— DINSRDC-77-@1/2 : | Dee Gerar ea ea re E pape = ioe we G iene EDN Som neprere ni cnterns S20. 77 OVE: c hTT2EOO TIEO O M0 1OHM, 18 DOCUMENT LIBRARY Woods Hole Oceanographic Institution SPA = Serco oeiaons 9 vedere seeds sin tt lar ns wre rN Ss eros ag arg 01pm | (18) DWE RDC UNCLASSIFIED ——E——————————————— ee SECURITY CLASSIFICATION OF THIS PAGE (Zien Dats Fatered) PORT DOCUMENTATION PAGE rm REP ONY NURI : 2. GOVT ACGaRsION MO] BAEC liq 77-B1i2 wa i TLE (aid Subéitie) Dass 8. = OF REPORT & PEMD CAvERED fl a INTRODUCTION TO. WLENZR-HOPF METHODS IN | Formal vepe v7) { ail ACOUSTICS ARD NUB2ATION a \ shed nL ATLL CATALCS RUBBER emer cs Bees SOT 6. PERF “82 ORD. KEYORT MUMBER ee eae 10 ‘David G./crighton_\ 9. PERFORMING CRGANIZATION NAIE AND ADORESS 0. PROGRAM ELESENY, AREA @ WORK Um Mena 4 David W. Taylor Waval Ship R&D Center a i dies Bethesda, Marylnd 200284 J.0. 4 13800-001 32 Vl. conPRotunntt ss “Vit. RePpey para vv iT Decontor 2377, Cato fre 24 — Z, Ganz Sor PAGES Beet e-e Vr iehaane 3) 94 V6. SECURITY CLASS. (of tala UNCLASSIFIED Dach AUN PICATION COUN GRADING SCHEOULE Té. CHSTRIGUVION STATEMENT (of thie Rapast) APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED 6 Em ne ee RT RT 117, CHSTRIGUTION STATEMEAT (of the abotract safored in Picck 30, If ciffercal fram rt t P16. SUPELOMENTARY NOTES H18. KEY wOROS (Castinas oa reverse esd tf necoosary ems tasntl iy kp bieck umber) Wiener-Hop® oetheds Strectural acoustics i Kalf-pisne flow probiens ECR ER neem i ree eer er, ome. ww ir. ABSTRACT en epwstco gids bf escevers/ "Seneigy by bisek ex=mbut) - The Wiener-Hopf technique is) 3jfirmly established as a powerful tool for } research in certain types of boundary value problem arising in acoustics. f Typical problems wiich may be solved exactly or asymptotically with this f techiiaue concern the sound and vibration levels generated by finite or seni- f infinite plenar or cylindrical surfaces, of local or extended reaction, | immersed in a compressible fluid and subject to acoustic o* mecnanical forcing. ; rooweyers even the simpleat of these problems involves complications watch are } EER ENE OE LYSIS EN ES BESO rant ete a ED EB ; Fon © 7) EXTON CF | uOV 69 18 BUSOLETS UNCLASSIFIED O84 en "Lod 6601 | iy isl eecuaiTY CLAD TICATION OF THIS PACH (Uhan Dare Enter) ye -~— Ae nn en REELS, SRL LE AAA, I IIIS 3 ET a Se ~-S NEN i i RE A a EE ant aS A IE Sia ata bt ete) ond « Raia ea ha are bee is tee 4 on i ay Peete hy ta SNe i ; AGUS inne Yeats : ‘aetna agit ater aera tec en tate pera ee nse ey peri is neues (oN ——JNCLASSIFIED ae reer eraser \ 20. ABSTRACT (Continued) CRARROUETE FEST VE ATOM. en eeenecceeeeen tenes q Ce ea ee cil oh ese eas OISTRIGUTION /AVAILARILITE GEOEB i Ti ae ‘4 irrelevant to an understanding of the Wiener-Hopf method itself ard its various extensions. Accordingly, this report was written in an attempt to display the operation of the technique in an even simpler physical and mathematical context, and thereby to encourage its more widespread use. The report deals with the application of Wiener-Hopf methods to one- dimensional wave motions on strings and beams, and in particular with the reflection and transmission from discontinuities in the mechanical proper- ‘ties of a string. Also included is a section illustrating how a generalized Wienexv-Hopf problem can be set up for a three-part problem involving a Two dimensional wave problems are then exempli- fied in a discussion of the acoustic field generated by a vibratir.g half- plane, and the effect of uniform mean flow cyer the half-plane ia included to show how different types of “edge condition” may be accommodated. The final section sets out in detail the properties of certain functions arising very frequently in application of Wiener-Hopf methods to acoustic problems. string of finite length. FO LIL COREY CP TS TEI MORE LY SSUES AT OSL ERT NSDL \ SECURIT? CLASSIFICATION OF TwIS PAGE(thor Deta Entered) \ AD SPS ESAT. CTE re Ge - —— eee eet Sena seb er ed a asic cee a Sil ae eh ta i A NR | ew RDA tn ste Shas Dae a i ad I eR I er te UNC! ASSIFIED a rr ee Oe SECURITY CLASSIFICATION OF THIS PA32(When Desa Ents rod “A «| 4 j oie GaN ar Way dk i i Des aii BENG tse i a a , —— SNE Laer ue ye wow one TABLE OF CONTEMTS ABSTRACT ice wie. oie oi coniicitiolites monroe: (epics etre}. Js4 eu erty ADMINISTRATIVE INFORMATION . . ~ - © © © © © © 1. INTRODUCTION... ++ 2+ + ee 2 © es © @ ® pl ei poe eS i a a BE OS REE RETA AMET TE Sch aS | ARPA a a ni Hs OES EM Cae a Na MeL ache ha ae alan Saabs ai Pat oke ey ra AD uel) SLE arc Soe ee ke 2. REFLECTION OF WAVES FROM DISCONTINUITIES ON A STRING. . 2. - 2+ se ee 2 2 se © eo 8 8 3. SOLUTION OF THE W-H EQUATION. . +--+ -+ = - 4. INVERSION OF THE FOURIER INTEGRALS .-. - - 5. DIFFERENT CONDITIONS AT x #0. . - + - = > 6. WAVES ON BEAMS . - © «© © © © *© © © © © © 8 7. GENERALIZED W-H EQUATIONS: THREE-PART - - BCUNDARY VALUE PROSLEMS 8, WO-DIMENSIONAL HALF-PLANE PROBLEMS. . . - 9. HALF-PLANE PROBLEMS WITH MEAN FLOW: WAKES AND KUTTA CONDITION. . 2. - »- © © © « {0. CONSTRUCTION OF W-H SPLIT FUNCTIONS. .. - REFERENCES «0 0 2 © eo -* © s © #8 8 8 8 oe 8s LIST OF FIGURES 1 - Reflection and Transmission from a Simple Discontinuizy in String Density. . + ++ + 2 - Complex S-Plane with Overlzpping Upper and Lower Half-Planes R,, R_ and Strip of Analyticity D. . 2.2 2+ see eee ee? 3 - Singularities Associated with Reflection Fica Discontinuities on Beams. . - «© «© « « 4 - Common Choices for Location of the Branch Cuts for the Square Root Function y. - + - Sa aro Page 30 62 71 87 &2 83 84 SARTRE GRE SETTLES TAIL ES AD A ERTS NEE RTE aitaaisiere Cie lente ail eal Sota aint ahe ABSTRACT The Wiener-Hopf technique is now firmly estab- lished as a powerful tool for researcn in certain types of boundary value problem arising in acous- tics. Typical problems which may be solved exact~- ly or asymptotically with this technique concern the sound and vibration levels generated by finite or semi-infinite planar or cylindrical surfaces, of local or extended reaction, immersed in a com- pressible fluid and subject to acoustic or mechan- ical forcing. However, even the simplest of these problems involves complications which are irrele- vant to an understanding of the Wiener-Hopf method itself and its various extensions. Accordingly, this report was written in an attempt to display the operation of the technique in an even simpler physical and mathematical context, and thereby to encourage its more widespread use. The report deals with the application of Wiener-Hopf methods to one-dimensional wave motions on strings and beams, and in particular with the reflection and transmission from discontinuities in the mechan- ical properties of a string. Also included is a section illustrating how a generalized Wiener- Hopf problem can be set up for a three-part prob- lem involving a string cf finite length. Two- dimensional wave problems are then exemplified in a discussion of the acoustic field generated by a vibrating half-plane, and the effect of uniform mean flow over the half-plane is included to show how different types of “edge condition" may be accommodated. The final section sets out in detail the properties of certain functions arising very frequently in application of Wiener-Hopf methods to acoustic problems. ADMINISTRATIV®. INFORMATION This work was performed under DINSRDC Contract No. N00167+76-M-8415, financed under DINSROC Job Order 4-1900-001-32. At the time, the author, whose permanent address is Department of Applied Mathematical Studies, University of Leeds, England, was a visiting professor in the Department of Mechanical Engineering, [Catholic University of America, Washington, D.C. 6S) PAE EMRE Ta a aS IO eam mr es ee FA RAE ASRS OE MN re ec ge apres ween — — i PEO NOCRIDN Let i a 9.8 mmaguticnsi ; ey i be ein Ly thal 1 Nae meer roe 1. INTRODUCTION The Wiener-Hopf technique was devised in 1931 {1} to deal with an integral equation arising in neutron transport theory, though its origins—and indeed its essentials—come from Russian werk in the 1870's on singular integral equetions [2,3]. During the war the tech- nique was extensively applied by Schwinger and his colleagues [4] to problems in electromagnetic wave propagation, and much of the sub- sequent developnent of the method has taken place in applications to wave diffraction processes. Standard books on the method are those by Noble [5], Weinstein [6], while several books (e.g., Carrier, Krook and Pearson [7], Morse end Feshbach [8]) have chapters which attempt to fatroduce the method. In such introductions, two-dimensional] boundary value problems involving a partial differential equation for come field variabi® are invariably used as the simplest demonstrafioa prob- lems (the Somaerfeld preblesa of plane wave diffraction ty a semi- infinite rigid screen being the best known). Such probleas, however, bring in at cnce a number of issues which are irrelevant to the exposi- tion of the W-H method; the introduction of branch cuts in the complex wavenunber plane is one auch igsue which—while it is an important one, and one which must be understood by anyone wishing to deal. with wave diffraction preblems—causes great difficulties for most students. Accordingly, an attempt will be made in these course notes to illustrate the W-H method in a wuch simpler context than usual. We shall study one-diwensional time-harmonic waves on strings and bars, and in particular we will study the reflection and transmission properties of changes in properties of the medium, abrupt changes of density giving ci ete rise to various hinds of standard and generalized forms of W-h equations. Not only is che derivation of the WH functional equation much simpler than usual for these problems, but its solution is also much easier, and the final inversion of a Fourier transform integral can be readily rer- formed and the resuits seen to correspond with familiar undergraduate ideas. The W-H technique is a method of solving certain types of boundary value problem in which, typically, we have information about the pressure, sny,on the half-plane x < 0 and about the velocity on the half- plane x > 0, and we cannot solve for the radiated acoustic field until we know the pressure all over the whole boundary, -™ 0), ef the unknown velocity (for x < 0) and of the given forcliug field arising from seme prescribed fo.-e or source or incident field. The transformsof the two unknown distributions are known (partly : because of information supplied by the anticipated physical behavior of the system under study) to have certcin analyticity properties as functions of the wavenumber regarded as a complex variable, and a certain crucial step (the W-H method) and the use of some fundamental theorems of the calculus of functions of a complex variable together enable one equation to be solved for two unknown functions. Then the field everywhere can be found in terms of an inverse Fourier integral which in aany instances can be estimated by stationary phase or steepest . deacent/saddle point techniques [e.g., 9,10) or by generalized function methods [11,12]. In the preblems to be discussed here such eleborate wethods are not needed, and an exact inversion of the Fourier integrals 3 Ra aE Ty a Lt IFRS RNA EN EE I A RA a a ERNIE AEE EN Se fa a eae can be obtained with the aid of elementary residue calculus (see, e.g., {7]). It is usual, and no doubt more logical, to start an introduction to a technique of this kind by first summarizing all that will be required in the way of results from the theory of functions of a complex variable. As our problems are extremely simple, both from the conceptual and froma the manipulative point cf view, 1t seems unnecessary to start here with such a digression, and we chall introduce the various ideas and theorems as they arise in the course of the problems. For a rigorous statement of the theorems the reader is referred to standard references (e.g., Titchmarsh [13}). We should warn the reader, however, that the degree of rigor which is usually adopted by workers in wave theory is not altogether a matter of mathematical pedantry. Certain problems, involving coupled wave-bearing media in particular, are extremely delicate, and require much wore attention to mathemacicel rigor than do the simple problema at hand here. ° ik Matern a Ob vind a waka Le: Ue Bie ‘anya 2. REFLECTION OF WAVES FROM DISCONTINUITIES ON A STRING : Consider a uniform string of line density Py lying along the portion -@ 0, understood through- out, waves in x < 0 have wavenumber ky = wil, while those in x > 0 have wavenumber k, = w/C,. We wish to determine the reflected and trans- mitted waves in x < 0, x > 0 respectively when a progressive wave with displacement exp(ik x) is incident upon the junction fro e <0. [See Figure 1.} This is a trivial problem to which the solution can be found by elementary methods and which can be generalized to cover a variety of different conditions at the junction. It and its generalizations are also suitable for introducing the W-H method very simply. We start by writing the total displacement as I yr exp (ik, x) inx <0 | and as (2-1) y inx? 0 The reason for this is that then y (which might de called the scattered fiela) must take the form of an outgoing wave as x + t@ and as x +-™, and thie enables us to state something about the Fourier transform of y. Ce Raat Rae e We have firet, however, to make t!.2 wavenumbers ke and ky slightly RE nT te ie ew Nh haw oa eh Me ernheci ap oils pee: fing \ To onal fal - Cpe ace pany ws ee ae i t i 4 if (2.2) where real end imaginary parts of both wavenumbers are beth positive and wuere the imaginary parts are small (and in the end vanishingly smali compared with the real parts. The rationale for this is as follows. We start by assuming a tine factor exp(-iwt) with w > 0. (We could just as well take exp(+iwt), but the choice of exp(-iwt) is helpful for reasons that should emerge later.) Then as x > + © the phase factor of an outgoing wave must be exp (+ik, x) where Kk, s wiC. Giving Ls, a swall positive tamginacy part therefore sakes an outgoing weve decay a5 x + +, like exp(-k, x). corresponding to the presence of eal] incernal dissipation in the string. Similarly, as x > -@en outgoing wave will have the pl ase factor exp(-ik x), and then will also be cxponentialiy daszped if we give Ke a small positive imaginary part. We therefore now know that for x > 0, y(ax) is a continuous function which decays like exp(-k, ,x) as x~+o, Its HALF-RANGE FOURIER TRANSFORM then hes certain preperties as a functicn of complex wavenumber s (that being a useful syabol for the vevenumber which dees not have any particu- lar significance, the symbol k for example often being associated with a particuler wavenumber). Define A ¥,(s) = { y(x) exp isx dx. (2.3) ° eatbormesn re: fe ty) x sada ne on SE ant EES Then XY, (s) exists as an analytic function of e at all points in the « complex s-plane for which the integral converges. The integral over any finite range of x certainly converges (and y(x) is certainiy inte- grable near x=0) so that the convergence is dictated by the behavior of ey y(x) as x ++, There y(x) exp isx ~ exp[-(s, + kx (2.4) where s = s. + is, and the integral up to infinity cherefore converges if 8, + ky > 0. It may also converge for some other values of s, but what can be guaranteed on the basis of the anticipated behavior of y(x) as x > + © is that Y,(s) is analytic in an upper half-plane Ry : Im s > “ki; (2.5) Moreover, Y,(s) has non-growing algebraic behavior as |s| + © every- 5 where within the domain R, if y(x) > finite at x = 0+ or has an tategrabl2 singularity at x = 0+, so that lv, ¢s) | = Olen) say for some A > 0 as |s| > © along any radius in the domain R,- If, as may occur in some problems, y(x) has a non-integrable singularity at x = Gt (that is, a singularity ac least as strong as x7!) then y(x) has to be regarded #3 a zeneralized function whose generalized half-range trans- forn ¥,(2) is 3till analytic in an upper half-plane R,, but now can have algebraic growth as | s| StaiCol f By analytic we mean that Y, (8) is singise-valued and has a uni, derivative eg a mannan are nmr nen i Wi i ye es ig PETS | Toten SUL re dO 0 Oe 1 | i 1 | { Y¥, (sth) - Y, (s) Qin + + nee renner C) ! s YX, (8) as the point sth approaches s along any path in the plane. In a precisely similar way, the half-range transform of the scattered displacement y(x) for x < 0 Y_(s) = fi y(x) exp isx dx (2.7) —c is an analytic function of s in a lower half of the s-plane, H <+ . R Im s Koa (2.8) and for a function y(x) integrable at x = 0- Y_(¢) has non-growing algebraic behavior at infinity in R_, so that ly_(s)| = 0(|s| “) for some 1 > 0 as |s| + along any radius in R.. (Algebraic growth of Y_(s) is permitted if y(«x) has a non-integrable singularity as x + 0-.) Since kot and Sr are strictly positive, the FULL-RANGE FOURIER TRANSFORM +20 Y(s) = Xs) + Y_(s) = f y(x) exp isx dx (2.9) =o exists as an analytic function of s in a strip D, D=R, QR :-ky,< Ime 0, H(x) = 0 for x < 0. The FOURIER INVERSION THEOREMS run as follows:- if C, is any contour from -®ro +@lying within che domain Ry and such that YX, (s) exp (-isx) is integrable over this contour, then 1 On Y,(s) exp(-isx) ds = y(x) (x > 0) + = 0 (x < 0) both of which are contained in y(x) H(x) = at Y,(s) exp(-isx) ds (2.13) 2m jc + The fact that the integral vanishes for x < 0 follows from Cauchy's theorem applied to a closed contour T consisting of the contour Cc, with its ends joined by a large circular arc in R,- Since the integrand is 1 analytic everywhere in Ry 1 f ¥,(s) exp(-isx) ds = 0 Jr ; p and the integral along the circular arc vanishes if x < © copes Seapets jeer Sahin SY UAT yee am sapere: ad SSO LARIAT? BO NAPUS sa ccc wn RE amen SEA se eh Ne eo lnermarin geist nes neve Arey geen stout arginine rn ee es ser OE LAE LIED HORST A because exp(-isx) is then exponentially small at infinity ia Ry and . ¥,.(8) is at least algebraically small there. (This result shouid really be proven carefully using Jordan's lemma; see [7,13].) Correspondingly, if C_ is any contour from -- to +~ lying within R_ and such that Y_(s) exp(-isx) is integrable over it, then § % ES fi % E E | | y(x) H(-x) = +/ ¥_(s) exp(-isx) ds (2.14) 27 “C Now if C i: any contour from ~~ to +~ lying everywhere within the strip of D cf overlap between Ry and R_» then we can identify C with C, for (2.13), C with C_ for (2.14) and obtain by addition of (2.13) and (2.14) rr ee the inversion theorem oes) oes { Ve)iexp (iss \eds (2.15) 2m +¢ We remark here that our convention exp(-ivt) for the time factor is consistent with the formulas (2.9) and (2.15), in the sense that we are rs: iy taking F.T.'s in space and time with the definitions f Y(s,w) = J { yix,t) exp(isx + iwt) dx de y(x,t) = aa | f Y(s,w) exp(-isx -iwt) ds dw (27)? Note that if a time factor exp(+iwt) were taken, and kook, were defined as La = wiCs k, = w/C,, chen we would have to give kooky small negative imaginery parts in order to secure a strip of overlap in which to take F.T.'s in x. 19 | | | | | aa se sits ares wy oJ sa 1) ba it sei Oe ; i The point of using half-range transforms is that conditions are different according as x 2 0, so that a two sided-transform cannot immediately be applied. For x > 0 we have the equatio:: of motion 2 oe tke y=e. (2.16) Multiply by exp(isx) and integrate from 0 to =. Then the last term produces 2 = ki x, (s) provided seR,» while the first term can be integrated by parts to give 2 co cE oy aes dx = oo) ares -is Fyenine -s*Y(s) lp ax dx a ° + From the anticipaved behavior, y ~exp¢ ik, ,x) as x ++ we see that dy: ers and esos both vanish as x ++ provided seR,. If we write dx P + im Lim y(t) = I yds yO) = OS ayy (2.17) then we have (s? - ki) ¥,(s) = -y'(O+) + isy(0+) (2.18) as a statement, for scR,, of the equation of motion in x > 0. Applying the same procedure, this time for scR_, to the equation ‘ rn a I TT TS i 2 oe +k y= 0 (2.19) x which holds in x < 0, produces (s* - ke) Y¥_(s) = + y"(0--) - isy.(0-) (2.20) 05 oR goksen! Be Hol amie a ¥ ‘y vou poten hat pale i ie nee Fess ITN OL TRG RETR TEN ONS Rr OCD Roe 9p rnanrecnenesene wo heey nt ea ne ee cS, eR RE ence ree te nO eA I er arpa TI! RN COTES The system (2.18), (2.20) 1s completed :hen the boundary condition . at x = 0 is specified. In the present simplest case, in which there can be no difference of transverse force from x = 0- to x = (+, the boundary condition is that 24m a ikox) | im dy : Ola aR RON vax 2 (2.21) 4 | so that { y'(O-) + ii, = y' (G+) (2.22) | while the total displacement of the string must also be continuous, so that i Lim ik.x Rim | BRIG er oy rey (2.23) | or y(0-) + 1 = y(0+) (2,24) Eliminate say y(0-) and y'(0-) from (2.20) by using (2.22) and (2.24), giving (s? - k?) ¥_(s) = y"(Ot) - isy(O+) + i(s - k)) (2.25) and if we now add this is to (2.18), the unknowms y'(0+) and y(0+) | disappear and we have | (8? - ki) ¥,(s) + (s* - 2) ¥_(e) = (a -k)) . It is convenient to divide through. by (s? - ko) which we may do because the equation itself is only meaningful in the strip D and the zeros Hk Jie outaide thet strip. Then we get a standard form of Wiener-Hopf < functional equation 12 i ER eee ae ‘ate to pees a clas hic Papas an i Weed. ay ine: se oaks | ” Ge: V Pw ne : ; Oe psueiel, ie ne Oh ol ma K(s) ¥,(s) + ¥_(s) = (2.26) where the kernel is 6 (2.27) The W-H equation relates a linear combination of the unknown half- range transforms ¥,(s) tc a functirn related to the incident wave field exp ik x: The equatio: holds in the strip of overlap D, and the coeffi- cients, K(s) and (s + ert are analytic and non-zero in the interior of the strip in the simplest cases, though cases In which K(s) has zeros or poles in the intesior of D can also be handled straighforwardly. . TH: the next section we show how the general situation of the W-H equation van be obtained by inspection and how the Fourier transform integrals can be inverted to yield explicit solutions for the trans- mitted and reflected waves. Following that we look at differences which arise when the beundary conditionsat x = 0 are changed, and when the strings are replaced by elastic beams. 3. SOLUTION OF THE W-H EQUATICN Assuming that we have obtained an equation (2.26) with the coeffi- cient of one or other of Y,(s) reduced to unity, the first crucial point lies in the W-H FACTORIZATION of X(s). In this we express K(s) = K,(s) K_(s) (3.1) as the product of two functions of which K,(s) is analytic and non-zero in Ry and of at most algebraic growth at infinity in Rus while K_(s) is analytic and non-zero in R_ and of at most algebraic growth at infinity ‘i oS va (alkane. a “ etontak Heta twee eh GG) ae eh (or atten 2 mk abel Sa, denny i ; ay fee if Sala) ; a rs ip ‘ ; Ke : iE joe 4s : ‘ i is iy TOP Raye! I A fj ya a , , i Hy at u ee vA Bi ‘ 2 , ae , r i | ett? Ee ites oN Tino I ; ny ead i ‘ 132 : | | | | | | | | Ste Mah Cae there. (Here we assume that K(s) has no zeros in the strip D, a point we shall examine again later.) It is remarkable that such a factoriza- tion exists for any K(s) which is analytic and non-zero in D but which may have any kind of singular behavior outside D; the proof was given by Wiener and Hopf {1}. Hera the truth of the theorem is obvicus; 8 +k, s-k, K, (8) = =) » K (s) = Sie E, (3.2) Oo. is one factorization with che required properties, and is such that K, (8) > 1 at infinity in R,- After we have completed the solution we chall return to the uniqueness or otherwise of this factorization. Because K_(s) is free from zeros in R_, and in particular in D, we can divide (2.26) through by it, to get Y (s) { K, (s) ¥,(3) + Ks) = Gk) KG) (3.3) The analyticity properties of the terms on the ‘eft here are know, while the function on the right is neither a(#) function nor aQ© function. Our next object is to write it as the 3UM of the two functions aralytic in Ry and R_ respectively, and of algebraic behavior at © in those half-planes. Again, the existence of this ADDITIVE SPLIT is assured by the W-H theorem [1], and here we can again: see how to perform the split by inspection (and in this case the method of inspection is widely useful and should be carefully noted). The function i/(s + k.) K_(s) is analytic in R_ except for the pole at s= wk, Near the role, the function behaves like i/(s + k)) K_(-k))» so that we can isolate the pole behavior by adding and subtracting this term, to give 14 1? fain Yom wr ASE WNP ORIG Bee, ror ES WIAA LER Nt CA RL OA Soe Cherm eran eae ell tt eR meet arte Rem yt AS pager mere pene tecter wer ne seencnteerbee rer re mcseeeeeee ee et a . His ise Young 8 Pe 3a. antago vote me ' DOU EN 9) a LES sin eels ES (gs + k) K_(s) (8 + asians ele (s + k)) K_(-k,) (3-4) The first term no longer has a pole at s ®@ “ky: for near there at 1 K_@) - K_Ck,) = (s + kx (function analytic at -k,) '; and it is therefore a@© function G (s) aay, while the correction term i/(s + k,) K_(-k,) is evidently a@) function, G(s) say. Nete that this cdditive split is not restricted to any particular form of K_(s), but turns on the presence of a pole term (s + oe only. 5; We now rewrite the equation (3.3) as Y_(s) } K,(s) ¥,(s) - G(s) = G_(s) - Rac) (3.5) and consider the function E(s) defined by E(s) = K, (a) Y,(s) - CG, (s) A (3.6) This is a functicn defined and analytic throughout R, and of algebraic behavior (algebraic growth or decay) at © in Ry. E(s) is not defined by (3.6) except in Re However, within D, E(s) can equally well be defined by Y_(s) E(s) = G (8) = KG) (3.7) and this definition then CONTINUES ANALYTICALLY the function E(e), defined originally only in R, by (3.6), through tiie strip D of overlap into the lower half-plane R_, and there E(s) is also analytic end of algebraic growth or decay at infinity. 15 a nS SS ASS SO SSSSSENSESOSNRSUEPSS PaamnOn PSY OTOGTIns SER RIE SEERA as “wi : i suotavonis iia: "i Bb & din soli + © is obtained by inserting | in (3.9) the asymptotic expansion of y(x) as x > OF and integrating } term by term. The process cen also be used in an inverse fashion to fina the behavior of y(x) as x > O+ by examining the behavior of Y, Gu) js NOT simply related to the behavior of the full-range transform Y(s) as u++~[141. Note, however, that the behavior of y(x) as x > Ot : | 16 ‘ PO pia if o } u for large values of s. If one knows Y(s), one has then to decompose Y(s) into YX, (s) + Y_(s) and then look at ¥, (2) for large values of s, a procedure explained in detail in [14]. In a similar way, the asymptoticsof y(x) as y + S- are determined by those of Y_(-iv) as v > +o, using Watson's lemna. Now in our problem we assume that the deflection y in the reflected or transmitted wave is finite as x > &. Then the leading order term in the expansion of ¥, (iu) is (Const) ip exp(-ux) dx = (Const)u7? (3.10) oO and similarly for Y (-iv), so that YG) are each O(s~!) at infinity in Ry» respectively. Ee) each tend to 1 at infinity, while G(s) are each O(s~?). It then follows from (3.6) that E(s), which is the polynomial P(s), is O(s~') at infinity in R,, and from (3.7) that it is Ofs'') at infinity in R_, and hence, because Ry have a common strip, P(s) is a polynomial which vanishes everywhere at infinity like s~. Therefore P(s) must be identically zero, and the ecinution subject to the conditfon of finiteness at the junction x = 0 is ¥, (s) = G(s) MK, (8) (3.11) Y¥_(s) = G (s) K_(s) and in explicit form this gives re — 17 {SiR RGR eeepc ee tesa nina a ae es A at aN al ER A A i SR EN i a er ce i ai iE oc ne ea ete iad ae oF a an fps yu He) cele ates Wie aie SI wy)! eet en apes Sst . , BEEN YT, COT SSS ce ae 2ik, Y,(3) * <---> + (a + k 1) Ck, + k) (3.12) Y_(s) = 1 21k, (; - k\ s+ ky * (s + ko) Ck, + k.) s- x) 3 Before proceeding to the inversion of the Fourier transform we return to the question of the uniqv: sess of the K (8). Let one specific facturization be K, (s) K_{s). Then in any other factorization, the [A,(s) K,(s)][A_(s) K_(s)} where, since K, <8) are analytic and free of zeros in R,» A, (8) must | factors can be written also be analytic and free of zeros in R,, respectively. Further, : + A,(s) A_(s) = 1 for seD, and because A_(s) has no zeros in R_ 1 AOS A_(s) again for se€D. Define F(s) = A,(s) seR, 1 F(s) = CAN seR_ Then F(s) is analytic throughout the whole s-plane (an ENTIRE function). Suppose further that the factors are required to have some . ,ecified algebraic behavior at infinity in that respective half-plane. Thea A, (s) each tend to constant values at infinity in Ry» and the entire function F(s) hes constant values everywhere at infinity. By Liouville's theorem 18 BORD RIT 5 the only such function is F(s) = Constant Fo so that if K,(s), K_(3} are one pair of factors, any other pair must be of the form : 1 Fo K,(s), ES K_(s) : Ia other words, a factorization with prescribed algebraic behavior at infinity is unique up to multiplication cf K, (s) by a constant Fo and division of K_(s) by the same constant Fo: Wote the importance of the restriction to algebraic behavior at infinity. In many applications, | part of K(s) can be represented as an infinite product from which the split K,(s) K_(s) can be effected by inspection. Usually, however, the infinite series of factors which gives K, (s) or K_(s) has exponential behavior at infinicy in R, or R., and then it is nECeSaary to divide say K,(s) by an entire function with the appropriate exponential behavior at infinity in Rs so that the resulting factor behaves algebraically, at the same time multiplying K_(s) by the same factor to eliminate exponentiai behavior at infinity in R_. Several examples of this are given in the book by Noble [5]. 19 BUR eea TEAR cement posesmcemmemrem mn i RGB IDNY: PR a NR a a sen LN ah i A a stile BCE cnt Mit TN WEI! Na Sh is ah REE EE i neem a al ie Lace le Be ede oS aly ie uo HEE eel a i am Bi ed A) lee 4. X.NVERSION OF THE FOURIER INTEGRALS From (3.12), the full-range transform of the scattered field y(x) is given by ¥(s) = ¥,(3) + Y¥_(s) , e ka £ 2 and wre! y(x) = xf ¥(s) exp(-isx) ds 3 Cc where C runs from -™to +@in the strip D. Since Y, (8) behave | algebraically at infinity in Ry, the convergence cf the integral is | dictated vy the exp(-isx) faccor. | For x > 0, close the contour C with a large semi-circle ia the | lower half-plane. The contribution from this semi-circle vanishes as the radius becomes infinite because exp(-isx) is exponen/ially small for x > 0 and Ims G) = on (-271) (Residue of ¥,(s) ats 2 -k) exp (ik, x) 2ik Si) egy exer, =) fo) 1 = T exp (ik, x) { | | I | | | where the transmission coefficient is { i | 20 | SiS nee ea aN a Rr eld ka (4.4) For x < 0, complete the contour C with a large semi-circle in R,. There is no contribution from the large semi-circle, and applica- tion of Cauchy's theorem now gives y(x < 0) -> (+ 271) (Residue of Y_(s) ats =+ k)) exp(- ik) (21k) s+i Gk), +) (+ ky - k.) exp (- ik x) = R exp(- ik x) with a reflection coefficient ky - k, R = ares ; (4.2) t¢ ds a crivial matter to check that these solutions for the reflected and transmitted waves satisfy the conditions (2.23) and (2.24). In the kinds of problems encountered in acoustics, the funcrion ¥ (33 usually has a branch point singularity in R_ as well as simple poles, and correspondingly Y_(s) may have one or more branch peints and simple poles in R,- The pole contributions are unaffected (except in certain critical citcumstances) and give rise to natural modes of the system, analogous to the reflected and t.ansmitted waves found here. When the dissipation factors are small, some poles will lie close to the real axis and give rise to propagating modes, as here. Others may lie close to the imaginary avis and give non-propagating modes, of the kind found in wave~guide problems and in the motion of plates and beams. In cther problems poles mzy be present in the complex plane 21 Tan a) AR ct sca en la ee ee ee ee a IR UTE PP RN ORT eT a SEN CNL TO / Ste a pee ea et : A ae) (ries i ve and yet may never be captured in the appropriate deformations of the integration path which will have to be made, and such poles then represent no distinct and identifiable structure of the field. Pole coatributions also serve, in accustics problems, to represenc the abzupt changes that weuld occur according to geometrical optics as one crosses boundaries between iiluminated, reflected, and shadow wave zones. These pole contributions have to be supplemented in more complicated problems by integrals arouud the branch cuts which join brench points. There are many techniques for Gatimatine the contributicns from -ranch cut integrals [9,10,11,12] including cases where various kinds of singularities come close together, and even coincide. ‘enerally branch cut integrals represent forced near-field behavior, which decays algebraically away from a junction or discontinuity of the two-part system usually studied by the W-lh techuique, leaving only the natural propagating modes at large distances. Sant pinta tdanaaond eel =) — Al ‘ttiaeged sents, 5. DIFERENT CONDITIONS AT x = 0 Suppose now that the strings are each stretched to tension T, but are joined by a mass m which is free to slide on a smooth wire perpendicular to the strings. The condition of continuity of dispi.ace- ment of the string remains in force, eo that as in (2.24) y(0-) + 1 = y(0+) (5.1) while the equation of motion of the particle is tT Zor - mh (y+ eK) (0-) = - mw? y (OF) (5.2) Wt since yCOL ens is the particle displacement. We now find that it is impossible to eliminate all of y(0t), y'(0+t) from the equations (2.18), (2.20), (5.1), and (5.2), which are statements of the equations of motion and the boundary conditions. The simplest W-H equation one can get, replacing (2.26), turns out to be i mu? y (0+) (s) = me K(s) ¥,(s) + Y_(s) aa K, + Ton ya (5.2) oO containing two unknown functions, YX, (8), and the unknown constant y(0+) also. Supposing y(0+) were known, however, we proceed as before. The additive split of i/(s + Le) Kes) = G, (s) + G (s) has been given in (3.4); for the other term in (5.3) we proceed in a similar fashion and get Litas dese J (ap Se H,(s) + H_(s) (5.4) 5 Ei(e2= ke)’ RCs) where 23 Se RR ieaeeToe ee ee emer rH ALD eh iris gat viehiiee ats ob i ‘ * Aaa ania. ity the oui 3 vind on. besaony a ca =. BLO ty maton: cacliend a ak nb aay te, 4 its ate mle = % y x es Ng = ee iS = = | i care ee RT NREL DRE ES Se Hs) = 7 y (0+) (are k,) (-2k) K_(k,) (5.5) 2 1 1 1 H_(s) = = y(or) — Lear eG) cacy T (s + ky (8 k)) K_(s) ( 2k) K_( k) The entire function E(s) is again zero and thus we find i mu? 1 K(s) ¥,(s) = ——- yo - y(t) ———— (5.6) + + (s + k)) K_( k) 2k oT (s + k) K_( k? from this equation we can now determine the value of y(0+), for we recall from $3 that the behavior of Y¥,(s) as g*+© in Ry is related to the behavior of y(x) as x > Ot — i.e., to y(0+). Ass 7+ in R, we have ¥,(s) 5 f y(O+) exp(isx) dx +... o = at). + .... (provided Ims > 0) (5.7) while K, (s) >+lass+min R,. Therefore the leading terms of (5.6) state that ay COR) ira SMU 2) mae On) a La (5.8) 8 sk (-k_) 2k T s K (-k_) y =~ "o ° -* “o which requires Coie ene a ee a 5.9 y =. QeaTIKUGKD) K (-k_) (5-9) On = KO -* “o The solution can now be completed in precisely the same manner as before. This method—of examining the detailed behavior, possibly to several terms, jn the expansion of ¥,(8) as 3 +o in R, ~<=-ig 24 ibe aie Pe ties saa balan ce WAR) Ne waite amrinad ws ororovaite ke dk am 4 ie iii ict, rp re ce dN ET A EN NNR me | | | frequently used to determine unknown constants arising from boundary conditions. Another common method, which is illustrated in the next section on waves in beams, involves obtaining a solution like (5.6) for general values of the unknown constants, and then arguing that unless the constants have certain special values, a plus function will have a pole somewhere in R, or a minus function will have a pole some- where in R. In complicated problems involving coupled eiastic plate/ acoustic fluid motions the plus or minus function which apparently has the singularity may not be the obvious Y, or Y_ function, but some more complicated related function. Wh Seo i a conlaledies 7 z ew Soy, eee a6 Ne iy inde a aOR, 4 x ¢ “ PRP NS Ste sissy SORA IO SRS INR er EN ie al TRACI” CAD BCRRRYIIENA br eth VAP ee ire operas seas nan’ ar oe pe nena 6. WAVES ON BEAMS P We now replace the strings of f2 by beams of specific mass PoP, and bending stiffness Bo? BY in x <0, x > 0, respectively. The free wavenumbers will be denoted by ko k, as before, for time dependence exp(-iwt), where k* = (p w?/B) , ki = (p,w?/B,). (6.1) For the moment we ieave conditions at the junction x = 0 unspecified and take a wave with displacement exp (ik x) incident from x = -%, denoting the total displacements again by y + exp (ik x) inx <0 and by y in x > 0. This ensures exponential decay of y(x) as x + + © and as x + - © provided In ko? In k, are given small positive values, fcr we anticipate that in x > 0 y(x) = A exp (ik, x) +B exp (-k, x) and (6.2) y(x) = C exp (-ik x) + -D exp(k x). . The near-field terms here, B exp (-k, x) and D exp (kx) » decay as x > + and as x + - ©, respectively, because ko and k, have positive real parts; they arise because the equations of motion are \ DS Se ok 0 in x <0 x (6.3) eS Un a aan AD ky} y=z0 inx>0O Taking half-range Fourier transforms of (6.3) gives 26 om pies ts on hina ‘oily, vou vate ots ” (s* - k}) ¥,(s) = y™ (Ot) - sy" (O+) - s*y'(O+) + is*y (0+) (6.4) (s* - k?) ¥_(s) = -y'™(0-) + isy" (0-) + s*y'(0-)-is*y(0-) (6.5) the first of these holding in Ry» the second in R.. Now the poles gs = +k,» s = +1ik, lie in R, (see Figure 3), so that YX, (s) will have ‘poles at +k,» + ik, unless we choose y™(O+) - ik, y"(O+) - ki y'(O+) + ik? y(O+) = 0 (6.6) y"™" (Ot) +k, y"(O+) + ki y'"(O+) +k? y(O+) = 0 (6.7) Similarly, Y_(s) will have poles at s = nk» 8 = ~ik) in R_ unless soluble set of eight equations for the eight unknown boundary constants. | | ! ~y"™"(0-) - ik) y"(O-) + ke y'(O-) + ik? y(0-) = 0 (6.8) | | aM CO s)icts kaa KO—)ae=rkeny* (O=)i ke y(O=))'="0 (6.9) | : ‘Whatever the boundary conditions at x = 0, conditions (6.6-6.9) must be satisfied. (Two analogous relations could have been deduced in §2, but it seemed unecessary to emphasize such a poiat at an early stage.) Four further conditions may be imposed at x = 0, at least one of these being a nonhomogenous condition, so that we shall have a uniquely \ | We shall not take any particular set of boundary conditions, as these tend only to lead to complicated expressions without any special structure. By addition of (6.4) and (6.5) we get | | | (s* - ki) ¥,(s) + (8* - bk?) ¥_(s) = Q, (8) + Q (2) (6.30) mona + Hay eS are | a eZ) | bata « Ne ‘ saath ‘ eae ae ‘ er ia a ae al aie id notin aS - ~~ oe pas re: ce ; ead ait hae a" “ltd where 6 Q,(s) = is°y(O+) - s*y'(O+) - fay"(Or) + y"" (Ot) (6.11) x Q,(e) s-is"y(0-) + s*y"(0-) + isy"(0-) - y™"(0-) and conditions (6.6-6.9) are satisfied, so that Q, (k,) = Q, (4k) = 0 tee S 1 1 (6.12) GQ) Gk em Qi(cik)) = 0 Equation (6..0) holds in the strip s <-k < < : 8 ky Im s < + koa Y For simplicity we can take koi 2 kig so that the strip D is symmetric about the real axis end the points +k +k, lie on the upper boundary of D, the points -ko» -k, on its lcwer boundary. Because kos is supposed to be very smail, the other points of interest, tik,» tik, and ~ik)» -ik,, lie well above and well below D, respectively. If ? therefore we work in the interior of D we can divide (6.10) through by (s* - k*) say, and get a W-H equation Qi(s) + Q,(s) K(s) Y, (8)4Y = 6.13 @)X, G4) > =) (6.13) in which the kernel is & Lh K(s) = 241 (6.14) 8° - ko , The W-H product split, into factore analytic, non-zero and of algebraic behavior at infinity in R, » respectively, is again obvious: 28 | 49 He 3 (orgar < Wo) niet (td) { Pe ; t a a». a“), sf shvenet ena er et hats thes oe ie foo ed + ag état mee. 493, inn on me | id, a To" oratenan ‘anwetiaas axosat ost: sahhaw saabora tah tages tage lf ceeotaoncian gh a Dati ch | . air mee i (9 +k )(s + ik.) (s - &,)(s - ik,) K,(s) = = 7 » K_(s) = = = (6.15) + (s + kts + ik.) (8 k) Gs tk) Then division by K_(s) (K_(s)40 in R_ and sogO in D) gives ¥_(s) Qi(s) + Q.(s) KS) ¥,0) + eG)” Ge kG + ike - k= ik) (6-28) } and we have to make an additive split of the right hand side. Now because of (6.12), Q, (s) must contain the factors (s + k,) (s + ik), | while Q, (s) must contain the factors (s - k)(s - ik,) so that we can write Sa) = (s + k)(s + ik,) (aos + bo) (6.17) Q, (s) =(s- k VCs - ik )(a is + b) where the coefficients ayy a, > bo b, are known when any particular 1 set of co;ditions is specified at x = ©. Now the right side of (6.16) has the form as+b () ° A a,s + b, -k)G-ik) Gtk)@+ik) which is already in the desired form G_(s) + G(s). The reason for this is that enalyticity arguments have already been used to remove pole singularities where they are not permitted and because pole singularities are the only kinds of singularity which are present in these one-dimensional problems this effectively means that the additive split must already have been carried out. 29 a arene ne oer shton tsa m paoanee at ana to, paaane ace imtnaneo aac (0) sMSERD 2 salsdides aah = ohh jem) exoate} etd nannies mane F iallivini vin: ities ‘tay 94. yaad val ele ge nant rn Wott estonia Hee Sin ih 6 armies ~neininadeiad pmsnaomtnoa’ | ‘i t of } 4 J Gt , Tl sat Siar y iF 7 y iV hi n f ae + , : SL Li! : “ nie 7 : bint ,! ui ‘ " & fi 1 ' ib oe Ht | ; a - rps : __ ri Liha Te Se SERIA EDDA lp teesrgcehaioneacoee-asssonoeerrtenecarctss nye eros ar al erevrn-rrene=ae, omens paneer nm anna ape tt St NRN SAS SIPPY CRSP nga esenanan tne nae aa ahem) Blas “Tesi By the usual arguments we then have the solutions K, (s) Y, (8) - G, (s) = P(s) : (6.18) : - XS s =z : WO) eo lt) . j where P(s) is a polynomial. And because G,(s) = O(s~*), K,(s) = 0(1) and Y,(s) = O(s ') (because y is finite at x = 0) at infinity in their respective half-planes of analyticity, the polynonial P(s) must be identically zero, so that a,s + b, | YO) 7 Ge eG) . (6.19) ays + b RR) i ear ny TTA H (s k,) (s ik)) | | The inverse transforms can then be performed explicitly, the poles | s = k and s = ik, in R gives rise to the reflected wave and a | decaying mode in x < 0, the poles s = -k, and -ik, gives the trans- mitted wave and a decaying mode in x > 0. Specifically we find : (x > 0) ACR (ik, x) ae PETC USED yin rani { (a, ik, 2 b,) | aay ee) (ajk, + b,) (6.20) y(x < 0) = 1 ———— _ exp(-ik x) k (1 - 4) | ° i (a ik, -b) ° EG. ED) exp (+k x) which is of the anticipated form (6.2). For any specified boundary conditions the values of y and its first three derivatives are known at x = 0,, so that the coefficients ay» bo» ay» b, in (6.20) can be 30 7 - a8 GO Pay ads + ates i. i es 4 3 (x, to) ae ae 2 * “4 > ae ne ct - Ma £ ae i, ome all asa wi er hn eh thet te A ett NN EA A A A ST obtained from s direct comparison of (6.17) with (6.11). But of course, the W-H was never intended to be used for solving such simple problems as these one-dimensional ones. We are using them as the simplest vehicle on which many features of common occurrence in the W-H method can be demonstrated. 31 a PGES Nee reat cee a et a A tnt ht tec TI SS SI a Deiat AREO NS Ce YN ate cease A Per Rare Ser ene ee et et 7. GENERALIZED W-H EQUATIONS: THREE-PART BOUNDARY VALUZ PROBLEMS The standard W-H equation, (2.26), arises in many, though by no means all, boundary value problems in which difterent boundary data are prescribed on, say, x < 0 and x > 0. Many problems of interest in acoustics fall into this cateyory; for instance, the problem of energy conversion from the elastic to the acoustic mode when a surface wave in an elastic plate encounters a junction in the plate across which the plate properties change abruptly can be mcdeled in terms of two semi-infinite plates x < 0 and x > O for many purposes. The boundaries concerned do not always have to coincide with Rose che x-axis. For example, the standard W-d equation arises in the diffraction of waves by an open-ended parallel plate waveguide, or by an open ended circular duct, provided these are both semi-infinite. On the other hand, dif- fraction by three paraliel equi-spaced semi-infinite plates is a com- pletely open problem, though diffraction by an infinite cascade of semi-infinite staggered piates Is a relatively simple standard W-H problem. If the waveguides referred to above have closed ends (diffrec- tion by a semi-infinite thick rigid plate, or by a semi-infinite solid rod) the standard W-H method does not Nead to a closed form solution, but to an infinite set of coupled linear equations, whose solution can only be approximated by the solution of a finite subset of the equations in the low frequency limit. Thus it is clearly difficult to give any general guidelines as to when the standard W-H method would work except to say that it will not work, without modification, in the case of three-part boundary value problems where data is given on, say, -*™< x <0; 0< x< 8; 32 9 ath ase treealins fe ae eng muta ay om teenaged cs amatsony oan 0 =e bs 0% + oa i ‘pe Qo. mekdieg ai icohmawe 103 termes oats eso} | | svaw pabi4wa & aay Spies shvevobe: eda 2 09) abaea sil ‘=o datite manor ately ada 8) cobwanf a Byssnedane sanity & : 4 |, Rete Me ania ath babksion sie nad ehaguete aqawads mr Webniiavog ait ata ig a € Oak 0? a wind wey. satan aii Sau fe dahu Whteatas at vied wyawta, sen Z mean 9 baiicieabie sit. mt aeetaa, nna Lal heiaerg. watt ag: eas ‘s6itye ined a oaks tees 20d whe seqits non | 203 a's) sadeiy eaten bus pena Desay leiteieg Bx389) we . p . hs sticass stkakins he ed) wobiaes tb Mgusits peadduny cove Wlsoatg li mgyreta aiquse Ween aLon el s098%q) boone tate “awed tb) te) keasts eat swede. hatystsx sobhewinney sqa $2 of © Li Baten. atid nubabes & vga aed mere Aa Kits slab onda *, we te oo im ti Somutien eotnit 6 20 HARKOR awe v8 Homaptnoges Bees ; TL pare e vet : ~~ rc nett tang Lactonang oe OVhy oo siugeh ees trata ok pe sunt | hi " aaesttbe ee ots anos Ww Divo hea St buna ad mukow Vaabnwes Myers Yo-ogas: of Ak mod TNE ALge sods te «tae ES OOo E> Hie et ee ee et ee te 2 > 1, one can suppose that to first order each edge is unaware of the presence of the other, so that one can start the approximation with two semi-infinite problems of standard W-H type. Then if tne incident wave amplitude is O0(1), the first inter- action of one edge with the other will te through a cylindrical wave emanating from one edge, due to the incident wave, and of amplitude (kt) 7) near the other edge. For the second approximation we there- fore solve another two semi-infinite problems, but now with more complicated forcing terms arising from the mutual interaction between the edges. While one can see how to continue the process in that simple case, it {s advantageous to derive a modified W-H equation whose approximate solution throws up these successive interaction problems in a natural way. The advantage is that one can see from the generalized W-H equation what to try in more complicated problems where the physical situation is less clear. For example, near resonance the "weak interaction" sert of approximation is quite inappropriate, as the 33 eoate Maes stat a « divtional YE te emt nfs >t “tellin, aoe lenaeuninny eal on beasties Aa % no Aaron i a sathewie rashes * we at ‘thar ua woven te. ti od ms a wasn ney Labtion an iba, eee ate ren © aa tase omits ail aed ad ae Been aus ened ned aed Ah | Seatt naa Hy se a wives Be Keane oor droite wil. 82 aieh cSiebe wah wired stsuaan isa wo sddonatsanurs Bites she elt 24 hese baila cio dene wer Sind (aie bin tirtey did mhanse cote pane orton ie uate Boe pore pncriet haasotgags | aes ‘nt aeaaay it: eunh-tno Ox. ot ‘wae. ae Sanja Bets Boke iit ie nab 99 i comised ac oen etd tne eis tee aes wie 010 Sioa. Priecnit ata voeenonie a, ae ax 19003 besarte) ot ike hither tnkioaias wide sesbidin ano ra cower site nls am manny wld sit in iia ilo’ tel et eae ae naiy Miekito sg’ baad oniaorss aris erbe. aes bs ees Ga ashe nasusopi wt ois wowenose: aber: SS aa 0 Aap Ne seul. ak sab ‘ntti tessa ‘i ea hidqoseiesi init ak. matomdaurai, Te, iam haps | ee TESTE ERE ea RR FT a ae TRE ERNE REIT Cane ond renee 4 “4 : whole phenomenon depends or strong coupling between the ends of the systen. We shall try here to ure our one-dimensional examples ro illustrete the possibility of generalizing the W-H technique to deal with three- part problems. For such one-dimensional problems, however, there is not generally any "weak interaction" approximacion that one can make, because waves on strings do not decay in the way that two and three dimensional acoustic fields do, so that in a sense one is always con- fronted with the strong coupling situation. Nonetheless, a number of interesting points arise in the string problems which have direct comparisons in more serious three-part problems. Consider a uniform string of line density Py in-»o inR, ¥_(s) ~ ~ 4 y(0-) which we also get from XY (s) ~ (; y(O-) + ...) eae (7.4) using Watson's lemma. The same differential eyuation holds in 2 < x -k oO + i because of the exponential decrease, exp{-k) x) of y(x) asx* +, However, YX, @s) does not now have algebraic behavior at infinity in Ry. To find the behavior we write 35 a RNR rancor me a i paso A i Seactinceas z i in CN an a od ES ORS wl ore SE in Bas we. Coat sais is sboey Saks) a a ER SOE Need ORS TSR T ROT ET | Rb Rat gue, Cay, NSE TOTS EST IS TESTO AUST | ARI ARE Cece ane eto nee. ¥, (8) i y(K 42) ofS 42) ay ie) = te y(x +2) ef®* ax fe) z ate [c [y(2+) + X y' (Lt) + ...] isk dX fe) ist . Aa ee lee y(2+) + 0 (: (7.6) =) s* ' as s>o in Ry. Thus ¥,(s) has e:ponential behavior, exp is&, at infinity in R,- From the differe tial equation we have (s? - K2) ¥,(s) = [-y'(@4) + is y(24)] ef (7.7) from which (7.6) is cbvious. Again, there is a pole at s = k eR, unless pe ~y'(24) + ik, y(L+) = 0 (7.8) | For the middle portion of the string we define Q Ya(s) i= { y(x) exp isx dx (7.9) ° Since the integration is over a finite range only and the integrand is Lounded and continuous, Y,(s) is analytic in the entire complex pl.ne (the Fourier Transform of a function with compact support — i.e., vanishing outside some finite range, is an ENTIRE function). As 38+© with Im s > 0, P.-) Ve) = J (y(O+) + xy"(0F) +...) e (2) See gene) (7.10) because when Im s > 0 only small values of x contribute to the integral, so that we can expand y(x) about x = 0+ and also extend the integration : 34 : | ) | | | an deh. ce snokwaisod tatsnonogss ani wy wut” ova 0, aokonui fans seas add: aoe Dt a ~ tik i ake ee ee a Cera eb cette el a a sia, aa) tog. Wek Sella aaah amine ot on . i= sai ae + he ih vale _ stn sone "i ety ahora x bd er pia e458 GA deta) Barris on ek jeaue if ‘aenned eo haa . 40 CW = xh Mace: § Chay vie ‘ oan : ‘eae ne SO a a: ree a PCO TER OO AS oer wees up to ©. As s + with Im s < 0 we need to write x = 2 - z, to get R Y,(s) = exp (isk) J y(2 - z) exp(-isz) dz ° and because of the exponential exp(-isz) we can expand about z = 0 again to get Rate) cap tee (i [y(-) - 2y"(2-) + Je 8? az ce) = exp(isf) {- y(2-) + oc (7.11) Thus the entire function Y,(s) is algebraically small, 0(s7!) or smaller, in Im s > 0, but exponentially large like exp(is2), in Ims < 0. The differential equation ayer +k‘y =0 dx2 1 (where y is the total displacement in 0 < x < %) gives (s? - k?) ¥,(s) = [y"(@-) e° - y"(o#)] sh - is{y(2-) et - y(o+)] (7.12) and confirms (7.10) and 7.11). Further, Y, (s) can have no singularities for any finite value of s, so that the right side of (7.12) must vanish for both s = k and s = “ki, giving ee % _ y(or)= 0, (7.13) [y'(a-) e@M1* — y"(ory] - ak, fy@a-) et ty'(a-) ei? ~ yt (ony + tk, fy(t-) eA” - yor} = 0. (7.14) 37 | | | Fa eee RCN GI NOON BIN Sonprecisemne inersrenseens et Stas meomreena cee nee aso ee eT MOR IDE PA a Ty oe) Ses SE area nant Gr Lae loa ety ake To make the algebra minimal we choose a definite set of conditions at x = 0 and at x = &, namely the simple junction conditions that there ig no change tc the total displacement or to the slope at x = 0, x = 8, the tension in all three strings being the same. Thus we take y(0-) +1 = y(G) y(2-) = y(R+) y'(0-) + ik, = y' (0+) (7.15) y'(2-) = y' (£+) If ko # k, the set of 8 equations (7.3, 7.8, 7.13, 7.14, 7.15) has a finite solution provided sin k,& # 0 (7.16) (excluding resonance of the middle portion) and then Ore (7.17) etc. This of course completes the swlution for these simple problems, for now that all constents are known, Y,(s) is known frow (7.7), Y_(s) from (7.2) and ¥, (s) from (7.12). A method which shows how a generalized W-H equation may be treated ignores the detailed solution (7.17), and instead eliminates the uaknown constants from (7.7), (7.2), and (7.12) in just the same way that the corresponding constants were eliminated in $2 to get a standard W-H | equation. Thus we write 38 j Te i 7 wind tare eet edie ge me ‘oaMhe sei # Ai “* hy Le eealsoia otawts, gears a nodsoLos 44), sacle ‘pedROS Me, ant i i wSiND- i awa a , ¥ ae in wamaqsada ie ails, | oehbet weil coe ofa C208) moat | sida a oe sobepans, tie nyshtasecns jee erate one toe & » i (e? - k2) ¥, (5) = eS [y"(2-) - te y(2-)] - [y' (G+) - is y(O+)] = — (si =k) ¥, (8) = (s, =k.) ¥\(s) + i(s - k) on use of (7.15), (7.2), and (7.7), so that i ¥,(s) + Y_(s) + K(s) Y, (s) oh ome K, (7.18) where the kernel is, as in ¢2, s* - k? eG) ees (7.19) . ef o4ke o This is the required generalized W-H equation, a single equation for three unknown functions Ys), Y_(s) and Y,(s), given the kernel K(s) and the forcing field (8 + kT Equations of this kind have been considered by Noble [5 p. 196]. Methods exist for solving such equations approximately in "high frequeucy" limits in which the finite part of the boundary is many waveienachs long, in soze appropriate sense. Generally these methcds rely on weak interaction between the ends x = 0 and x = 2, though as remarked before, the ends are rather delicately coupled in cases such as the resonunce of a finite open-ended waveguide, and metheds have also been developed to deal with such cases. Here, because only pole singularities are involved, it ig possible to solve (7.18) exactly. BRS eA BTR CR RSC SRNR ante mpir encom mvwemeentate eres ne a en pee de a We carry the analyses through in a general form as far as possible, to indicate the procedure which has to be followed in more complicated problems. First of all we extract a factor exp is% from Y, (8), writing Y,(s) = exp(is2) 2, (s) (7.20) so that z,(s) = 0(s_') at infinity in R,. The necessity for doing this will be apparent in a moment. Then write K(s) = K, (s) K_(s) as usual, and divide through by K_(s) to get et8*2 (5) ¥_(s) KG. | eyo 1 (s + k)) K_(s) The second term on the left is analytic in R_ and O(a!) at infinity there, the third is analytic in R, and 0(s7!) at infinity there. The term on the right can be split in the familiar way as (s + i) Kacey 7 cae CnC) (7.21) and we also make an additive split of the first term in this way, isk e® Z, <3) aiXaCe)ienk = U,(s) + U_(s) (7.22) using a general theorem to be given in a moment. G, (s) are each CQ(s7") at infinity in R,, and we assume that is also true of U,(s). Then we have U,(s) + K,(s) ¥, (s) - G(s) Y..(s) = G (s) - KG) (s) - U_(s) 40 hi) “oti: a6 0 as: a at ak fom ak Rat ase con if “5 te x ‘ a on _ - v 7 es eh no, i and each side is the representation, in Ry or R_ as the case may be, 1 of a single entire function which behaves like s ° everywhere at infinity, and is therefore identically zero. Hence K, (s) Y, (s) - G, (s) + U,(s) = 0 (7.23a) Y_(s) G(s) - Ks) ~ U(s) =0. (7.23b) Now return to the generalized W-H equation, and this time, instead of dividing by K_(s), we divide by exp(ist) K, (s) to get -isk Z, (8) e X¥_(s) ase eo ist — + —————— K, (s) K, (s) +e K_(s) Y,(s) = (arena) (7.24) a + The first term is analytic in R, end 0(s7*) at infinity there whil,: the third is analytic in R_ and 0(s7') there only because of the factor exp(-is2). If we had not divided by exp is& we would have been left with a third term which was analytic in R_, but exponentially large at infinity in R_ (see Eq. 7.11) and the finction theoretic argument would not go through (in particular, the entire function would not be zero, but would be exponentially large in R_, and there is no general way of constructing functions of this kind). The function appearing second on the left in (7.24) has mixed properties, so we again try to split it as -isk e Y_(s) K, (8) s v,(s) + V_(s) (7.25) SARIN Tae rere nim receet nee taetonn ake mt owatonna ee ras 4 a aed with V,(s) = 0(s"") at infinity in R,. On the other hand, che function EOF PRG Pag an occurring on the right of (7.24) is a) function—but it is exponentially large, like exp (s,%) in the upper half-plane, and so it is necessary to make a split ee (+k) KG)” Hts) + H_(s) (7.26) with H,(s) = 0(s-?) in R, in order to remove this exponential increase. Now we can split (7.24) into(+)and() parts, each of which is 0(s-*) at infinity in R, and each of which therefore vanishes identically. Thus we get | H_(s) - V_(s) - e *™* RK (s) ¥,(s) = 0, (7.278) | Zz, (a) : | Etey * YaCe) - He) = 0 (7.27b) The equations (7.23a,b; 7.27a,b), obtained by the W-H argument, are not, in general, solutions to the problem, but they constitute a pair of integral equations which are in a form suitable for approximate solution in the "high frequency limit" (i.e., for ke >> il, k,& >> 1 in general). To see this we have to use the general formula (see Section 10) for expressing a function F(s), analyede in the strip D and with suitable 1 behavior at infinity in D, as the sum of functions analytic in R, and O(s') at infinity there. If F(s) = F(s) + F (s) where the functions are to have the ecated properties, then = nN ii aay eg Oh ee oc eee ini 7 seni et ee ed ui ste “' cae ome eqnseagte WH edd yd. oon atbade tts Lae iy eeyers sive r : te stay & wyustténag Youle aie smabitens sib ca seshouson "pena tioe sanntnorgae rel side ttue rol & at hae “doit SAibd ; vilawenaa > ES OM, a Me Ker Ld ‘tet ec! ‘tail i tae sila , Se = P,(s) = 3 EO) at : (7.28a) sieeh F(s) => | tia, (7.28b) ee where in (7.28a) the path runs from - ~ to + © in the strip D passing below the point t = s while in (7.28b) the path passes above t = s. Applying (7.28) here we have, for (?.23b) and (7.27b) in particular, etth Y_(s) 1 2, (t) G_(s) = carta | = aGIEscne = Q (7.29) —y— Ze(aynrd igh itlvest wie (e) “HY (s) +—— K, Kt) al K@yct = 6) dt=0, (7.30) —— the forcing fields G (s) and H, (s) being known, in principle. Clearly (7.29), (7.30) are a pair of coupled integral equations for the unknow functions ¥_(s) and Z,(s) which determine the reflected and transmitted fields in x < 0, x > 2, respectively. Once these are known, the field ¥,(s) in the middle portion of the string can be found directly from the generalized W-H equation (7.18), for example. To see the structure of these equations, suppose that the integral term in (7.29) were zero. Then we would have ¥_(8) = K_(s) G (8) and noting the definition (7.21) of G (s) and comparing with (3.11) we see that this Y (s) is precisely the field in x < 0 if the wave exp (ik x) were incident upon 4 semi-infinite, rather than finite, etring to the right. Similarly, the situation Z,(s) CS K, (s) H, (a) is the solution for a semi-infinite problem of reflection at the 43 aes 4 pide eovang ian ‘edi AAAS: ik ti i iia > etehetoug et ane.ty bas gHes.«) mat aye we seit iat th went. ery os Abehraven hayek: ne te: ton & om (OL ¥, e © Weastoeges bars haa liey ‘edd nora diod ites ), A ‘bea fy). % . bier’, ‘enhy eon aan acne wos ‘Leite eatbe nba w etait: eine wit: thes ere eh Ba oufoion, aba e ith: bean = 1 elects at vee. v) ‘Wo esioR ae estan — Wea ) Soman wd Samy Faas wD in vail a srg ae ons Hom or 7 yet | Suki bbey asimaht \ceeee “CRY a Ud uy ta 3 Chie): tate Bitsngeny aw 609 2) ene oly te a > gph beKosh ene) by Lis. < “woke te we MANES Mit veLS4H9 wateiaclane 4 ew fey «, ee wobiauste, sae Ms teste yn ita Mi 7 en oy et), te ibisaatses ‘Ve wnddeng . ) rite ihe: x0)) eerie ott a » eee hte ATES Aree de Pan aeema parton mete Be Se ES RRS GAY BCD DAN TRY PE GER RE ASDIT Ute Qe ORNS RIE UIC Me YR RSAC CREATE HORT LONER AN Me foe SOND STM SINR junction x = 2, the left hand portion of the string now being regarded as extending tox =- ©. The integral terms repzrese~* interactions between the ends x = 0, x = 2, giving rise to a sequence of reflections and transmissions. Methods have been devised for dealing approximately with these interactions both when they are weak and when they are strong, as in resonance situations (see Noble [5 [Ch.5]). Here it is of course possible to solve the coupled integral equa- tions completely. First, however, we show how they may be uncoupled when the kernel K(s) is even in s, as it is here. It is then possible to work in a strip D which is completely symmetrical about the real axis, so that Y (-s) is a(+) function with the same domain R, of anelyticity as Z,(s), while Z,(-8) is a) fuaction. It also follows from Sec- tion 10 that K, (-s) = K_(s) for seD. Consider then (7.29) and (7.30) fer seD, and choose the path —/\— above s to be a straight line from — +ia to +> +ia, a > 0. Choose —__,—to be the image of this path in the real axis, going from -- -ia to +™-ia. In (7.29) change s to -s; the integration path can bs chosen to run above both s and -s, and so tk Ye(-3) 1 f et” 2.(t) F,, (s) * ona K (t)cetsy 9 7 9 (7.31) eel aN G (-s) - In (7.30) change the integration variable from t to -t; (-t) runs from tia toto +ia -~ag ¢ runs from -ia - to -ia +m, sa that the path for (-t) is the same as that in (7.31). This gives Z,(s) 1 ¢ ety _(-t) de { 1 | ~H, (s) + KG) By o nL J K_(t) (1 (t+A) a 0 r) (7.32) — ; | 44 a s Asaq woOhsetgesol ws js- 09 @ sganis (2.1) ai .ety "+ 09 ato oe wend a eG Soe ,4- bie « diod avods tur: of saneds of peo . ; a _ (De ¢ 3k, ‘ : bond? 7 . op al i | a jt tat 8 om Een' i (it. i 0: 23h 7 a 5 a oF Ga ta} 9 esl ces é seus (i) p9~ 69 9 Mur]. oidedtiev molvaxgezct od? eynads (02.1) oe wad Gach om , + Bh 02 Oe Bl~ tre) bunt 2 che ~ Bit og =* ah+ cond | " wewhy utay sf2E,X) od Inds eo nee mis wk {2-7 203 ane | Ay GED O° ears |e 55 ba ie A et Pa a a Now add (7.31) and (7.32) defining the difference : 3,8) = 2,3) ~ Y_(-s) 5 to get ick Dt) dt eG hey 1 [G_(-s) -H,(s)] +E K, (6) ore j R(t) —— =0, (7.33) { while if we subtract (7.31) from (7.32) erd define S,(s) = 2, (s) + ¥_(s) we get S,@) f ert® s(t) de KG) ai J Kye +s) ” ele [=Gi(—s) He) + Facey) <7.34) so that (7.33) and (7.34) are a pair of similar uncoupled integral equetions. Let us now look at the forcing functions in these equations. The additive split of G(s) in (7.21) is simple, giving i tO + (s+ k.) K_(-k,) (7.35) | cae oe eee o2) eikikg (tes 5) For the function H(s) in (7.26) we have to use the Cauchy integrals, which give te ith EMO) al (+ = yK@e-.s) 4 (7.36) —— i 5 We cannot complete the contour with a large semicircle in Ry» because the factor exp(-it%) is exponentially large there. Instead we complete the 45 a ASTRO Rater ir ec amos em nentazeecremcaR MENT racer tare eerretameum ne testaonn paerenewomsecerentttiscomete-beswnerpusarca arsanmet nonctrrcmenattrateoner renee amtoanee ea erent tees ener contour with a large semicircle in R_, along which the integral vanishes because of exponential smallness of exp(-it%) there. To interpret the meaning of K, (t) in R_ we write it as X(t)/K_(t), so that itd oe feb Kae Hy (S)eorG (t +k) K(t)(t - s) ac a) Sw, The pole at t = s lies outside the contour, K_(t) is analytic within the contour, and t- k, (t +k.) K(t) = ea, (t + k) so that the only singularity within the closed contour is at t = -k,» and therefore i iethi* x (-x,) nadie H,(s) = Sri (7274) Eye k, = (e+k,) » (7.38) aga) oie) S) -isf ik, 2 ie ie) i ist ik, POC) Fin (ont) we) GBie ELC? peoblee. 62 v8 This instructive example shows the importance of removing exponential growth at infinity; although the original H(s) is analytic in R, it is not algebraically =mall at infinity there, and use of the Cauchy integrals shows how it can be split into H, (8) + H_(s), with H, (s) = 0(s-!) at infinity in R, as in (7.38) and (7.39). Coming now to the integral terms in the integral equations (7.33), (7.34), we can this time complete the contour with a large semicircle in R,, because along that semicircle the factor expit%) will be Si ovicliibers dbs iin else Leese att igiaitte i ay CMe A a SRI Ketpareh eta-doziy quote. kMt ata batdion Mme md eee BangveRt OF . vaste) (S9a-)eke Xe auted tae Eel ae) * [24 : ni erm ya ; a a, CH) 3h Op ae Sk ede ot 8 ah Cae . . amet © | 3 ae Vee: } | (ti,.%) ; an {n= 5) ie ae ae I J i Badahe obey tah Gk (5) ¥ yewsaaes Bhd ohterye raniit a ir ag " ts 3 | r. , ad f F (2450 je (2a. ota | \a / or 7 2. Sao) Nwosaod beadtis ade BG ab be ee.t) ! aa ee Pe ee ~ a <2 ~~ 1s ARPT LEE ep or at a i (88, 0) ween . Cee gatvoncy 20. gocetreqal Ads sveits Sivnects anh joupieg aka. ; deities ee Capt abit aD the Sete hak ee iors ‘at yaaongee 5 as 10) d0ts bon. evans yrineiat ap. biker: ithentaeligsi zen ab 2h a ak dzdlw: ate) +. Cad.G odet S¥kus a4 dos Sh wed mies elargwont oud. yi COZ. bow COIS) wh on Pe ykee hen oe Peo - be ' off2.0) Waedsicpe Lesgugat O42 he eens: toSeaiad ale a en nia, Speloloen oye. 4 doiw xuerass. uit) mS Luray 4 a Di (Pique »: Sook wdd eked adincn: daily duome inated. gS ale | 3 ee wie Obatened, og BE eb {ene a doar. - mama TT Psiealuhe abies exponentially small and the integral along the semicircle will contribute nothing. The functions D, (t), S 4 (t) are analytic within the closed contour, and the function 1/K_(t) = K, (t)/K(e) has a pole at t = k, only. Therefore (7.33) gives D8) ky = ky) enna zc) * Tia tks) DES (G_(-s) - H(s)] + Ey (7.40) and (7.34) gives S.(s) (ky - &) eM? 5. (k,) —_—_—_—_—_—-o OS = «OO pie ee) it Rae. GeereeD (7.41) The unknown constants D,(k,)» S,(k,) are found by putting 6 = k, in _ (7.40), (7.41) and then we have completely determined the functions D,(s), S,(s), from which Z(e), Y_(-s) can be found, and hence the whole solution is determined. While there are no difficulties of principle, the algebra is very tedious and there is no point in giving it here. The only remaining point of interest concerns the inversion of the Fourier integral for this three-part problem. We have : ‘ y(x) = al (Y_(s) + Y¥,(s) + ¥,(s)] exp(-isx) ds (7.42) with the integral along a path from -~to +™in Pf, and we recall that ¥_(s) = 0(s"") as |s| >> in R_ Y (s) = 0(s') as |s| > ink : ah 2 (7.43) = 05) as |s| += in R_ ¥,(s) = 00—) as |s} +> ink 47 : finttneitinsiy. te ae oR. awe A * s. S ankiova: x Sebat digs oe “naive wf handoranaty ¢ i Enaahghies dyad aw ag “ang eeannil bia “vis oh ao: @, bi , ; os wantticgthaab on sep aorta ae, i. rm : os a * i ~ ROIOVRR 1S Kaboom ThidtarOh Po geiko ; ies _ eee as ‘rea baengy Yaa (As) * ; tik trad dyigns feat aaa: Ligoks ome bows eOont eh re atin ¥ Hone, ' fat ee Le (t8,%) ® ad Se i | ee # at we teh ae PSEA OB i gatas ecreana ven cmarnrtieceinyretee cen er enere te ame ee that Y,(s) are analytic in Ry» respectively, and that ¥ (s) is analytic everywhere (an entire function). For x < 0, deform the contour into the upper half-plane R,- The contributions from the large semicircle in Ry to the Y, and XY, integrals are zero, because exp(-isx) is exponontially small and Y, and Y are at least as small as s ! at infinity in R,. Further, Y, and Y, are analytic within the closed contour [NS ;, and so their contour integrals vanish. Thus for x < 0 y(s) = = J ¥_(s) exp(-isx) ds (7.44) which we may try to evaluate by completing the concede in R,-—theugh note that we know nothing in advance about the behavior of Y_(s) at infinity in K,. Ifx>2 we complete the contour with a large semicircle in R , along which, elthough Y, (s) is exponentially large, the product ¥, (s) exp(-isx) is exponentially small when x > 2. Similar arguments then give yx) = 5 J Y,(s) exp(-iex) ds (7.45) Finally, if 0 < x < 2, close the contour for Y_(s) in R_, that for Y, (8) in R,, and we find that the integrals of Y (s), Y, (8) vanish, leaving i 1 a y(x) = | Y, (s) exp(-isx) ds (7.46) Because of (7.43) it is only possible to close the contour here in R_, ¥, (s) exp(-isx) being expenentially large in R, when 0 < x < 2. These arguments apply generally in three-part boundary value probiems. Here the integrals (7.44-7.46) can all be evaluated by 48 * ey = nt staat shew #43 gt at, a esd settee att sae ‘Ehebaacongts ak dunt) genad otee at8 | - a : ciel ‘ 2 wt esiabiar eae eer | seast 1 bet: se wands or bees . we Dywosnes bavolo ots aseane aie ac i ¥ hes O > ee eat? porn d (08.0) eb (net-)ene (ie) ¥ \¢- ; Aguada find aut yoo ait etre: sathancs wd aasufsve 2 a9 fap. Ras solvenod rod twos Souevha ot pobtiog aun F _ eS ee eistaee sg085 o itty imasaes. ads sisiquas we Asa ™, : Pauieoig: avi 4 MER ae tjneyeexe sk te), ¥ tigate | " - el . eonmiaeeh selale 680 < & pstiv Laws vfles aeoneae ek fvat ¥ y j a | | ab ietdese tt | oe = Gg gaye oe ae (a). ¥ t61 thodaus oi septs .2 > = +6 %4 ettenkt (a2, w@)-¥ Bo elergnrel ody “guls ead wd’ Drv «yf nk (7 - => ¥ (d¢.%) De > O'tibie 8 Bt sah CrEaksannoynn: pated (xnt-)qes he alan pines Fenq~eaNts ‘a2 eLtaaensg Ligh ame SROdT ea Ere ly aes COALS h) Ch wiasyeten ony eal .oimkdorie wa residue calculus, since only simple poles are involved, leading again to results which can be confirwed using elementary methods. The purpose of this section has not been to solve a particular string problem, but to illustrate how the W-H method, applied co three- part problems, leads in general not to a solution, but to a pair of coupled integral equations. For an even kernel K(s) we can decouple the equations, and deal with a pair of similar iadepencent integral equations for the functions Dts), S,(s). In our case the integral terms cen actually be evaluated explicitly in terms of unknow constants D,(k,)s S,(k,), which can then be determined by setting s = k, in the integral equatiors. A precisely similar situation exists in "near-resonance" problems, such as the scattering of acoustic waves by a long tube, open at both ends, at frequencies near resonance. There it is argued that. although the function K (t) in the integral equation (7.33) has branch point singularities, the dominant contribution near resonance comes from a pole term. Well away from resonance it is anticipated that the dominant contribution to the integral comes from a branch point singularity which represents the rather weak acoustic interaction between the ends of the tube. The functions are expanded about the branch point, and the integral term can then again be evaluated (approximately) as the product of D, (-k) say (where -k is the branch point) and an integrul which is expressible in terms of Whittaker functions (which can be further epproximated in most cases). Again the constant D, (-k) can be found by setting s = -k in the integral equation. 49 ( Ne s sboiiyge is eauntois dete: : ‘ Mebunt aed « ce ave. bo ens ban “ia cs ans i 1 na “ bus ties I eal a sit weet seaxdeuke a3 dese mat “ey arte? in sgeaat ofa ‘ben tne ths iu a ic 2 Gi 2. 8 soeean02. oon 30 ‘ging? wh ela lakhines: ars ene ‘am adn _ nage ais ‘gat ed Raevaie. a ore: oid s biugae at hater Mashsay & Best notahepan’ 3 2 em ae aqaewe: aga, for .0% payee FBS Fay Haein a2 GF nh SOE Oa yi ‘HORE OSA OR Sy wax DOES REDE DAG: tans ENS Hi ' Rh 2e0F Rete nbakeen at 33 SOGKAGRSS GT) yore et it A ‘ _ fi He Mine a ES Bae te eek 4 ay ia are MRS aRW Ret) shsesiod los Yarieer dd ween nasa ‘ talog flabesd Shs cil k ibis RS EN Ain Hd ee (Pies aye) bokeh ava Jad BLASS osAS lens was PS RTS BE ates Aicagageak We hes Congo ds the ¢ ae WS DORM) AOR oad Tens hay Bh GMa tee tab Moai an pie TRG EY Foie i] a ; “0 aap (#-) 11 HGS waht tase, SROKA RUDE LAR A oH yng ath APTI eh a NOMI 5 as FT at FEDORA GRE Rete eirsengsae le ones igen cetera eee ele ea 8. TWO-DIMENSIONAL HALF-PLANE PROBLEMS We go on now to lock at a simple problem involving a half-plane embedded in an acoustic fluid. In the first place the fluid will have no bulk motion, whereas later we shall allow the fluid to flow over the half-plane with uniform subsonic velocity, leaving a wake behind the plate if the edge is a trailing edge. The issue we want to examine is the following one. Suppose that the plate were infinite in both the positive and the negative x-directions and a wave were forcedto propagate along the plate with some prescribed frequency w and wavenumber q> the wave- Number q being real and greater than the acoustic wavenumber ko = wie, Thus the velocity in the positive y-direction is prescribed in the form v(x,t) = Y, exp(iqx -iwt) . (8.1) Then the soluvion for the potential $(x,y) in the fluid in y > 0 is (dropping the factor exp - lwt) Vv o(x,y)= - a exp(iqx - Yqy) (8.2) q where a ey Pez vi ne 2.2 Yq (q k)) ’ (8.3) for this makes makes 6 + Oasyrt+o ) and makes = (x,0) = Yo exp(iqx) . The field described by (8.2) is that of a subsonic trapped surface wave. No radiation takes place across any plane y = const. because the pressure, 50 “A LAR 8 (distal ge 28 Yan, ita. iog aM! Ad ay te i4 Coxe: Meh te hibacllle eS Se ; bite sy i ae ‘i shasaiy . eh tdet Sind eal state meiian dite gpoalowt fan ¢ is v ai. 3 ‘ yobitenrd & Gk ahs: eee Se sh ix ; bs ete ‘sz t agoquue wie, atwollo® bili sete jy Pomgen ed bee hunawew J Mpa Jae as easyer aaa bw uIRty etia j 7 asf rors bom Ise? goted p veda truolkay ace aut ) - { i ¥ hin ,Gd Si9 matt sl ey Onkegodh) ato 3 As 8: 2 geste MS gS oe eae a tae aie Lika wey eoksalbey ot Cavite. Ge: “ae , ia aes | aauawes p = piud, and the velocity 9$/dy are 90 degrees out of phase. The energy is locked in a thin layer oi thickness o(ya") adjacent to the surface, and none escapes as sound. Suppose now that the surface is semi-infinite, occupying (- = < x < 0, y = 0), all dependence on the z-coordinate parallel to the surface edge being excluded. Let the surface - ~ < x < 0 stili be forced to move with the prescribed velocity v(x) = Ve exp (iqx). Then (8.2) cannot be the solution, because it is easy to see that since 0o/dy is prescribed on y = C, for some range of x at least, 4% must be an odd function of y and since > must be continuous across the extension (y=0, 0 < x < ~) of the surface, $ must be zero there—whereas (8.2) is not zero. Clearly, no single mode like (8.2), nor even any discrete set of modes of this kind, is capable of making 0/dy have the value ve expicx on y = 0, x < O and of making ? = 0 ony = 0, x > 0. The E solution for $¢ must therefore contain a continuous spectrum of modes like (8.2) with all values of the wavenumber. In particular, it must contain modes with wavenumbers a, say, with a < ky and for these | modes the exponential decay exp {- (q? - ke) 7y} must be replaced by | oscillatory behavior eo tte = ee the choice cf + 1 (ke - a2)2y rather than - 1 (k* - a?)Zy peike dictated by the radiation condition, that the phase factor exp {1(ke - a?)%y - He) Be that of an ovtgoing wave : as y > +™(when y < 9 we take ~ i(k? - a?)7y). Energy is radiated across a plane y = const. by such a mode—and we say that the energy which was trapped in the subsonic mode (8.2) on an Anfinite plate has 51 ~ gequce oateivi laksa Wd aa P ey ide a , ar ket tering whoahendae a ‘no Saerbigiey Loe 5 18 oe ie od ite Ry + oo saahae ei. 48d isu ‘aia — TN eam sale ee enkeo law Gai iseadyy. ta bei ‘i 1 he steeax ot ate Ay oande aan sad on ren, et ae Saussiad, gE silo ott we 000. wa ; mK : we sau e fens wh * 36 nine ‘he Ms AGT i any as bodisz0eng 4 i : - natwnscas aa: wero 6 Pion erter ee hei, 4 Fa) oaks bak va a wi i | Gay Beesodieerody ap ao avan t Vem Weau gits tn ve =a » a | Sinton yas nove foe 4 $8 ay ont i eee gixnie 9 ches | | madey ony weg vbhas hieesen Sa a Pegaso: ab nals “ebrly * “F si? a 4 - QO ay, aii ‘Oe » want ate AS Ta oF it oe WN as bea ta | Seta _ayowitrne i.e mentgas sedaneny peur, ot sil a wh TI6q th.) heathen ibis Sar 2G. sbivtn t dy yt eo ite ast Yad hie , a 2 H sha bse a ve oe mene 6b ‘aha tn “se dosnt od Dice” wy a ay “ae Wonk (Mensagem: ik ‘athe vit*a ” oe *. 25 panies aefit alt Bah é yes iste, pa ine re fads aBkatbGeD gad anihe~ a8 va Peahsath sate, a sey gabesine or Aes vu or ifsrob mig ta DSeandbwr ah gan ty cs . #4) “aes Be 0 2 aarti Chien wd darts aida Besa a’ ‘fue ae en stat ester. sh nes beth a] = AP eee poe 5 PE ERS ca Mtr rican been scattered into other subsonic modes and also into supersonic e eis radiating modes by the discontinuity in the surface y = 0. To analyze the process of scattering, or wavenumber conversion, by the plate edge, we define a scattered field ¢ by Vv oO Pr rotal a exp(iqx - Yq”) +o (8.4) for y > 0. Because the derivative 0 /d3y has the same value on total y= 0,5 Oe eeat must be an odd function of y, and so it is enough to tee eer EE TERETE consider only y > 0. The scattered field ¢ is a solution of 3? 92 (f+ Fe +8) ¢20 (8.5) with =? ony =0,x<0 (8.6) and with v o- = exp iqx = 0 ony=#=0, x > 0. (8.7) "q This is a typical two-part mixed boundary value problem which we may expect to solve by the W-H technique. Two further conditions are needed, however, to get a unique solution for ¢. One comes from con- ditions exnected to hold as |x| +o, and defines the domains R, of ehalyticicy of half-range transforms and the strip D of overlap. The other comes from conditions at the plate edge, x = y = 0, and determines the behavior at infinity in the transform s-plane, and hence determines the entire function arising in the W-H method. We shall leave the matter of eage conditions until we need to look at it in detail. For the moment we just assume that all functions with which we deal have at most integrable singularities at x= y = 0. 52 As to conditions aa x + + ~, we give K and gq small positive imaginary parts, k. = k, + ik,» qs. 4.20, (8.8) and then (8.7) gives @ = O(exp - q,x) as x >+° y= 0. Away from y = 0 we can expect that as x ++ ¢ will take the form of arn outgoing cylindrical wave, -. ; o~r 2 exp (ik r) £(6) = O(exp - k,x) (8.9) It then follows that ail()functions linearly related to ¢ and its y~derivatives will be analytic in R : ims > - min(q,,k,) (8.10) As x +=—, contains an exponentially growing part which we have total split off in (8.4), so that » should behave like an outgoing cylindrical wave, $ = O(exp - k, |x|) (8.11) as x*+- ©, Then ail© functions will be analytic (and with algebraic behavior at infinity) in R_: Ims<+k,, (8.12) and the strip D is D: - min(q, »k,) at niim od Ele etbrs.. i ha a i iy i , ‘I i i i i 1 av 4 { = > , i H { rear ts ee. ie ‘ (oka : f i “ 7 J = J A 1 te ‘ , - gt a is ri iy 7 ' } i i rf ' Ce Sas Raia rer cccinerenes a a ee For y > 0 we can apply Fourier transforms to (8.5), and can integrate the 076/dx? exp isx term by parts twice with no contribution from x = + © provided s lies in D. This gives 2 (5 28 ") o(s,y) = 0 (8.14) where 1 Y, 7 (s? - K2)? : (8.15) introducing the square root function whose behavior in the complex 8-plane holds the key to many aspects of acoustic diffraction and scattering processes. The function Nea (s - Kee +k)? has branch points at s =+ ky» and branch cuts must emanate from these points to form a barrier which mist not be crossed. We can either make a cut from + ko to ko» or we can make a cut from + ko to © (in any direction) ang cut Tae -k, to © (in any direction). If the values of (s + ee (s - kee are specified at any point in the s-plane (not necessarily the same point for the two functions) and the branch cuts are fixed, then a unique value of (s + Ko, (s - ee obtained by starting at the given point and moving to any desired point without crossing any branch cut, and insisting that the function change continuously from its initial value. Figure 4 gives various possible choices of branch cuts. In eat addition to the choice of branch cuts we shall take each of (s + ks i 1 to be the branch which behaves like + s* (rather than - s*) when s is large and positive. Now in our problem we know that $ must be analytic in D, and since the general solution of (8.14) is 54 ta a Te palbaudbasine ‘oy iba euliaos wits ab josvtled pean, nokogaud, t602)-ohmaRD, whey ra bas Wmkieswiih otsauove 20 atosqas’ Crem of wax bib Seihod a © ¥, : eo ity t= ae 36 tat 5 t+ ah aa : ! it etobeg daaaig end Cat a} C2 #} * Ashionw’s 9a reskin Fabvsed sod os tice AREAS soy? Sd BAND aw lye fonaad bas! i nate ea sf eee ys 8: Selene: verti bi Al: abide on 200 tae ‘ ; : eR i" max? duy eos Giabaosc th wna nh Oy ‘ayy . @ ; i | ik) . eo ies 83 i 4 POE hat * eh ah a #4) 4o paudew olla 22 dena: saree er) won, a @ ails ylixsuesoon dod) Suatarcs aria ud ero, ie Aish sabhauege ory & Hey (bextl ote bays voneyd ty bon (enokt agus alla Pree ve. antog ) a ak ewe one ay aes qd bsnks3ds wt oo a 4 y ve de eutey I Tei fted 2 wos? eke sans a ap hsdaet, ana naif antoarend a at .atuo apes Io sedtods oldknpoy enchead! als, A wsuget souiy 4 + ‘iat #) i¢ toss suka Made an dogo dead tc sated ata o2 pokssbhe P ee ony (as yet \osh92) *, 4 pat i sayeH yf pee ciicteatiad ots sa hdl Hones haw 4% nh ohslane od TEiay PA | seeMe @(s,y) = A(s) e 8Y + B(s) e's” (8.16) that will only be pessible if the branch cuts from + Le do not enter the strip D. Thus the cut from +k, must go to infinity above the strip, that from -k, to infinity below the strip. No further specification of the cuts need be made at this stage, because we shall find a solution . for s in D, and the values of Y, are already fixed for s in D by the 1 a requirement (s + k)? ~+s? as 5s ++ and by the general location of the cuts. We can now see that 0 < arg(s + k)) <7 (8.17) -7™ < arg(s - ky) <0 for all s in D, and therefore T + T _= Pon 2k oe 2 < arg(s ko) ] or equivalently Re Me > 0 for ail s in D (8.18) which is the essential property of ¥,. It is possible to choose the branch cuts so that (8.18) holds throughout the entire complex (cut) plane, but there is no need for this since at the moment we are concerned only with values of s in D. Then the only possible form for B(s) in (8.16) is B(s) = 0, otherwise $ would be infinite as y++o, Thus $(s,y) = A(s) exp(-y,y) (8.19) 55 ‘sb00 300 3b, ahi ach es acd atten a ARR A ray tn 2. il ac “ReRiviCn 6 batt reba: ie ‘phokon yet abit, hao ha pms eth tbh View: (BL.6) Tat 6) 1p sgt 9s ihm os Way W40arle.0d stiteodq eb 31 ye yariqetl rina od, ab . (awa) wollonoa) wxtana ana; ub rigirn ny abled end Soa a ava bertgones map yy srenon ota 1 Sale abet: ae ‘bene ‘aa. Br: otede id eo: (oya 402 tro} Sidbbnod xiv ada ‘aaattt if at ry vf ‘eta wast et oe wan eat bot tak ad bie: Es) wapietasieo A Gal (ie) te, ibaa wn * oe ee Rn a a es or ae ey ance a = CE RRR IN SOS OOF PRAIA RATER RET EYE RT SEAR ENED NOD Us ee renncrmcom aarntne | AAs ne ee eee and the original radiation condition of outgoing waves is seen to be equivalent, for k, > 0 and s in D, to the condition that $(s,y) + 0 as yr+o, The boundary condition (8.6) involves $'(s,0), where the ' indicates d/dy, and therefore we differentiate (8.19) to get the pair of equations (equivalent to the differential equation plus the radiation condition) $(s,0) = ©, (8,0) + (8,0) = A(s) (8.20) @°(s,0) = 0) (8,0) + 6'(8,0) = - yA(s) where, for example, @ (8,0) = [ $(x,0) eS* dx , -—-@o ) (8,0) = I, 32 (x, 0) tS dx, etc. In (8.20) two of the functions are known. From (8.6) we have $'(s,0) = 0 (for s in R_) (8.21) while from (8.7) we have “0 (8,0) = AC =) (8.22) for Ims > - q,, i.e., for BER, . Eliminating A(s) between the two equations in (8.20) and using (8.21) and (8.22) gives - iv, a oO cE EEE K(s) 0, .(s,0) + ® (8,0) TAC + q) (8.23) 56 ee > > ae na' oO HOODS BA SG uw Be Ge fey O53) Ms a ’ a , 7. oO Sesenlt a2) Sea? { ene t : eta 3 a bk ama “7 PIAMBON MOLIS2 hE Ae va ¢ Lao) Tw L. ) c ee pet he ee ‘kino oolsze Eee Lietg resend baw oy ai @ bee OS ah 4 eee ae ee a - ¢ get x0" {eboosod we fw SB ee oe a; } bes ie (00.8) nt anokoeuas alee MURR FE ay Gal a ae EUR Re separa neice porn pn rn ar a standard form of W-H equation, with 7 1 K(s) = + = (2? - k2) * (8.24) Y ° 8 : The factorization of K into factors analytic and non-zero in R and of algebraic behavior at infinity there is again obvious: : ih = K,(s) = (6 +k) 7, K (8) = (s- ky)”, and after division of (8.23) by K_(s) we again have the additive split of the function 1/(s + q) K_(s) to make. We thus arrive at the equation iv ' {e) K,(s) 0!(8,0) + Tkcoer (8.25) PS) ave 1 1 KG) ¥,@8 +4 |K_(s) ~— K_Cq) = an entire function E(s). We anticipate that E(s) will be a polynomial, and consider the implica- tions that the degree of the polynomial be N, E(s) = ase. + a, s\ +...tay- 1 Rs PENS NaF EST TET eee Be We ey eee Pte OF ON TE ge Sey ee Vee 2 1 iv (s + k)) N e AR ite kO what EIOM Liky 2 91 (8,0) 1k. Cae ya) + (s + k)) {a's toed ay ae ~ 0(s 2) + O(a,8"*) ass >© in R,- The first term vanishes algebraically at ™ and is therefore the half-range transform of a function with at most an integrable singularity at x = 0+. In fact it follows from Watron's lemma that the 57 sor aromet| eat mage af Katee aad int de: i an Hynes Hane ning ny ceali orth ait shot a at (0) ag (Say ‘ta io ; 7 cs . iP 7 vi - | | ae: =— | | ake a ae es Bi Wen Pits a ps 5 ra fe aye OST Ga, aitang * es spat vita nibteges: Ree. ieee hi a ba be a) tah se wt ey ae iat? Fe Lg By Ser «t as a) whtasleniigts, petile psa seats, oii Abe gntieh ew Anew aN ete wiRagnut © Ae amddunied by Ss ‘t seach ia mts ae ao onl oon 38 at at A eee rt tz 1 corresponding value of ¢$/dy on y = 0, x > O is O(x *) as x + OF. The second term grows as s + ™, and must arise as the generalized Fourier transform of a function which has a nor-integrable singularity »* x = 0. According to Lighthill [11,p.43] & (1-A) sgns fe exp isx dx =e (-A)! [s|A-? ° for real s, and the correct interpretation of this in Ry is TL > a -A) e (-r)! oo (8.26) where the branch cut for the function si-! is to go from 0 to @ in the lower half-plane, 1 Hence 1 (s,0) = ocs\*?) 3 <=> = (x,0) = o(x 8-2) as x > O+. 3 Therefore the velocity has 4 singularity at least as bad as x near x = 0, and the kinetic energy in a small region around x = 0 will diverge to infinity. We argue that this singularity is unacceptable, and choose the solution correspcnding to E(s) =0, f thus giving the least singular behavior—like x %—in the velocity at x = 0. Note that it is impossible to impose a Kutta condition, that the velccity be finite (except by abandoning the radiation condition, or allowing @ to be discontinuous, and that cannot be permitted in static fluid). To see whet kind of « pressurm field exists near x = 0 58 a O mast og ont ‘Ae; noksonet bdo 10% 05 suciand se Let soiipnans ; MAS i i vas ee Pan rs ae a he ae: cs teead, o i. an a: . _ tt . eae Ga, 0. « - oe 0 > en we ve bad sh aeaot a ctisionnes é + si vptnaen ata : Aw om ht beuore “elsen tty pt ak ox 8 4 abt vay exyods. beg veldirqsosaitl si, mextabtgsnt it E as egies aT abi Ee + pbtaooaiies ) a nn a : i | we have iv Sara We ®,.(8,0) Ye +4 iv, 1 O(a 50) eee UL ee Sait (s + q) + : Yq (s -k,)? K_(-a) Qn y = 0, x > O we know from (8.7) what $ should be, and this can be confirmec from the expression for ®,. On y = 0, x < 0 we close the inverse Fourier integral path in the upper half-plane. The pole s = -q lies outside the contour and makes no contribution, so that we only need to examine the second contribution to @ ass +®inR_. For this contri- bution 3 @ (8,0) ~s* and so PS) $(x,0) ~ (-x) (times some coefficient) as x + 0-. Thus the pressure and the pressure jump both vanish like Cae near the plate edge. Note that although the pressure jump does vanish at the plate edge (which would be regarded in aerofoil theory as the satisfaction of a Kutta condition), the velocity is nonetheless infinite at the edge. In summary, the least singular solution hes 1 = 0(x”) ru (8.27) Vo = O(x 2) aa near the edge, and conditions of this kind are often imposed et the outset as edge conditions. It seems preferable not to anticipate the edge behavior in advance, but to follow the W-H method threugh as far 59 a sae aie vt at we al eo 03 onbatane bane set Mg. sg - ae et iit joa: yee: bie gems aust (0 oo « (sta) ; | | f | f Fi ig | | | a ee le eines wae hod nit at ‘i ge ae Ha 5 ws Is RCo ESR Gat Mramenrmrescamenesmeatein oer as (8.25), and then to see in each particular case what behavior must hold near the edge and what freedom exists for minimizing singular behavior (as we shall do in $9). In some cases, in particular in recent work by Rawlins [15] cn diffraction of an acoustic wave by a bane pane which is "sound-hard" on one side and "sound-absorbing" on the other, the edge conditions are not at all obvious, being in fact 1 = 0(x") 3 Vp = O(x *) To justify acceptance of a solution with certain edge conditions one has to go beyond the simple linear inviscid wave equation used here, or beyond the simple zero thickness model of the boundary. For example, one can look at linear acoustic propagation with viscous effects dacluded, or one can look at inviscid propagation around a surface which is thin compared with any other relevant length scale but which has a smoothly rounded edge. It can be proven (though the proof has not yet been published) that our solution with conditions (8.27) is the urique one which can be matched to an "inner solution" in which either viscous forces or the continuous curvature of the boundary lead to finite velocities everywhere. That is a rather special kind of yroof, however, and we shall refer in §9 to the unavailability of a comparable vroeof when there is uniform subsonic flow past the radiating half-plane. This discussion of edge-conditions completes the formal determina- tion of the field as AGS { Sie REE ey da (8.28) am J Yq K_(-¢) (8 + @)(s - ke 60 i eR RT 4 cause 28 ae bo toe stun aoe oo waa te ksh meandaitets ston wigain aay nf - - i “seit maga dato mene atiauass Taonkt 4a soot 189 nti st stbnas its oxtiaaton ue aia ti ie i‘ a stan 0 bib aioe’: wt ie saan wwouadanes oa: * ‘ mr, ets 4 twits Lulowen seul & us SoH sermci 2 ie fae Asie atleratives. ap (en LeLLaves ait oe ane at wade tii ng where c runs from - © to +@ in D, iene above 8 = -q, above the branch point s = aoe and below the branch point s = tk,- This holds for y > 0, while for y < 0 we use ¢ (-y) = Te poraiOD- The integral here can total actually be evaluated in closed form, in terms of Fresnel integrals, though the details are complicatec, and are nothing to do with the W-H method. We refer the reader to [5,10] for descriptions of the details, and remark simply that the distant radiating acoustic field can be estimated asymptotically from the fornula F(s) exp(-isx - vey) ds (8.29) c 1 2k m\? ee 2 ) exp(ik r - 7i/4) sind F(-k, cos@) in which x = r cosd, y = r sind, O< 6 <7. (This formula may give apparent infinities in particular angular directions, and near those 5 here we have te interpret (- 7k. cos® - ko xz, which we take as { | angles it is necessary to use more refined approximations. ) To apply “7a + peer because the arg of -k, cos§ — Ik. is equal to -n for | all 6 between 0 and T. We also need | 1 z z K_(-q) ? (-4 = k)) = -i(q a ky) r again because arg (-q - k)) = -7T. Then we have 1 $ (4): ae 1 gtkor -7i/4 sind /2 (8.70) i (q- k= ° (q - k, cos6) ; which gives the level and directivity of the scattered acoustic field. In the next section we look at the same problem, but with sub- sonic mean flow over the surface, 61 eee ee le ke + oe Hod sGtuana ae GOR a) 09. sete a sete a bra Pen sani doesent’ aay Sgate ‘ebyaty Auster be c i pt ee iteatoaehdie Peuawret hy bea Naw (8809 ft Wot Our « x strom ~'te ¢ ns Pa WA ea a - =) ais iy 1 | Stott ardeunon Sista wi, F v3 as seth a pau Bake aed’ iti oan, vols to a ‘ 1 A ‘i Pe iro vi yh ~ satisfies the convected wave equation total 3 2 A 92 92 -w+uB) - (2. +F) soca? @.1 and the boundary condition on the plate is that of continuity of displacement (not of normal velocity), giving I) r) = total (-10 +U a) n By where n = v/(-iw) is the surface displacement. Thus ag ter __total dy = (-iw + iq) ve ony = 0, x < 0. (9.2) Because of the presence of mean flow there is a new possibility on the extension of the plate. A wake can exist there, across which there may be a discontinuity in tangential velocity 06 /ax as long as the total normal displacement (ard hence here also the normal velocity) and the pressure are continuous across the wake. Take a single Fourier con- idx ponent Ae of erat for y = O+, x > 0; then for y = 0-, x > 0, below the wake, the corresponding component is Ae | because ¢ . must total be an odd function of y. The pressure jump across the wake associated with this particular compenent of potential is p(x,0+) - p(x,0-) = (-0)(-tu +U 2) (ane) i ne en a ON ee ee A A SALES TE IT ete ta a Be Fx gf Wan ir =) Ve. seve ase ptiiaaad it ae ak ioe ca [ASE ener gearee ' and this vanishes either if A = 0 (which implies continuity of oral and jnence abgence of the wake) or, for any value of A, if A = w/U. Thus the general condition on % otal across th2 wake is that iux/U rotal (kr0t) & Ae (9.3) dux/U pea for some value of A. The tangential velocity is (twA/ude the wake, (-tuwa/0) ex/U below it, and hence the wake js an oscillatory vortex sheet of strength 2wA/U, modulated by the phase factor exp [ ied - ue)| » Which shows that any element of the vortex sheet propagates downstream at the flow speed U. We wish to solve the two-part mixed boundary value problem posed by (9.1-9.3), subject here to the restriction to subsonic flow (M = u/c, < 1) and te a radiation condition, but leaving open the ienue of edge ccnditions. We first write Vv Seotal ” 7 E(eqy DD exp(tax -ECa)y) +6 (9.4) where we shall write generally rz E = {2* - (k, + na)?| (9.5) AGU Na De = 2 tole 1 8 wW k oO D(-q) = 1 - (9.6) The function [= replaces the aie in the no-flow case, while Ds is a kind ef Doppler factor for wavenumber s. The object of writing Pe eal in the form (9.4) 13 that the incident field associated with the velocity 63 stat tt 0 Sn aa ) Js es, Le henion =o enmiene ests Steading oF [poksitedens oa; or pani sealiion ae 68) we wished sae ae. tvast, audi jastatinoe ‘not zetbes ao ime 01? en 7 aw dents att: saabatos ayhe san a . ot (etp2)d~ eebigse ee; a tase | Ps Wiind 4 tind. + nade e tate ie ae " ul bald oe of Miter enna vitae wo. Bh | dt adobe bs a mh ey eatatiy aefeutte: an “a ‘ashi paibotey ats Make bse? Stic Tat. suabbai wt ms cS + 0.8 a: EQ: MGUY Set Secret ey! DIP OEASS nae e tre my pn samym mye emasvenensewr nea PANE ri nn SIE a GE OL GRRE RRM RET re net TUR ERS, H n t m Y TRRAN P EM SHIN HO RAT OID ISN COMA NAS SETTER AMET emrne Vv erat on an infinite plate has been split off; that field is exponen- tially large as x > -~, and the $ that remains is assumed to be just an outward propagating wave. We now have the following problem for 6: diya) 2) ala Om | [cis +U ce ce (=. + | ¢=0 (9.7) xe (y = 0, x < 0) (9.8) v D(-q) g=A exp") + ay exp(iqx) (y = 0, x > 0) (9.9) and we have to examine conditions as x ++. Write k, = k, + ik) s w/t = (k, + ik,)/M, q=4, + iq,- As x > - ~, @ must behave like an out- going acoustic wave, whose decay will be like that of a plane wave propagating against the flow, so that iwx gd ~ exp {nine - | F k, [x] ~ exp To (9.10) As x > + ©, } may behave either like an acoustic wave propagating with the flow iwx ¢ exp |- swe + | k,x ~ exp {- | (9.11) or may behave as it does in the wake, 64 i 10, RD ey, aul Ai eet aye et ais ae t ayt a he; at i atniey ies ot lu gittagaqosg pivey: abi euane) is 9b | ) Ces Pe aly | . | i ni : Kx d ~ exp - [="} (9.12) or in the way associated with the forcing field exp iqx, H (9.13) ¢~ exp {- q, x} It follows from these that k, R, is Ins >-nin( a.) k R_ is Ims<+t TS and that the strip is k, \ i, ° = a 0 to get o(s,y) = A(s) exp a) + B(s) exp(+é -y) (9.15) Es is defined in (9.5). We take a cut from s = + k/Q - M) to infinity above the strip D and one from s = “k/a + M) to infinity below D, and es 11 define [s - (k +Ms)]* to be the branch which behaves like + (1 - M)? s* ro) 1 when s is large, real, and positive, [s + (k,+Ms)]” to be the branch Poon behaving like + (1 + M)? s* for large real positive s. With these branch - cuts and choices of branch, 2° — > 0 for all s in D, just as for ve in $8. Then since (9.15) only holds in D we have to have B(s) = 0, 2 and now we can differentiate (9.15) with respect to y, put y = 0+ and eliminate the functicnA(s). Thus PROPS,» rte gee rege ney ee ° 65 % h £ ee a Fe | ee & & = & | ae ) ae : é& = ee ce S ee Nets ks tt a til a ae oe ea cc. ar. wire aut Lonel x - "I =| k Spats f - a i) : yt We ed ee Peet eg \,P “ee pale oe ing 40g) 08 FE ong Pbk “(t. ae ae soem) swt a = . Ne a eae oh | 5 | (ane (21,2) . a vito! bys. + (i. pes sel {oa sail nde ; Satan ot i ty), i ca =? mod dup. rs piles ie aes hinds a a ; re ‘a woled selatint Ch a ro + DA, i * c oe, a cule wt ¢ he re : | Sot - ay + ike (sive aah Hana F oa shat (4h 6 tte 2090 « = Kot Vink? of ovina aig q a aptea, co (tt) ei “ Ww % an % $2 apie rida oe 2) mabe ak | pint “hala aoa it oe : = ‘3 ¥ é PSOE eee me BREESE BOIS ROE A eo ot asc nperinemennnnne oene 0 (3,04) + 9° (s,0+) = - E19, (6,0+) + @ (s,04)] From (9.8) we have o' (8, 0+) = 0 while from (9.9) we have iv) D(-q) 1A ©, (s,0+) : Gir k. 7 + Fee) and hence we get a W-H equation lias iA A D(-q) Ez ) (e,0+) + _(s,0+) + te + kM) + E(-a)(s + 4) = 0 (9.16) We define the factors 6 (s) E_(s) by a E,(s) = [s + (k, + Ms)]? ; (9.17) E_(s) = [8 - (k, + Ms) and by the usual route arrive at k iv) D(-q) 1 . iA _o. ae eel nee ory nO} ®)(8,0+) + + KI) ECA) + E(-q)(s + q) &- V iA S = -&(s) %(s,0+) - Tip 36-08) - EG 4D fe) iv, D(-q) 7 eeayG@ ay wee) Hea) (9.18) = an entire function E(s), which must be a polynomial because of the algebraic behavior of all functions involved here. In fact we must have E(s) = 0, for if E(s} were even a constant, the@ part of (9.18) would give 70 nner gre ee ren a creme mt ne tr me ne oe a ab owe 1 1 (8, 0+) = 0(s*) at infinity $ which corresponds tc the strong singularity V¢ = O0(x 2). With that choice of E(s) the solution is still not unique, because the wake strength A is undetermined. We can argue that conditions are such that there should be no jump in tangential velocity across the extension 1 1 of the plate, in which case A= 0, Vo = O(x 7) and $ = O(x?) near the ee en cm OE PSTN TTY LE ELS OT PET EIR , * plate edge and we have a situation essentially the same as was examined in §s. Because the radiation is emitted into uniformly moving fluid the "stationary phase" formula (8.29) is not immediately applicable, but we shall show in a moment how it can be generalized to the moving fluid case. We can alternatively argue that a uae will adjust itself to eliminate the high velocities which would otherwise exist at the trail- ing edge, and that a Kutta condition of finiteness of the velocities at the edge should be imposed whenever possible. The physical basis for such a condition in unsteady trailing edge flow is a matter of contreversy at the moment, but that does not concern us here as our interest is merely in seeing if and how such a condition can be appiied in this model problen. Expand theG@) part of (9.18) as 3 + @ in R,- We have é / k, dv,D(-a) &_(-a) ee > 46400) ~ (ue ae + mie (1 + M)* 8 } ; 3 + 0(s 7) (9.19) 1 the term given explicitly corresponding to Vo = O(x 2) the second *o i Vo = O(x?). Thus we can impose a Kutta condition, that the velocities POINT neers 67 ‘ k z = S: : ia & 7 coca ra 88 nl a “patosiie sd waoton dtuate cae was tame oa.= 4 0 nos ae sphaiei tse: whiten om 6 athena be. aveed Arti a ian seve Sense ad a its sts ae - a avinth, aa ; LG > By, to-j0 vee Pa kt +3} ox sa (21.8) Se eek eh St AT ont or RE eet ye RABE PLAID eta rare NL auch tease merce an rom SS be finite at the edge (note that the velocities in the part of otal 2: ete split off in (9.4) are finite, but non-zero, near the edge) by choosing a wake strength come eT ST en pe re TT NITES: | TREES NT ve D(-q) (OO (9.20) 2 e(- #2) £,(-4) It is easy to see that the © part of (9.18) contains terms like (s + qDsts (s + kay which (as in $8) make no contrinution to > for x < 0, y = OF and that with the choice (9.20) the first term in the expansion of (s,0+) ass *>™ in R is o(s %), This corresponds to 3 > = 0(x?) ; mear the edge, though the pressure is 0(x2) because p = (2 +U 2) >. The expression for the potential in y > 0 is found to be >= ml. F(s) exp(-isx - 2) ds with k iv_ D(-q) | = L eet An nO AONE eat oe TD SIE) ( ray uM’ * = Cae +a) (9.21) s + M In the exponential factor we write 1 -isx - [s? - (ky + Ms)?)? y a Mk \2 k2 cf } so that if we define { 68 1 1 (8, 0+) = O(e*) at infinfty 3 which corresponds to the strong singularity Vo = O(x 2). With that choice of Z(s) the solution is still not unique, because the wake strength A is undetermined. We can argue that conditions are such 1 | | that there should be no jump in tangential ver ocrty across the extension | of the plate, in which case A = 0, Vo = O(x 2) and ¢ = O(x*) near the plate edge and we have a situation essentially the same as was examined in és. Because the radiation is emitted into uniformly moving fluid the "stationary phase" formula (8.29) is not immediately applicable, but we shall show in a moment how it can be generalized to the moving fluid case. } We can alternatively argue that a wake will gatuse itself to | eliminate the high velocities which would otherwise exist at the trail- ing edge, and that a Kutta condition of finiteness of the velocities at the edge should be imposed whenever possible. The physical basis for such a condition in unsteady trailing edge flow is a matter of interest is merely in seeing if and how such a condition can be applied | | | | controversy at the moment, but that does not concern us here as our | in thie model problem. Expand the@ part of (9.18) as s + © in R,- We have 4 k,. _ iv D(a) &_(-9) ie A o" (8,0) LAE (- =) Ea) (1 + M)* 8 < 3 | + 0(s *) (9.19) 1 | O the term given explicitly corresponding to Vd = O(x 2) the second to i F Vo = O(x*). Thus we can impose a Kutta condition, that the velocities : 67 OTe tinea nianec memes me Le z 1-M~ 1 x=R cos (i), Q - M7)? y=R sin(@} Mio is, exp|-1 Temz FB cos lle (c+ Mk, ) x 21 lo i-M7 ATs eee PE ee exp io R cos (A) (Oo K)) R sin@}eo which is now capable of asymptotic estimation by the formula (8.29) for | | | | | | | | then | | | | | | | | static fluid. To make the algebra less complicated, suppose the Mach number M is small so that M? can be neglected compared with unity. Then the field associated with the second contribution in (9.21) turns out to be 1 2 iv, (1 - Mq/k, ) Ga) AT) (9.23) see isind/2exp {ik, Gop ikiMx = > 1 ( ay [q - k,(cos® - ™)] rer there being no difference between r and K or between 9 and @) if “2 << 1, The field (9.23) is a trivial modification of (8.30). The distant field associated with the first term in (9.21), with the Kutta condition value (9.20) for A comes out as 1 iv, (1 - Mg/k ey 1 2 M e au + (Fz) kN ~~ (l= M cos6) Se Ae an es SR 69 By end oe “Bown k Sct eae “e ae - Laer oa vol Rare e fs of Soy hes ft we 102 eo hora es é voli ae DARONgtigoe. do. > ee 4 voi ato aeoaua eabvrnskig eeah step isi Wied Banat wyiako. ily bapigens basa 88) 4 sii a Rha errs cena tisey ) Henvottaaes # forse ” Hoiwinge s £E.2) ce 24 cai - 2. ih Ka Nita ——— MS tee, a. ay 11D nen © ania en ahs gait 0.8) 40) aos: saat thea detvtsy 5 nas (es, ie ood dey + (1s, Ohmi are sari) dy Ad Ber Bares ne et sew! $0: abi cred : | ee 142 une Sais 2 DAS Sore oe aye Parte. tps See) ae Ie EOE IE SOE AEE TSE TLIN: WR Bei nmeerere cee ree eo oe a ae ee Ne ter ce NR A REN Nee ReTN FU | HE Pete and the ratio of the wake generated field (9.24) to the field (9.23) which would exist in the absence of the wake is (since q > k) roughly Htc) (9.25) This shows that if the plate wave travels at speed w/q between the flow speed and the sound speed, the wake contribution to the far-field is negligible. If, on the other hand, the plate wave travels aore slowly than the flow speed, Mq/k, exceeds unity and the sound field when the Kutta condition is imposed exceeds that in the absence of any wake. This is not to be regarded as a general conclusion, for in other problems the wake sound very nearly cancels the primary edge field. That in fact happens here, for when q becomes close to ki/M it can be seen that the fields (9.23) and (9.24) have small but equal and opposite | : | | | values. Whether the extra wake field dominates, or mainly cancels the primary edge sound field thus depends very much on details of the basic excitation. ATR TETIY anntin= RN ET RNAI Ie GTR LCR RS PT RORT REIT A minoe ee e e a a — —= eetameeeeeconeee AC Rtapatersekrssm cp ee iJ L$ RESISTOR ERTIES ee 1 . 10. CONSTRUCTION OF W-H SPLIT FUNCTIONS In this section we first outline a general method for effecting either the additive or the multiplicative decomposition of a function analytic in a strip, and then we set dow sone properties of the functions which arise frequently in acoustics problems. Finally, we record the corresponding properties for strictly incompressible flow problems. A. Gauchy Integrals Let F(s) be analytic in some strip D; E Ey» R_ the domain Im s < Eos Suppose also that |F(s)| + 0 uniformly as fast as |s|~ for some \ > 0 as |s| + » within any closed region within D, i.e., as |s| + © with 1 2 Then F(s) «= F(a) + F_(s) (10.1) for e in D, where he Bid AG sh F) "ont | ree ot 2) —— is analytic and bounde”’ in Ry» F (s) = st Fir} dt (10.3) a Nae . is analytic and bounded in R. The path —~_—— runs from ~ © to + @ in D below t = 8, while —/\—runs from - © to + © in D above t = 8. Suppose further now that |F(s)|+0 uniformly as fast as le}? for soma 1 > 0 as |s| + 1in the strip D. Then { F(t)dt converges absolutely, end 7 eon _ a. : (8,02) cE.08)- ple aa 350% a6 ict o + = ) a RE IND Re stoaevaeureatra PAA Tp SI MSS ARSAY COMET ame nr tre rae PINE AENEID MIL HE ALIPAC AWAY were PPR yey nl API aie te as |s| +> in R,, while J ty ei t ies ‘ i ih as s seeone rhe Va $0) ts J Lisile, F (s) ~ “ar | F(t) dt = O(s °) ee ; as ja| +> in R. 1939 Without going into fine details, (10.1) is proved by applying Cauchy's theorem F(s) = a F(t) dt 2ni t- 8 rr nner IIIT to a contour lying within D and enclosing the point s in D (F(s) being defined only for s in D in the first instance). The contour is then’ deformed to consist of a rectangle with —-—_,-—~» as its lower side <—/\— as its upper, and with the ends of these sides joined at 1 infinity by short sides parallel to the imaginary axis. In the limit these short sides (of finite length, less than E, - €,) make no contribution to the integral, so that ami t- 8s t-s8 EN F(s) = uJ Fe) dt - > | Eye where now both ee of integration run from -™ to +™, But then, according to the basic theorem of complex variabie analysis, the first term defines an analytic function as s varies without crossing the integration path, i.e., it defines an analytic function in the upper half-plane above the integration path. Similarly, the second term ! defines a function analytic everywhere below the integration path eosemnnl Gh Nocremes T@ For the behavior at infinity we have 72 | | | poltey een 1 F, (s) = ( 2ris f F(t) | a = aie {: a —S- Without any real loss of generality we can take the path of integration to be the real axis, and 8, > 0 in Fy» so thag SF) ae | < del] _TF(e)] de —U— Let |s| + @ in R, along a rey with s, = Ks,, and divide the range cf integration at pointst = +M. M is chosen so that |s| >> M, but so that for |t| > M, |F(e)| < ele] ot for some constant c and some A > Zz. Then on (-M,+M) fe} [Fce) ae 1 ft Sie — (t)| de fe aie 3 - z bai i | | | | and the integral is fiuite and independent of s, while on (M,~”) say we put t = Ks, + s,tand to get { ae M Vit ~ Ks,)* + 6 2 | -{. _sec 0 Ae ee co =) (K + tand)> and again the integral is convergent and independent of s. Thus under these conditions F, (3) = O(s !) at infinity in R,- The above is hardly a proof, but it cen te rigorized. In any application the behavior at infinity should be cnecked out carefully in each case. 73 The Cauchy integral formlas (10.2, 10.3) enable the product decomposition K(s) = K, (s) K_(s) to be effected by taking logarithms. Suppose K{3) is analytic in the strip, and that |K(s)| + 1 uniformly as |s| + © in the strip. is Se ae F Se é i | [ | | Suppose further that K(s) #0 in the strip. Then F(s) = 2£n K(s) is analytic in the strip for any branch of the logarithm, and can be decomposed as in (10.1). Define K, (s) «= exp F (s), K_(s) = exp F_(s) . (10.6) K(s) = K,(s) K_(s) (10.7) for s in D, and K,(s) are analytic and non-zero throughout R,, respectively. This decomposition is unique up to multiplication of say K,(s) by a nen-zero entire function and division cf K_(s) by the same function. It may be necessary to use this freedom to remove non-algebraic behavior at infinity of K,(s) in certain cases. Noble [5] gives several examples of this minor difficulty. When K(s) is even, the factors K,(s) as defined by (10.6), (10.2), and (10.3) have the property K,(-8) = K_(s) (10.8) If the split is achieved by some way other than use of Cauchy integrals it may be necessary to adjust the functions before (10.8) holds. For 1 exan.'e, if K(s) = (s* - k?)? then the “obvious” split is 1 1 K,(s) = (6 +k)”, K(s) = (s-k,)”, : 1 | but K,(-s) = erui2 (8 - k.)*, so that we need to redefine K,(s) as ' 714 (8.02) : i ~ ; ; ; , eee pialientaaaedinaiiine mete ities ne fcr Shree avtianirie 1 nn re ee mm ry -71/4 z Ti /4 1 K,(s) =e (s +k)’, K (s) se (e - k,)? in order that (10.8) will be satisfied. B. Decompositions Related to the Square Root Function ve 1 As just noted, the multiplicative decomposition of Ye = (8% - k?)* into factors analytic, non-zero and algebraic at infinity is 1 K,(s) = a(s + ke K (8) = aM(s - k,)? (10.9) for any constant a. If it is useful to require that K, (-8) = K (s) then a should be chosen as els. For the convected wave equation (with subsonic convection veloci- ties) Me is replaced by 2 = 2 232 e. {s (k, + Ms)*) for which the multiplicative split is yt + Soy K, (8) = a(l - M*)* (3 + Ta 1 K_(s) = & "(8 - 75)? (10.10) for any constant a. Here of course we cannot uske K, (-8) = K (s). Because the factors K, (8) K_(s) for K(s) are analytic and non-zero, the split ‘or 1/K(s) is given by Q/K, (s)), (1/K_(s))- The additive decomposition of Nie end functions related to it arises very frequently. Noble [5] gives several ways of calculating the split functions and several representations of those functions. Here we will just verify that if eae) a’ A) asf, he : Meee? “hee ry os Fan ita “ | ny a * alae Cabs | % “td mile: * 8 (otsot): alee asa Poreray Lo a ) a he E ace a SATIS EE BAT ai an RP Ene 8 Coren ue Veweae, es P,(s) == cos (8/k,) E 3 (10.11) : 2 uB) ee e : P_(s) 7 cos ¢ 3/k,) bs a i then é as Pi(s) + Ps) = ¥, (10.12) for s in D, and P,(s) are analytic in R, with a certain behavior at infinity which will be determined shortl;;. Firstly it is necessary to define the function cos"! (s/k,? for complex s. If s/k, is reat, cos! (s/k,) is defined as the branch for which cos! (s/k,) = 1/2 when alk, = 0, i.e., cos? (s/k,) lies between O and 7 when s/k, is rea) and between +1. Let 6 = cos? (s/k,)- Then s/k = ere? (o) 2 and so au Cea a) 4B oe Gene fo) ° sty = 8 Qn ( i: ° 1 where Viens (s? - ke) with the branch cuts 3s already discussed. The logarithm here is defined to have its principal value, i.e., &2£n z is such that —7 < Im £n 2< +7 (10.13) with a branch cut along the negative real z-axis. Now take s = 0; the corresponding value of y is ik, and hence if = £n{+(-1)} = Fu, sc thac B =< 1/2 if we choose the lower sign. Hence 58- ¥ Ba-i fn aaa (10.14) ° 76 Soh dosbad eax an bonita of (hap das * cheawuseks ak $/a2 <8) gautay fey) which can be rearranged as follows. We have (s + Y,) ¢s - Ys) = ks Bertin eo Sita sty °) hence The definitions (10.14) and (10.15) of the function cos’ (s/k,) are not those given in most books on mathematical functions. In those books the square root function is usually understood to have a branch cut from ake to +k)» and is quite different from the function Ye which occurs in wave epper cect ong: In particular, if the cut goes from =k to Ma (s? - K2)? behaves like s both as s +> + © and s + - ~, whereas Y; behaves like s as s > + © but as - s whens > - ™, With the definition ist+y cos~' Gy =+i hm (= (10.16) ° ° Consider the functions p(s), P_(s) defined in (10.11). In the first place, no new branch cuts are introduced by the logarithm. A branch cut would be needed only if s + A = 0 were possible for some value of s, and no such value of s exists. Thus the only singularities are the branch points at s = + ko: Consider the function Pi (s) near the point s = + ko writing s = ko + u where u is small. We have 1 1 1 1 du2(2k_ + u)? u2(2k_ + u)? Piet 4) = 2 Oat Ik doaee sh aaa (e) (0) 1 1 1 1 12? (2k)? u? (2k)? ; ee gc ia + 0(u2) (10.17) 77 mi i “east vn ae Tine ener H eeu Ev! hodzesaba “eae, a ‘cok 009 nscy i) ebay 7 mis ie i‘ LOL) at boeetah a as u > 0, so that P,(s) is in fact single-valued near s = ko The branch point singularity of the cos! cancels that of Ne at s = ko» and hence Pi (s) is analytic in R,- Similarly P_(s) is analytic in R_, the branch By; point singularity of Ne ats2- k, being cancelled by that of cos"! (- s/k,) there. is To verify (10.12) we have, for s in D, : ae [ os eee : Pi (s).t Pe (s) i sa e k aa aaa re ty, mseahs £n(- 1) = Ys since with (10.13) &n(-1) = - in. To find the behavior at infinity we note that as |s| + © in R, Ke scar reson ata (10.18) if the approach to infinity is below the branch cut from s = + ko? 1 k? k" Y,e~ 78 fs -3-32+...| (10.19) if s goes to infinity above the cut. Since P(s) has no singularity at s=z+ k, it does not matter which of these is used, and we find k2 \ &2 iL Ol 2s\ __o Pts) == le aE +.) | (22) at | i 28 a te in (i ) + 0(s-? £n s) (10.20) oO These properties of the P, functions arise in a great many applications to wave problems. Corresponding results for the convected wave square 78 i cn et, ‘ i H { | root g3 can be derived by making a change of variables to transform bs into Ys? @s was done in 9. C. Incompressible Flow Problems Incompressible flow results follow from taking the limit ke 0, which has the urfortunate effect of reducing the strip of analyticity D to a line on which the functions are continuous, but not necessarily analytic. The branch cuts from + ko also join up to form a complete barrier along, say, the imaginary axis. To avoid possible difficulties stemming from this it is usual to work with finite ko and then let ke > 0. This, however, makes for unnecessary complications in much of the werk, and it is useful to be able to tackle the incompressible problem directly. To this end we imagine the branch cuts as starting from 0 + i0 and going to infinity in Im s > 0 and from 0-i0 to infinity in Ims < 0. Then the limiting form of the function Xe is real and positive on the whole real s-axis, i.e., it is there the function |s|. We shall write (2)? for this function in the complex plane with the cuts as indicated; it is the continuation to complex s of the function js| on the real axis, and can also be defined as s(sgn Res) where sgn x = + 1 for xx 0. Thus 1 es (s2)? = 8 (sgn Res) as k, 70 (10.21) The multiplicative split is 1 1 1 (s2)? = st BG % (10. 22) 1 1 1 : where s? means the branch of s* which behaves like s* as s + +with a 1 1 cut from 0 - Oi in the lower half-plane, while s? behaves like s* as s ++ © but nas the cut from 0+0i in the upner half plane. 79 2 : ue . ; 4 Sl #6 akan ‘its anboute te 10M + sonst at mad Dies $Eeanwssaen Jen, 0 ai saouatanos a | wee: ie e a ae ae aad) One a anal aay er Wi ever ‘ee dt £ ees ines Sit3 to iow ea aucksnanto; \esieesernn rot | oie A rai od «3 ‘ i il - ae ti / ehspea walldons ‘phabuse araont ana abies a stds oa ete a a8. hee anion be oe + Gi aaa aoiriads er alia soma va pesado! ag - Sites, bres tea! 6h a wadeeanns sta ie ort aa ; eae ed: al the een sity: aid a) at “et sre Se Sb oat ca is a1 ada cr Shy sada alquos: ‘ona at gota aa. a on wis ee ep caksgaut bitd Bo, » ‘wotoaaa oy nat ubesacs a. at a yomneatnd ae WF Dk ey age aiede $0) Presi y a6 ‘baat So nee ern Tae is cy : rt 7 oo yy AS tis 13,01) Oe) oe (a voto oe ts. 04) £ EE ASST 6 Rite ORV TE NUTT SEER SA (Raa note snare ee ee * Sy 1 The additive split of (8*)? can be found by taking the limit of (10.11) as ko +0. Define &n,8 to be the branch of 2£n 5 with a cut from 0-01 in the lower half-plane, and with ins real and positive when s is real and greater than 1. Define &£n_s similarly except that it has the cut from 0 + Oi in the upper half-plane. Then we can see that £n.s =nin e*= 0 if Res > 0 in, s -2£n.s = 2in if Res <0, srt’ pee URE swear cen A AMT HORE IEA is é and both of these are covered by fos - £n_s = 2in H(-Res) where H(x) ie the Heaviside function equal to 1 cr 0 according as x >OQcrx <0. Since sgn x = 2H(x) - 1, we can write this now as fn,s - $n_s = in - in sgn Res and multiplyiang by s and using (10.21) gives 1 sins -s ins = ins - in(s?)* (10.23) i Now define 8448 P,(s) 2 + 5 ns | (10.24) 8 is P_(s) = a es £n_s J and then it follows from (10.23) that 1 (s?)? = P,(s) + P_(s) (10.25) and P,(s) is enalytic in R, (Im s > U), P_(s) is analytic in R_(Im s < 0), | | 80 | casvouy: ean, | Ae, iO > a) t ol nine on 4 as required. Further, by careful consideration of the branches it can be shown that these P,(s) still heave the property P,(-s) = P_(s) (10.26) enjoyed by the function (10.11) for ky ?# 0. Use of these functions enables incompressible problems to be more complicated compressible problems. but we should stress that because there is no strip of overlap ali procedures should only be regarded as i H | svlved much more elegantly than by taking the limit as ke +o of the | H formal, and the results should be verified by independent checks. 1 81 | i] | KE ; .% : : : Sapna rire arene -yemeqong it ove san (eigesoa Aqysueg 3utiys ut Aqynuyauosstq atduyzg e worzy uoysstusueay pue uoTIIEeTJay - T PaNnsTy by yg haha Oyen 10.) «Og _ 82 yo eg gee CID zc et Vs Ro ee ee ara (x31) do eee | (x’3)dx0 t ————— ——— | (x°m1—)dxe y ay gaan 9 Yb a neers | Oy a soy d kazozakteuy zo dzaag pue yx ‘ty SaUueTd-JTPH 19Mo7T Pue aaddn BuyddetiesQ yITA seueTd-S xatdmog - Z ain3ty ALISILATVNG 400 divls S wy] 83 ce ° Se erm an (oat Ga amines pia Sean Nast PRR eran enseeancetieen tamer minecrnnernn Se tS MES naar sueeg UO SdTJIFNUFAUOISTG Wory VUOTIDeTJOYy YUITA payefoossy saz 3yaetNBuys - ¢ dan3Ty ee rn rn 84 ethan Sennapninzeemeiicd I ee a ae en Vee ae “Ainletpr eae lame gm od SiS 2 MaMa seas pe, SPENT RE Foi SA a itd enbs 5 A voz Aouny Jooy 21 py aa BIND) YOueIg 2YI JO UOTIFD07 103 sadToYD u ay. 10Z 2: ORE Sel ce avd RAC RIS aaa Ree! etharhiaatatemener nears OR SY eT ADR aA a Sat Co al Ss pe aes LA Ra ne A Ct el A < Mo x - ois bs Hiei ni eae see Sekar a ise iace oe 3 AL ea ENS x 85 NOU SRA Si ERBEETOD adaene yove CHOTEES Yor, fo: 4 ETests ¢ = & Sea, ' ; ® REFERENCES (1] N. Wiener and E. Hopf (1931) S.B. Preuss. Akad. Wiss. 696-706. {2] N.E. Mushkelishvili (1953, "Singular Integral Equations," (trans- lated by J.R. M. Radok) Noordhoff, Groningen. (3] F.D. Gakhov (1966), "BoundaryValue Problems," Pergamon Press, New York/London. [4] J. Schwinger and-D.S. Saxon (1968), "Discontinuities in Waveguides," i Gordon and Breach, New York. [5] B. Roble (1959), "Methods Based on the Wiener-Hopf Technique." Pergamon Press, New York/London. [6] L.A. Weinstein (1969), "The Theory of Diffraction and the Factorization Method," Golem Press, Boulder, Colorado. (7] G.F. Carrier, 4. Krook, and “.E. Pearson (1966), “Functions of a Complex Variable - McGraw-Hill, New York. [8] P.M. Morse and H. Feshbach (1953), "Methods of Theoretical Physics," McGraw-Hill, New York. + [9] J.D. Murray (1974), "Asymptotic Analysis," Oxford Univ. Press, (Clarendon) London and New York. [10] L.B. Felsen and N. Marcuvitz (1973), "Radiation and Scattering of Waves," Prentice-Hall, New Jersey. (11) M.J. Lighthill (1958), "An Introduction to Fourier Analysis and Generalized Functions," Cambridge Univ. Press, England. {12] M.J. Lighthill (1960) Phil. Trans. Roy. Soc. A, 252, 397. [13] E.C. Titchmarsh (1948), "Theory of Fourier Integrals," Oxford Univ. Press, London and New York. (14) D.G. Crighton (1972) J. Sound Vib. 20(2), 209-219. . (15) A.D. Rawlins (1975) Proc. Roy. Soc. Lond. 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