Two-dimensional acoustic scattering, conformal mapping, and the Rayleigh hypothesis

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1 Two-dimensional acoustic scattering, conformal maing, and the Rayleigh hyothesis P. A. Martin a) Deartment of Alied Mathematics and Statistics, Colorado School of Mines, Golden, Colorado (Received 28 June 2012; revised 18 July 2012; acceted 27 July 2012) Methods for solving two-dimensional scattering roblems using conformal maings are investigated. It is shown that their convergence relies on a maed form of the Rayleigh hyothesis. It is concluded that methods based on conformal maings offer no advantages over established methods in which the scattered field is exanded as a series of circular-cylindrical outgoing wavefunctions. VC 2012 Acoustical Society of America. [htt://dx.doi.org/ / ] PACS number(s): Fn, Bi [AMJD] Pages: I. INTRODUCTION There are several effective rigorous methods for solving the exterior boundary-value roblems of acoustics in two dimensions. For simle geometries (circles and ellises), we can use searation of variables. More generally, we can use a boundary integral equation or a T-matrix method. Our urose here is to discuss certain methods in which conformal maings are used. The basic roblem is to solve the Helmholtz equation, (r 2 þ k 2 )/ ¼ 0, exterior to a simle closed curve C, with boundary and radiation conditions. Alication of a conformal maing can simlify the geometry (C can be maed to a circle) but at the cost of a more comlicated differential equation: the constant k 2 is relaced by a function of the indeendent variables; see Eq. (24) below. This basic aroach has a long history, going back to Garabedian. 1 More recent aers include Refs At the heart of our discussion are singularities of the analytic continuation of the (hysical) exterior field inside C. The relevance of these singularities to scattering roblems is well known, esecially in the context of the Rayleigh hyothesis. This concerns the use of circular-cylindrical wave functions when the scatterer is not a circle; see Sec. III. In Sec. II C, we give methods for calculating singularity locations inside C, with emhasis on the role of the Schwarz function. 5 This function has been used in other alied areas, such as vortex dynamics (see, for examle, Ref. 6 or Sec. 9.2 of Ref. 7). It can be constructed by conformal maing (Cha. 8 of Ref. 5) and by several other methods. In Sec. IV A, we introduce transformations (changes of indeendent variables), with conformal maings as a secial case. It is observed that solutions of the transformed Helmholtz equation can be obtained simly by introducing the same transformation into known solutions of the Helmholtz equation. Then, in Sec. IV B, we consider ublished methods in which a conformal maing is used to ma C onto a unit circle. It is demonstrated that doing this offers no advantages (and has the additional disadvantage of having to find the aroriate conformal maing). The new methods a) Author to whom corresondence should be addressed. Electronic mail: amartin@mines.edu are tantamount to trying to solve the roblem of scattering by C using circular-cylindrical wave functions (for which several numerical strategies are available), which means they are subject to (a maed form of) the Rayleigh hyothesis. II. SCATTERING PROBLEMS AND SINGULARITIES We consider acoustic scattering in two dimensions. Thus, we want to solve the Helmholtz þ k2 / ¼r 2 / þ k 2 / ¼ 0; (1) in the unbounded region, B, exterior to the smooth closed curve, C. Here, k ¼ x/c, c is the seed of sound and x is the frequency: the suressed time deendence is e ixt. We use Cartesian coordinates x, y, with the origin, O, inside C. The total field is / þ / inc, where / inc is the given incident field. For simlicity, we assume that C is sound-soft, which means / þ / inc ¼ 0onC (2) In addition, / must satisfy the Sommerfeld radiation condition at infinity. It is well known that the boundary-value roblem for / has exactly one solution. A. Circular scatterer When C is a circle, centered at O, we can solve for / by searation of variables in olar coordinates. Thus, as is well known, radiating solutions of Eq. (1) are w n ðx; yþ ¼H n ðkrþe inh ; n ¼ integer; (3) where x þ iy ¼ re ih Then, we write /ðx; yþ ¼ X1 and H n H ð1þ n is a Hankel function. c n w n ðx; yþ; (4) where the coefficients c n are to be found. If convergent, this reresentation for / ensures that Eq. (1) and the radiation condition are satisfied. For the boundary condition, suose C has radius a and ut f(h) ¼ / inc evaluated on r ¼ a. Then, Eqs. (2) and (4) give 2184 J. Acoust. Soc. Am. 132 (4), October /2012/132(4)/2184/5/$30.00 VC 2012 Acoustical Society of America

2 f ðhþ ¼ X1 c n H n ðkaþe inh ; < h : (5) As this is a Fourier exansion, we obtain c n ¼ 1 2H n ðkaþ ð f ðhþe inh dh: Let u n ¼ max{jc n j, jc -n j} for n ¼ 0, 1, 2,. As H n ðjþ ði=þð2=jþ n ðn 1Þ!; n!1; fixed j; the series in Eq. (4) will be absolutely convergent for (6) r > lim n!1 2nu nþ1 ku n : (7) For a simle examle, consider an incident lane wave / inc ¼ e ikx ¼ X1 where J n is a Bessel function. Then c n ¼ in J n ðkaþ H n ðkaþ i n J n ðkrþe inh ; (8) and u nþ1 u n ðkaþ2 4n 2 as n!1; using Eq. (6) and a similar estimate for Bessel functions, J n (j) (j/2) n /n! for large n and fixed j. Hence, from Eq. (7), the series in Eq. (4) converges absolutely for r > 0, for an incident lane wave, Eq. (8). In other words, / can be analytically continued inside the circle C, using Eq. (4), all the way to r ¼ 0, where there is a singularity. Indeed, there must be at least one singularity inside C, assumed now to be a simle, smooth closed curve, because the only radiating solution of the Helmholtz equation in the whole of sace is identically zero. These singularities can be induced by the incident field or by the shae of C. The singularity locations are intrinsic to the boundaryvalue roblem: they do not deend on k (it is enough to consider k ¼ 0), the choice of coordinate origin, or the method used for solving the boundary-value roblem. They do deend on the shae of C and on the incident field (or the boundary condition on C). B. Two more examles Suose that the circular scatterer of Sec. II A is dislaced so that its equation is ðx x 0 Þ 2 þ y 2 ¼ a 2 : (9) The analytic continuation of / will still have a singularity at the circle s center, (x, y) ¼ (x 0, 0). If we use Eq. (4) for /, with wave functions centered at O, the series will converge for r > jx 0 j; this region will include all of the exterior, B, rovided that jx 0 j < ð1=2þa. For another examle, consider an ellise, ðx=aþ 2 þðy=bþ 2 1 ¼ 0; 0 < b a: (10) The analytic continuation of / will have singularities at the foci, (x, y) ¼ (6c, 0) with c ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 b 2 (see Sec. II C). If we use Eq. (4) for /, with wave functions centered at O, the series will converge for r > c; this region will include all of B rovided that a=b < ffiffiffi 2. C. Methods for finding singularities One way to find the singularity locations roceeds as follows, adating a method used by Keller 8 for grating roblems. Thus, suose that the boundary curve C is given by F(x, y) ¼ 0 for some function F. Write C as z ¼ SðzÞ; z 2C; (11) where z ¼ x þ iy and z ¼ x iy. Hence, x ¼ð1=2Þðz þ zþ ¼ð1=2Þðz þ SðzÞÞ and y ¼ð1=2ÞiðSðzÞ zþ, so that S is given imlicitly by Fðx; yþ ¼0; with x ¼ 1 2 ðz þ SðzÞÞ; y ¼ 1 2 iðsðzþ zþ: (12) Assuming that C is an analytic curve, S(z) willbeananalytic function of z in a region that includes C. We are interested in locating the singularities of S(z) insidec. The reason is that a searate argument (given by Keller 8 ) shows that, generically, these locations are where the analytic continuation of / inside C will have singularities; see also Ref. 9. (In general, there may be singularities outside C too, but these are not relevant when our goal is to solve scattering roblems.) The function S(z) is known as the Schwarz function of C. Much is known about roerties of Schwarz functions. 5,10 Let us give some exlicit examles. (1) Dislaced circle, Eq. (9): S(z) ¼ x 0 þ a 2 /(z x 0 ). Note that S has a simle ole at the center of the circle. It is known that if the Schwarz function of a closed curve C is a rational function, then C must be a circle (. 104 of Ref. 5). (2) Ellise, Eq. (10). With F(x, y) defined by the left-hand side of Eq. (10), Eq. (12) gives ða 2 b 2 Þðz 2 þ S 2 Þ 2ða 2 þ b 2 ÞzS þ 4a 2 b 2 ¼ 0; (13) with solution [see Eq. (5.13) of Ref. 5] SðzÞ ¼ 1 c 2 ða 2 þ b 2 Þz 2ab ffiffiffiffiffiffiffiffiffiffiffiffiffiffi z 2 c 2 ; c ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 b 2 : Note that S has branch oints at the foci of the ellise, z ¼6c. (3) Carl Neumann s oval (. 20 of Ref. 10). Barnett and Betcke 11 define an inverted ellise arametrically by xðsþþiyðsþ ¼e is ð1 þ ae 2is Þ 1 ; 0 s 2; where a is a constant, 0 < a < 1. Eliminating the arameter s gives ðx 2 þ y 2 Þ 2 ½x=ð1 þ aþš 2 ½y=ð1 aþš 2 ¼ 0 (14) from which J. Acoust. Soc. Am., Vol. 132, No. 4, October 2012 P. A. Martin: Acoustic scattering and conformal maing 2185

3 " SðzÞ ¼ z ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# ð1 þ a 2 Þþð1 a 2 Þ 1 4az 2 2 ð1 a 2 Þ 2 : (15) z 2 þ a We note that S has branch oints at z ¼6ð2 ffiffiffi ffiffiffi a Þ 1 (outside C) and simle oles at z ¼6i a =ð1 a 2 Þ (inside C). For Eq. (14), seeeq.(3.8) inref. 10. In lane olar coordinates, Eq. (14) can be written in the form r 2 ¼ A þ B cos 2 h, called theroser 2 by Davis; he gives Eq. (15) [Eq. (5.16) in Ref. 5] and some generalizations. Keller 8 did not identify S(z) as the Schwarz function of C but he did suggest a way of finding the singularities of S without having an exlicit formula for S. We have Fðx; yþ ¼F 1 2 ðz þ zþ; 1 iðz zþ ¼ gðz; zþ: 2 On C, F ¼ 0 and z ¼ SðzÞ whence g(z, S(z)) ¼ 0, z C. But g is analytic in z and z so that g(z, S(z)) ¼ 0 for z near C. Then, differentiating with resect to z gives @z S0 ðzþ; see. 114 of Ref. 5. Thus, S 0 is @x z þ SðzÞ ¼ 0 with x ¼ ; y ¼ 2 z SðzÞ : 2i (16) Combining Eqs. (12) and (16) determines the singularity locations, z. This combination of conditions on F [but not the use of z ¼ SðzÞ] was also found by Maystre and Cadilhac. 12 For an examle, consider the ellise, Eq. (10). Equation (16) gives (a 2 þ b 2 )z ¼ (a 2 b 2 )S. Using this to eliminate S from Eq. (13) then gives z ¼6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 b 2, as exected. The methods described above assume that C is given in the form F(x, y) ¼ 0. However, there are situations where C is defined arametrically by x ¼ xðsþ; y ¼ yðsþ; 0 s 2; (17) or, in olar coordinates (when C is star-shaed ), by r ¼ qðhþ; 0 h 2: (18) From Eqs. (11) and (17), we have x(s) iy(s) ¼ S(x(s) þ iy(s)). Differentiation then shows that S 0 will be infinite when x 0 ðsþþiy 0 ðsþ ¼0; (19) solving this equation for s ¼ s 0, say, the singularity of S(z) is at z ¼ x(s 0 ) þ iy(s 0 ). Similarly, Eqs. (11) and (18) give qðhþe ih ¼ SðqðhÞe ih Þ; so that differentiation gives q 0 ðhþþiqðhþ ¼0 (20) as the requirement for a singularity; solving Eq. (20) for h ¼ h 0, say, the singularity is at z ¼ qðh 0 Þe ih 0. The condition Eq. (20) is also obtained from Eq. (16) using F(x, y) ¼ x 2 þ y 2 q 2 (h) and h(x, y) ¼ arctan (y/x). We note that Eq. (20) is similar to a condition found by van den Berg and Fokkema; 13 see their Eq. (14). For the ellise, Eq. (10), we can take x ¼ a cos s and y ¼ b sin s. Then Eq. (19) gives tan s 0 ¼ ib/a. We also have q(h) ¼ ab(a 2 sin 2 h þ b 2 cos 2 h) 1/2 ; Eq. (20) gives e 2ih 0 ¼ a2 b 2 a 2 þ b 2 and qðh 0 Þ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 þ b 2 : Either way, we obtain the exected singularity locations. For another examle, suose that C is the rounded triangle of Barnett and Betcke, 11 defined by xðsþþiyðsþ ¼e is þ ae 2is ; 0 s 2; 0 < a < 1 2 : Equation (19) reduces to e 3is ¼ 2a, with three distinct solutions. The corresonding singularities are at z ¼ z n ¼ 3 2 ð2aþ1=3 e 2ni=3 ; n ¼ 0; 1; 2: These are all inside C; see Fig. (7a) in Ref. 11 for a sketch when a ¼ 0.2. In summary, the methods we have described determine the singularity locations based solely on the shae of C. Again, we note that there may be other singularities caused by the incident field: these will not occur for incident lane waves, for examle. III. RAYLEIGH HYPOTHESIS The discussion in Sec. II is helful in understanding the Rayleigh hyothesis. Thus, suose we write the solution of our scattering roblem as in Eq. (4). This infinite series converges outside a circle, C 0, of some radius, r 0, centered at O. The circle C 0 is at least as small as the circumscribed circle (the smallest circle, centered at O and containing C), but it may be smaller. Thus, it is ossible that C contains C 0,so that the series in Eq. (4) converges everywhere outside and on C: this is an assumtion known as the Rayleigh hyothesis. It is known that the validity of the Rayleigh hyothesis deends on the shae of C and the location of O. Secifically, C 0 must contain all the singularities of the analytic continuation of / inside C, as discussed in Sec. II C. A clear overview of this well-known material is given by Millar. 14 We emhasize that the Rayleigh hyothesis is concerned with the convergence of infinite series. It says nothing about numerical methods which, inevitably, deal with finite series and aroximations. For examle, we may truncate the series in Eq. (4) and write /ðx; yþ / N XN n¼ N c N n w nðx; yþ: (21) Evidently, / N satisfies Eq. (1) and the radiation condition, so it remains to find the coefficients c N n in order that the 2186 J. Acoust. Soc. Am., Vol. 132, No. 4, October 2012 P. A. Martin: Acoustic scattering and conformal maing

4 boundary condition, Eq. (2), be satisfied. One way to do this is to aly Eq. (2) at discrete oints on C; see, for examle, Ref. 15. Another oular way is to imose Eq. (2) in a leastsquares sense, minimizing Ð S j/ N þ / inc j 2 ds; see, for examle, Refs. 14 and 16. For comarisons and connections, see Ref. 17. It is interesting to note that singularity locations have been found to lay a role in the behavior of the method of fundamental solutions (MFS). In the MFS, sources are laced outside the hysical domain along a fictitious charge curve C and then their strengths are determined by fitting the boundary condition on C. For interior roblems, Barnett and Betcke 11 show numerically that the success (numerical stability and hence high accuracy) of the MFS relies on a choice of charge curve which does not enclose any singularities of the analytic continuation of the solution, meaning that there should be no singularities between C and C. IV. CONFORMAL MAPPING As noted in Sec. I, there has been interest in the literature, dating back many years, in using conformal maing in the context of two-dimensional scattering. We shall discuss some of this work in Sec. IV B. A. Maings and their effects Let us start by introducing an arbitrary, smooth invertible change of variables, defined by x ¼ xðu;vþ; y ¼ yðu;vþ: (22) When this is used in Eq. (1), together with the chain rule, a new equation is obtained for /(x(u, v), y(u, v)) ¼ U(u, v), say (see the Aendix in Ref. 18): jruj 2 2 þ 2ðru 2 r2 r2 v þ k 2 U ¼ 0: (23) The maing in Eq. (22) canbechoseninmanyways. For examle, we may choose it so that the boundary curve, C, in the xy-lane is maed into a simler curve in the uv-lane. One otion is to choose a conformal maing. For such maings, Eq. (23) simlifies substantially. The result 2 U þ 2 k2 JU ¼ 0; (24) where J(u, v) is the Jacobian of the transformation, defined by : (25) (The Cauchy-Riemann equations can be used to write J in other ways.) Equation (24) is Eq. (20) in Ref. 3, for examle. Searated solutions of Eq. (24) are investigated in Ref. 18. Return to the general equation, Eq. (23). Evidently, if /(x, y) is any solution of the Helmholtz equation, Eq. (1), then U(u, v) ¼ /(x(u, v), y(u, v)) is a solution of Eq. (23). This result follows by substitution, nothing more: the change of variables does not have to come from a conformal maing. One alication of this result is that w n ðx; yþ ¼w n xðu;vþ; yðu;vþ ¼ W n ðu;vþ; (26) say, solves Eq. (23), where w n is the outgoing cylindrical wave function, Eq. (3). We see solutions of this form as Eq. (4.2) in Ref. 2, Eq. (27) in Ref. 3, and Eq. (24) in Ref. 4, obtained there by much more comlicated rocedures. B. Comarisons In order to comare with revious work, we consider using a conformal maing. The simlest examle concerns the dislaced circle, Eq. (9). In the hysical z-lane, we have z ¼ x þ iy ¼ re ih. In the maed f-lane, we have f ¼ u þ iv ¼ re iu. The maing z ¼ x(f) ¼ af þ x 0 takes the unit circle jfj¼1 to the dislaced circle jz x 0 j¼a. It also takes f ¼ 0 to the center of the dislaced circle, z ¼ x 0, and f ¼ x 0 /a to z ¼ 0. From Eq. (25), J ¼ a 2. Let us assume that x 0 > 0. In the hysical domain, we know that the infinite series of wave functions, Eq. (4), converges for r > x 0, and this includes the whole of the exterior region if x 0 < ð1=2þa. In the maed domain, we follow Eq. (24) in Ref. 4 and write with Uðu;vÞ ¼ X1 W n ðu;vþ ¼Hn ð1þ kjwðfþj B n W n ðu;vþ (27) wðfþ jwðfþj n ¼ H ð1þ n ðkjaf þ x 0jÞe inhðr;uþ : Each term in Eq. (27) is singular where jaf þ x 0 j¼0, that is, at f ¼ x 0 /a. However, the analytic continuation of / inside the maed circle, jfj¼1, will be singular at the center, f ¼ 0. The convergence of the series in Eq. (27) deends on the size of jf þ x 0 /aj, the distance from f to the exansion center, x 0 /a. This distance must be greater than the distance to the farthest singularity which, in our case, is the (only) singularity at the origin, f ¼ 0. Thus (assuming no other singularities have been introduced by the boundary condition), the condition for convergence of Eq. (27) is jf þ x 0 /aj > x 0 /a. This is exactly the same as we found in the hysical domain, r ¼jaf þ x 0 j > x 0. C. Discussion The simle examle just described is tyical. Its study leads to the following conclusions. First, the Rayleigh hyothesis is still resent in the maed roblem, even though the maed geometry is a circle. Conformal maing has merely disguised this fact. J. Acoust. Soc. Am., Vol. 132, No. 4, October 2012 P. A. Martin: Acoustic scattering and conformal maing 2187

5 Second, as the functions W n are not orthogonal on jfj¼1, some kind of numerical scheme will be needed to find the coefficients, B n, in Eq. (27). Exactly the same observation can be made of the series exansion in the hysical domain, Eq. (4). Consequently, one may as well work directly with a truncated form of Eq. (4), which is Eq. (21): nothing is gained by introducing a conformal maing (which would also have to be found). Third, if C has a more comlicated shae, one would have to examine the curves jx(f)j¼constant in the f-lane in order to understand the convergence of Eq. (27). These curves may or may not enclose singularities of the analytic continuation of U inside jfj¼1. Making this determination is not worthwhile: it is better to remain in the hysical domain. 1 P. R. Garabedian, An integral equation governing electromagnetic waves, Q. Al. Math.12, (1954). 2 D. Liu, B. Gai, and G. Tao, Alications of the method of comlex functions to dynamic stress concentrations, Wave Motion 4, (1982). 3 D. T. DiPerna and T. K. Stanton, Sound scattering by cylinders of noncircular cross sections: A conformal maing aroach, J. Acoust. Soc. Am. 96, (1994). 4 G. Liu, P. G. Jayathilake, B. C. Khoo, F. Han, and D. K. Liu, Conformal maing for the Helmholtz equation: Acoustic wave scattering by a two dimensional inclusion with irregular shae in an ideal fluid, J. Acoust. Soc. Am.131, (2012). 5 P. J. Davis, The Schwarz Function and its Alications (Mathematical Association of America, Buffalo, NY, 1974), D. Crowdy, A class of exact multiolar vortices, Phys. Fluids 11, (1999). 7 P. G. Saffman, Vortex Dynamics (Cambridge University Press, Cambridge, UK, 1992), J. B. Keller, Singularities and Rayleigh s hyothesis for diffraction gratings, J. Ot. Soc. Am. A17, (2000). 9 R. F. Millar, Singularities and the Rayleigh hyothesis for solutions to the Helmholtz equation, IMA J. Al. Math.37, (1986). 10 H. S. Shairo, The Schwarz Function and its Generalization to Higher Dimensions (Wiley, New York, 1992), A. H. Barnett and T. Betcke, Stability and convergence of the method of fundamental solutions for Helmholtz roblems on analytic domains, J. Comut. Phys.227, (2008). 12 D. Maystre and M. Cadilhac, Singularities of the continuation of fields and validity of Rayleigh s hyothesis, J. Math. Phys. 26, (1985). 13 P. M. van den Berg and J. T. Fokkema, The Rayleigh hyothesis in the theory of diffraction by a cylindrical obstacle, IEEE Trans. Antennas Proag.27, (1979). 14 R. F. Millar, The Rayleigh hyothesis and a related least-squares solution to scattering roblems for eriodic surfaces and other scatterers, Radio Sci.8, (1973). 15 S. Christiansen and R. E. Kleinman, On a misconcetion involving oint collocation and the Rayleigh hyothesis, IEEE Trans. Antennas Proag. 44, (1996). 16 Y. Okuno, The mode-matching method, Analysis Methods for Electromagnetic Wave Problems, edited by E. Yamashita (Artech House, Boston, 1990), Cha. 4, T. Semenova and S. F. Wu, The Helmholtz equation least-squares method and Rayleigh hyothesis in near-field acoustical holograhy, J. Acoust. Soc. Am.115, (2004). 18 R. A. Dalrymle, J. T. Kirby, and P. A. Martin, Sectral methods for forward-roagating water waves in conformally-maed channels, Al. Ocean Res. 16, (1994) J. Acoust. Soc. Am., Vol. 132, No. 4, October 2012 P. A. Martin: Acoustic scattering and conformal maing

Applied Mathematics and Computation

Applied Mathematics and Computation Alied Mathematics and Comutation 217 (2010) 1887 1895 Contents lists available at ScienceDirect Alied Mathematics and Comutation journal homeage: www.elsevier.com/locate/amc Derivative free two-oint methods