We consider the diffraction of the dominant plane wave mode which propagates out of the end of a semi-infinite waveguide. This waveguide is made up of a half plane with a Dirichlet boundary condition and a half plane with a Neumann boundary condition. This semi-infinite waveguide is symmetrically located inside another infinite waveguide one of whose infinite plates has a Dirichlet and the other a Neumann boundary condition. A closed form solution of the resulting matrix Weiner-Hopf equation is obtained

Rawlins, A D

English

Brunel University Research Archive

                     TR/09/94                                                    June 1994 A TRIFURCATED WAVEGUIDE PROBLEM II   A.D. Rawlins                                                                    w9253398        A TRIFURCATED WAVEGUIDE PROBLEM II   by    A.D. Rawlins  Department of Mathematics and Statistics  Brunel University      Summary We consider the diffraction of the dominant plane wave mode which propagates out of the end of a semi-infinite waveguide. This waveguide is made up of a half plane with a Dirichlet boundary condition and a half plane with a Neumann boundary    condition. This semi-infinite waveguide is symmetrically located inside another infinite waveguide one of whose infinite plates has a Dirichlet and the other a Neumann    boundary condition. A closed form solution of the resulting matrix Weiner-Hopf     equation is obtained.       1.    Introduction  In a previous publication, Rawlins [5], a mathematical model to predict noise in             an exhaust system was proposed. This model resulted in a matrix Wiener-Hopf problem. Fortunately the matrix could be explicitly factorized. In the present work we consider another possible mathematical model of an exhaust system. This again results in a         matrix Wiener-Hopf problem which is solved exactly. For the justification of the mathematical model see [5].  The trifurcated waveguide problem under consideration is shown in fig. 1. The plates which make up the waveguide have Neumann or Dirichlet boundary conditions    on them. The plates are symmetrically positioned relative to the centre line of the system. A fundamental mode is assumed to propagate out of the month of the semi-infinite waveguide. Although some related trifucated waveguide problems can be solved as a result of their symmetry by a direct application of the normal scalar Wiener-Hopf technique, see Rawlins [5], the present problem results in a non-trivial matrix Wiener-Hopf problem. Fortunately we are able to solve this matrix Wiener-Hopf problem explicitly.  To be specific we shall consider an acoustic problem and in section 2 we shall formulate the mathematical problem that we intend to solve. In section 3 we shall solve the problem formulated in section 2. The solution will be expressed as complex contour integrals. In section 4 we shall analytically convert these integrals to infinite series of modes which propagate in the waveguide region. In order not to disrupt the flow of the solution in the main text an appendix has been included at the end of the paper. This appendix includes analytical details required in the main text.   2  Formulation of the boundary value problem  We shall consider the acoustic diffraction of a plane wave mode propagating out of the open end of a semi-infinite duct; this semi-infinite duct consists of one plate which is rigid and the other which is soft. The semi-infinite duct is situated symmetrically between two infinite plates one of which is soft and the other rigid. The geometry of the problem is shown in fig. 1. fig. l The sound source field, which is located at y = y0, x = x0 (x0 < 0, -a < y0 < a) and propagates modes along the semi-infinite duct. We shall introduce a scalar potential function (x,y,t) which defines the acoustic pressure and velocity by φ ,t/p 0 ∂∂−= φρ and u = gradφ  respectively, where ρ0 is the density of the undisturbed medium. The incident sound field is assumed to have time harmonic variation e-iωt, where the wave number k = ω/c, and c is the speed of sound. In the remainder of this paper  we shall drop the time dependence, it being tacitly understood that φ (x,y,t) = e-iωtφ (x,y). To this end we require a representation for the solution φ (x,y) of the two-dimensional Helmholtz equation      (1) 0k 2yyxx =φ+φ+φin the trifurcated duct system which satisfies the following boundary conditions (see fig.l) 3  ;x,by,0;0x,ay,0;0x,by,0;x,by,0yy∞<<∞−−==φ<<∞−−==φ<<∞−==φ∞<<∞−==φ   (2a,b,c,d) where i t  i s  assumed that  b  > a .   To these boundary conditions we add those conditions at infinity which are relevant to the nature of the propagating modes which various duct regions can sustain. From Rawlins [5] it is not difficult to show that For x → , -a ≤ y ≤ a ∞−   )e(0a4)ay(sineRa4)ay(cosy)y,x( 211 ixi2xi α−α−α +⎟⎠⎞⎜⎝⎛ π++⎟⎠⎞⎜⎝⎛ π+=φ   (3) where   ,....2,1n,a4/)1n2((k,)a16/9(k,)a16/(k 221n222222221 =π−−=απ−=απ−=α −  If we restrict π/4 < ka < 3π/4 then  α1> 0 and  ,k)a4/3(i 222 −π=α  so that Imαn > 0, Reαn = 0, n > 1. Thus in the semi-infinite duct region -∞< x < 0, -a ≤ y ≤ a can only sustain the lowest incident and reflected mode. For x → , -b ≤ y ≤ b ∞+),e(0b4)by(coseT)y,x( xixi 21 αα +⎟⎠⎞⎜⎝⎛ π+=φ                             (4) where ....,2,1n,))b4/)1n2(kˆ,))b4/3(kˆ,))b4/(kˆ 221n22222221 =π−−=απ−=απ−=α − For x  → , a ≤ y ≤ b ∞−),e(0)ab()ay(sineT)y,x( x2^ix1^i3α+⎟⎟⎠⎞⎜⎜⎝⎛−π−=φ α    (5) where  4  ,)))ab/(πn(k(~,)))ab/(π2(k(~,)))ab/((k(~ 221n2222221 −−=−−=−π−= −ααα  n = 1,2,... . For x → , -b ≤ y ≤ -a ∞−    ),e(0)ab()ay(coseTeT)y,x( xxi3ikx321 α−α− +⎟⎟⎠⎞⎜⎜⎝⎛−π++=φ   (6) Where  ,)))ab/(n(k(~,)))ab/(2(k(~,)))ab/((k(~,k~ 22n2222210 −π−=α−π−=α−π−=α=α   n = 0,1,2,... . Finally, in order to ensure uniqueness of the solution to the problem we need to specify the "edge condition" at the end of the semi-infinite planes, that is     .0xas)x(0)a,x(and)x(0)a,x( 2121y →=±φ=±φ −   (7) Solution of the boundary value problem For analytic convenience we shall assume that k = Rek + iImk(Rek > Imk ≥ 0). A suitable representation for the total field φ (x,y) in all space by,x ≤∞<<∞−  which satisfies (1) and (2) is given by: );x,ayb(,d)()ab(κsin)yb(κcosei21)y,x( 1iixi ∞<<∞−−≤≤−ααφ−κ+π−=φ−τ+∞τ+∞−α∫         (8) ααφ+κκ+−κκκπ−=φ−τ+∞τ+∞−α∫ d)}()ay(cos)ay({sina2cosei21)y,x( 2iixi )9();x,aya(,a4)ay(cose xi 1 ∞<<∞−≤≤⎥⎦⎤⎢⎣⎡ π++ α  )10().,bya(,d)()ab(sin)yb(sini21)y,x(ii 2∞<∞−≤≤ααφ−κκ −π=φ ∫τ+∞τ+∞−  In the expressions (8) to (10), κ  = (k2 - α2)1/2 and the branch cuts are taken to be from k to i∞  and from –k to –i∞ .  The cut sheet on which we shall work is defined by  5  0 ≤  argk ≤ , see fig. 2. The real parameter T is restricted by requiring that the            πasymptotic behaviour (3), (4) and (5) be achieved. This necessitates that the contour of integration in (8) to (10) lies in the strip:       max{-Imk, -Im -Im ,} < ,1α 1~α τ  < min{Imk, Im ,} 1~αfig.2 It can be shown, see Rawlins [5], that the inequality (11) is certainly satisfied by -Imk < T < Imk. No singularities of the integrands of (8) to (10) lie within the strip -Imk  < T < Imk. The unknown functions (−φ1 α ) and ( ) are functions which are −φ2 αanalytic and regular in the region Im  < Imk. We must now ensure that two remaining αboundary conditions are satisfied; namely that the field and its normal derivative are continuous across y = -a and y = a, (x>0) respectively, that is      0x),a,x(y)a,x(y),a,x()a,x( >∂φ∂=∂φ∂−φ=−φ +−+− .                        (11) By substituting (8) to (10) into (11) we get )0x(,0d1ka2cos)(ka2cos)ab(sin)()ab(cosei21121xiii>=α⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧ α−α+αφ+−κκαφ+κπ−ατ+∞τ+∞−∫   (12) ),0x(,0dka2cos)ab(sin)()ab(cosa2cos)(ei211121xiii>=α⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧ α−α−κ−καφ+κκ+καφπ−−ατ+∞τ+∞−∫   (13) where = . 1κ )a4(/π6   A solution of (12) and (13) is given by              )(a2cos)ab(sin)()ab(cosa2cos)(11121 αφ=α−ακ−κ−καφ+κκ+καφ +−−    (14) )(1a2cos)(a2cos)ab(sin)()ab(cos2121 αφ=α−α+καφ+κ−κκαφ+κ +−− ,   (15)  where are functions that are analytic and regular in the region Im  > -lmk.   )(),( 21 αφαφ ++ αThe equations (14) and (15) constitute a coupled system of Wiener-Hopf equations which we can write in matrix form as:  )/(D)()(K)( 1α−α+αψα=αψ −+     (16) where ;1D,)()()( 121 ⎟⎟⎠⎞⎜⎜⎝⎛ κ−=⎟⎟⎠⎞⎜⎜⎝⎛αφαφ=αψ ±±±     (17)  .1)ab(sin)ab(cos)ab(sin)ab(cos1a2cos1)(K⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛−κκ+κ−κ+κκκ=α     (18) The usual Wiener-Hopf technique can be applied in a straightforward manner if the                system (16) can be uncoupled into separate Wiener-Hopf equations. However the                   present problem requires that the matrix function K )(α  be factorized as a product of two matrices, one analytic in Im  > -Imk and the other analytic in Ima < Imk. This is a                    αnontrivial procedure and it is not always obvious how to achieve it explicitly, at least in      the classical sense, Noble [4] (where the elements of the matrix factors have algebraic behaviour at infinity). The matrix K(α) given by (18) is of a special form that can be factorized by a method of Daniele [1], and see Rawlins [6]. Without going into the             details we obtain           )(K)(K)(K αα=α −+       (19) with [ ] [ ][ ] [ ] ⎟⎟⎠⎞⎜⎜⎝⎛ακακκακκακα=α±±±±±±)(t2/1cosh)(t2/1sinh1)(t2/1sinh)(t2/1cosh)(K)(K    (20) where )ab(sina2cosd2cos)(K)(KKdet)(K 2 −κκκ−=αα==α −+    (21)        ).(t)(t4/bcos()4/acos(4/bcos()4/acos(enk1)(tiα+α=⎟⎟⎠⎞⎜⎜⎝⎛π+κπ−κπ−κπ+κ=α −+Dl   (22) Explicit expressions for  and ±K )(α ±t )(α  are derived in the appendix. They are: [ ][ ][ ][ ][ ]22n2n23n423n421n4221n423n423n421n422n421n2)u1~()1u~()u1()1u()u1()1u()u1ˆ()1uˆ()u1()1uˆ()ab(akbn21)ab(i)(K−−−−−⎟⎟⎠⎞⎜⎜⎝⎛−π−=α −−−−−−−−∞=− ∏      )(K)(K α−=α −+      (23) )i(n1)u11u)(u11u)(u1ˆ1uˆ)(u1ˆ1uˆ()u11u)(u11u)((u1ˆ1uˆ)(u1ˆ1uˆ(n1)(t3n43n41n41n43n43n41n41n43n43n41n41n43n43n41n41n41nκκ−α−κ−⎥⎦⎤⎢⎣⎡−++−+−−+κ−=α −−−−−−−−−−−−−−−−∞=− ∏ ll                              )(t)(t α−=α −+      (24) .))}ab/(n(k(,)ˆk(1ˆ,)ˆk(uˆ,)))b4/(n(k(,)ˆk(1ˆ,)ˆk(uˆ;)))a4/(n(k(,)k(1,)k(u,)k(1,)k(uwhere2/1222/1nn2/1nn2/1222/1nn2/1nn2/122n2/1nn2/1nn2/12/1−π−=α+=α+=π−=α+=α+=π=αα−=α+=α−=α+= The Wiener-Hopf equation (16) may now be solved by using (19) to rewrite (16)             as )/(D)](K)(K[)()(K)/(D)(K)(K 11111111 α−αα−α−αψα=α−αα+αψ −+−++−+−+−− .(25) The left-hand side of the above equation is analytic in Imα  < Imk, the right-hand side                  is analytic in Im > -Imk. Consequently each side is equal to an entire function E(α α ),  8  that is, a matrix with polynomial entries.  Hence letting D)(Kccc 1121 α=⎟⎟⎠⎞⎜⎜⎝⎛= −+      (26) we have  .              (27) )](E)/(C)[(K)( 11 α+α−α−α=αψ −−−The determination of E(α) depends on the asymptotic behaviour of various functions.              In the appendix it is shown that as  ∞→α  in Imα < Imk:                     )(0ni)(t),(02)(K 11 −−−− α+αα−=αα+=α l .              (28) Hence ⎟⎟⎠⎞⎜⎜⎝⎛+αα+α−α+α+α=α −− )1(0)(0i)(0i)1(021)(K 2/112/12/12/1.   (29) The asymptotic behaviour of (α) may be found from the edge conditions (7); it is not .ψdifficult to show that ,)(0)(),(0)( 2/212/11−−−− α=αφα=αφ                               (30) as  ∞→α  in Imα < Imk: It is now not difficult to show that by substituting (29), and (30) into the expression (27)         that the entire matrix E(α) is given by E(α)             (31) ⎟⎟⎠⎞⎜⎜⎝⎛−=01ic2in order that the asymptotic behaviour (30) be obtained from (27). The solution can be written after a bit of algebra in a very simple form by introducing         the following functions  F(u,l) = ,luul1 +                                                       (32) 9  ,)(Kk2e)iul(e)u,l(P)(2/14/iα−=−α−κπ  .)ullu)(ullu())u1()1u)((u11u)(ullu()ab(akbπn22ab13n4^3n4^1n4^1n4^2n2n3n43n41n41n41n2/1−−−−−−−−∞= −+−−+⎟⎟⎠⎞⎜⎜⎝⎛−⎟⎠⎞⎜⎝⎛ −= ∏  (33) Thus one obtains: ⎟⎟⎠⎞⎜⎜⎝⎛κ−−−−−+α−α=αΨ−/)]l,u(F)u,l(P)l,u(F)u,l(P[)l,u(F)u,l(P)l,u(F)u,l(P)()1,u(P)(111 .     (34) It is not difficult to show that P(l,u), P(l,-u) have neither zeros nor poles in Imα < imk. They do have branch points at α = -k, but the particular combinations in (34) are invariant under the transformation u → -u which means that ψ-(α) is indeed analytic in Imα < Imk. We now substitute the expression (34) into the integral representaitons (8) to (10) and obtain the field representations for the different regions as: Region A (-b < y < -a, x < 0)   ∫ τ+∞ τ+∞− α α−α α−−+−κκ +κπ−=φii1xi11 .)(d)}1,u(F)u,l(P)l,u(F)u,l(P{)ab(sin)yb(cosei2)l,u(P)y,x( (35)  Region B (-a < y < a, x < 0)  ∫ τ+∞ τ+∞−αα +κ−−π+π+=φiixi11x1i )l,u(F)u,l(P)ay(cos)ay(k{[(sinka2cosei2)l,u(P]a4/)aycos[(e)y,x( .)(d)/)]l,u(F)u,l(P))ay(cos)ay((sin1α−αακ−+κ+−κ+   (36) Region C (a < y < b, x < 0)       ∫ τ+∞ τ+∞− α α−α ακ−−−−κ −κπ=φii1xi11)(d}/)]l,u(F)u,l(P)l,u(F)u,l(P{[)ab(sin)yb(sinei2)l,u(P)y,x( . (37)  10  Region D (-b < y < b, x > 0)  ∫ τ+∞ τ+∞− α −−π−π+−π=φiixi11 )1,u(P/)1,u(F)4/bκcos()4/aκ{[cos(κb2cos)yb(κsinκei2)l,u(P)y,x(   .)(d}κ/)]l,u(P/)l,u(F)4/bκcos()4/aκcos(1α−ααπ+π−−         (38) In the expressions (35) to (38) the pole α = α1 lies above the contour of integration, that is T  < Imα1. We also note that the term in the curley bracket { } of (35) to (37) has no singularities in Imα > -Imk. Thus the only singularities in Imα < Imk of the integrands of (35) and (37) occur at the zeros of sin (b-a) = 0, that is α = κ κ ),))ab/(n(k( 22n~ −−= πα−  n = 1,2,... . The only singularities of the integrand of (36) occur at the zeros of cos2ka = 0 that is α = -α2n-1 = √  are the only singularities in Imα < Imk. ))a4/)1n2((k( 22 π−−The only singularities in Imα > -Imk of the integrand (38) occur at the zeros of cos2kb = 0, that is α = √  n = 1,2,..., and also the pole α = α=−1n2^α ,)))b4/()1n2(k( 22 π−− 1.  4. Mode field representation An application of Cauchy's residue theorem to the complex integrals (35) to (38) then gives the field in the various regions as a sum of waveguide modes. Region A (-b < y < -a, x < 0) )k)(ab(keu2)0,k2(P)l,u(P)y,x(1111 ikxα+−−=φ− .)}u,l(F)l,u(P)u,l(F)l,u(P){yb(κcos)()(e)ab()l,u(Pn~n~n~n~n~n~n~n~~n1n n~1n~nxn~i11 −++αα+α−−− ∑∞=α−    (39) 11  Region B (-a < y < a, x < 0) ]a4/π)ay[(cose)y,x( x1i +=φ α    ∑∞=α−+−−α+αα−+1nmmmmmm1mmnxmi11 )u,l(F)l,u(P))ay(κcos)ay(κ{[(sin)()(ea2)l,u(P       )]},u,l(F)l,u(P))ay(κcos)ay(κ(sin mmmmmm −−++−+ (m=2n-1). (40) Region C (a < y < b, x < 0)   .)}l,u(F)u,l(P)u,l(F)l,u(P{)()yb(κsin)(e)ab()l,u(P()y,x( n~n~n~1n~nn~n~n~n~n~1n~n~nxn~i11 ∑∞=α−−−−αα+α−−−−=φ (41) Region D (-b ≤ y ≤ b, x > 0) ),4/)(aκ(cos)l)(,u(P)l)(,u(F)()yb(κsinκe)(b2)l,u(P)y,x( 1nm^1n m^1nm^m^1nm^1m^m^m^m^xm^i]2n[11 π−+−−−−−−=φ +∞= ++α∑ααα(m=2n-1) (42) where [x] denotes the largest integer ≤ x.  Conclusion We have solved a new diffraction problem in closed form by using matrix factorization. The present approach would not be substantially changed if we included a flow in the region -a < y < a. This would be a possible model for an exhaust system, where exhaust gases flow out of the system. It is hoped to present this solution in the near future.  We also remark that if we considered the same duct geometry and boundary conditions, but with the incident wave propagating in the region x > 0, | y | < b from z = ∞, see fig. 3, then we would obtain a similar matrix Wiener-Hopf problem which can be solved exactly. As a special case of this latter problem by letting a → 0 one would obtain the solution to the problem considered by Lüneburg and Hurd [3]. 12   fig. 3   References [1] Daniele, V.G., On the factorization of Wiener-Hopf matrices in problems solvable with Hurd's method. IEEE Trans. on Antennas and propagation, AP-26, 614-616 (1978). [2] Kranzer and Radlow, J., An asymptotic method for solving perturbed Wiener-Hopf problems. J. Math. Mech. 14, 41-60 (1965). [3] Luneburgh, E. and Kurd, R.A., Two waveguide bifurcation problems. Z. Angew. Math. Mech. (ZAMM) 65, 551-559 (1985). [4] Noble, B. The Wiener-Hopf Technique. Pergamon, London, 1958. [5] Rawlins, A.D., A trifurcated waveguide problem. Proc. R. Soc. Lond. (submitted). [6] Rawlins, A.D., A note on the factorization of matrices occurring in Wiener-Hopf problems. IEEE Trans, on Antennas and Propagation, AP-28, 933-934 (1980). 13  Appendix Here we shall carry out the explicit factorization of (23) and (24), and give their asymptotic behaviour as | α | → ∞. For K(α) given by (23) we can write it in a more convenient form )ab(κsin)4/aκcos()4/aκcos()4/bκcos()4/bκcos()(K 2 −π−π+π−π+−=α  and if we let )a,(K)a,(K)4/aκ(cos)4/aκcos()a,(K αα=π−π+=α −+      , )ab(,(S))ab(,(S)ab(κsin)ab(,(S 2 −−α=−=−α −+ α)ab,(S)a,(K)b,(Ki)(K −= ±±±± αααα  Now from Rawlins [5] it has been shown that   ∏∞=−−−−+− +−⎟⎟⎠⎞⎜⎜⎝⎛π−=−= 1n 1n41n41n41n42)ullu()ullu(k)1n2(a)a,(K)a,(K αα  )ullu()ullu( 3n43n43n43n4 −−−− −−× ; And that   in Imα > -Imk and Imα  < Imk respectively. ∞→=α± ||as)e(0)a,(K ||a ααFrom the well known product for the sine function ∏∞= ⎟⎟⎠⎞⎜⎜⎝⎛π−= 1n 222nz1zzsin  it is not difficult to show that .))ul()lu((kn2)ab(1)ab())ab(,(S))ab(,(S 22n~2n~21n2 −⎟⎠⎞⎜⎝⎛π−−=−−=− ∏∞=+− αα   Thus 14  ∏∞= −−−−−−−−−−−−−⎟⎟⎠⎞⎜⎜⎝⎛−π−= 1n 22n~2n~23n423n421n421n423n4^23n4^21n4^21n4^22))ul()lu)(()ul()u)(()ul()u(())ul()lu)(()ul()lu(()ab(abkn21)ab(i)(K α .  Now since K(-α) = K(α) then by Radlow and Kranzer [2] we have since K(α) = 4+0(1) that K±(α) = 2 + o(l) . The factorization  of ⎥⎦⎤⎢⎣⎡−π+π+π−π=π)aκ4/cos()bκ/cos()aκ4/cos()bκ4/cos(en1κ1)(tiα  follows directly from Rawlins [5], as ⎢⎢⎢⎣⎡+−−+−++−−+⎟⎠⎞⎜⎝⎛ −α−−=α ∏∞= −−−−−−−−−−−−−−−−−1n3n43n41n41n43n4^3n4^1n4^1n4^3n43n41n41n43n4^3n4^1n4^1n4^)ullu)(ullu)(ullu()ullu()ullu)(ullu)(ullu()ullu(n1κ1kikinκ1)(twhere t+ (α) = t -(-α) and ,)k(l,)k(l,)k(u)k(u 21n2121n21αααα −=−=+=+=  2122n^n^n^n^n^2122n )))b4/(n(k(),k(l),k(u,)))a4/n(k( π−=−=+=π−=α ααα . The behaviour of t -(α) as α → ∞ in Imα < Imk can be shown to be )(0)(ni)(t 1−− +α−= ααα l .        

A note on the factorization of matrices occurring in Wiener-Hopf problems.

A trifurcated waveguide problem.

An asymptotic method for solving perturbed WienerHopf problems.

On the factorization of Wiener-Hopf matrices in problems solvable with Hurd's method.

The Wiener-Hopf Technique.

Two waveguide bifurcation problems.

A trifurcated waveguide problem II

A trifurcated waveguide problem II

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