πG(0p n)1(pπsin)ψ(1na  (8) So that  f inal ly  on subst i tut ing (8)  into  (5)  we get      -7-   )pN2qmp2(G]pN2qmp2[HNqm)k;P,P(qpG 00010π+π+θ−θπ+π+θ−θ−π−==π ∑∑   /q)mpπ2θ(θcosycoshdyiy)π(y/p)G(coshqm /q)/p)mpπ2θ((θsin/p)π(sin/q)mpπ2θ(θsinπP100010 0 +−+−− ∞+−+−− ∑= ∫   ∑ ∑−=− ∞+θ−θ+−−⎩⎨⎧+θ−θ+θ−θ+π− − ∫1021 0qmpn )q/mpπ2cos(ycoshdy)iyπ(G)p/y)npcosh()p/)q/mpπ2sin(()p/πsin()q/mpπ2)(1nsin((P1000   ∫∞⎭⎬⎫+θ−θ+−−−+θ−θ++θ−θ−0 000 ,)q/mpπ2cos(ycoshdy)iyπ(G)p/y)n1pcosh()p/)q/mpπ2sin(()p/π)1nsin(()q/mpπ2(nsin(    (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 G(ζ ) is given by equation (2).  In conclusion we remark that if we replace G( ζ ) in the expression (9) by ))(k(Horcosikre )2(0 ζζ R  we get the appropriate Green's function for a plane orcylindrical wave source, respectively. If we replace G( ζ ) by 1/ (R ζ ) we obtainthe Green's function for Laplace's equation, which has applications in incompressiblefluid mechanics, and electrostatics. Inversion of this problem will give theGreen's function for a len's problem as well. The author hopes to pursue thisaspect in a later paper.                       REFERENCES Bromwich,  T,  J,  1'a.  1915.  Diffraction of waves by a wedge.  Proc. Lon. Math. Soc.(2) 14, 450-463. Carslaw,  H.S.  1919.  Diffraction of waves by a wedge of any angle.  Proc. Lon. Math.  Soc.(2) 18, 291-307. Garnir,  H.G.  1952.  Fonctions de Green de l'opérateur métaharmonique pour les  problémes de Dirichlet et de Neumann posés dan an angle ou un diédre.  Bull. Soc. Roy. Sci. Liége 21, 119-140, 207-231. Macdonald,  H.M.  1902.  Electric Waves.  Cambridge: Cambridge University Press. Macdonald,  H.M.  1915.  A class of diffraction problems.  Proc. Lon. Math. Soc.(2) 14,  410-427. Rawlins,  A.D.  1986a.  Plane Wave diffraction by a rational wedge.  Proc. R. Soc.Lon.  (To appear). Rawlins,  A.D.  1986b.  Cylindrical wave diffraction by a rational wedge.  Proc.R.Soc.Lon.  (To appear) Rawlins,  A,D.  1986c.  Spherical wave diffraction by a rational wedge. Sommerfeld,  A.  1896.  Mathematische Theorie der Diffraktion. Math. Ann. 47,  317-374. Sommerfeld, A.  1897.  Uber verzweigte Potentiale in Raum.  Proc. Lon. Math.  Soc. (1) 28, 395-429. Whipple,  F.J.  1917.  Diffraction by a wedge and kindred problems.  Proc. Lon.  Math. Soc. (2) 16, 94-111.  

A class of diffraction problems.

Cylindrical wave diffraction by a rational wedge. Proc.R.Soc.Lon. (To appear)

Diffraction by a wedge and kindred problems.

Diffraction of waves by a wedge of any angle.

Diffraction of waves by a wedge.

Electric Waves. Cambridge:

Fonctions de Green de l'opérateur métaharmonique pour les problémes de Dirichlet et de Neumann posés dan an angle ou un diédre.

Mathematische Theorie der Diffraktion.

Plane Wave diffraction by a rational

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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