FULL-PLANE IMPEDANCE HELMHOLTZ PROBLEM

In this appendix we study the perturbed full-plane or free-plane impedance Helmholtz problem, also known as the exterior impedance Helmholtz problem in 2D, using integral equation techniques and the boundary element method. We consider the problem of the Helmholtz equation in two dimensions on the exterior of a bounded obstacle with an impedance boundary condition. The perturbed fullplane impedance Helmholtz problem is a wave scattering problem around a bounded twodimensional obstacle. In acoustic obstacle scattering the impedance boundary-value problem appears when we suppose that the normal velocity is proportional to the excess pressure on the boundary of the impenetrable obstacle. The special case of frequency zero for the volume waves has been treated already in Appendix B, since then we deal with the Laplace equation. The three-dimensional Helmholtz problem is treated thoroughly in Appendix E. The main references for the problem treated herein are Kress (2002), Lenoir (2005), Ned´ elec (2001), and Terrasse & Abboud (2006). Additional related books and doctorate ´ theses are the ones of Chen & Zhou (1992), Colton & Kress (1983), Ha-Duong (1987), Hsiao & Wendland (2008), Rjasanow & Steinbach (2007), and Steinbach (2008). Articles that take the Helmholtz equation with an impedance boundary condition into account are Angell & Kleinman (1982), Angell & Kress (1984), Angell, Kleinman & Hettlich (1990), Cakoni, Colton & Monk (2001), and Krutitskii (2002, 2003a,b). Interesting theoretical details on transmission problems can be found in Costabel & Stephan (1985). For more information on resonances of volume waves we refer to Poisson & Joly (1991). Eigenvalues for the far-field operator are computed in Colton & Kress (1995). The boundary element calculations are performed in the report of Bendali & Devys (1986) and in the article of Bendali & Souilah (1994). Hypersingular integral equations are considered by Feistauer, Hsiao & Kleinman (1996) and Kress (1995). The use of cracked domains is studied by Kress & Lee (2003), and the inverse problem in the articles of Cakoni et al. (2001) and Smith (1985). An optimal control problem is treated by Kirsch (1981). Applications for the Helmholtz problem can be found, among others, for acoustics (Morse & Ingard 1961) and for ultrasound imaging (Ammari 2008). The Helmholtz equation allows the propagation of volume waves inside the considered domain, and when supplied with an impedance boundary condition it allows also the propagation of surface waves along the domain’s boundary. The main difficulty in the numerical treatment and resolution of our problem is the fact that the exterior domain is unbounded. We solve it therefore with integral equation techniques and the boundary element method, which require the knowledge of the Green’s function. This appendix is structured in 14 sections, including this introduction. The direct scattering problem of the Helmholtz equation in a two-dimensional exterior domain with an impedance boundary condition is presented in Section C.2. The Green’s function and its far-field expression are computed respectively in Sections C.3 and C.4. Extending the direct scattering problem towards a transmission problem, as done in Section C.5, allows its resolution by using integral equation techniques, which is discussed in Section C.6. These techniques allow also to represent the far field of the solution, as shown in Section C.7. A particular problem that takes as domain the exterior of a circle is solved analytically in Section C.8. The appropriate function spaces and some existence and uniqueness results for the solution of the problem are presented in Section C.9. The dissipative problem is studied in Section C.10. By means of the variational formulation developed in Section C.11, the obtained integral equation is discretized using the boundary element method, which is described in Section C.12. The boundary element calculations required to build the matrix of the linear system resulting from the numerical discretization are explained in Section C.13. Finally, in Section C.14 a benchmark problem based on the exterior circle problem is solved numerically.

Direct scattering problem

We consider the direct scattering problem of linear time-harmonic acoustic waves on an exterior domain Ωe ⊂ R 2 , lying outside a bounded obstacle Ωi and having a regular boundary Γ = ∂Ωe = ∂Ωi , as shown in Figure C.1. The time convention e −iωt is taken and the incident field uI is known. The goal is to find the scattered field u as a solution to the Helmholtz equation in Ωe, satisfying an outgoing radiation condition, and such that the total field uT , decomposed as uT = uI + u, satisfies a homogeneous impedance boundary condition on the regular boundary Γ (e.g., of class C 2 ). The unit normal n is taken outwardly oriented of Ωe. A given wave number k > 0 is considered, which depends on the pulsation ω and the speed of wave propagation c through the ratio k = ω/c. which is also satisfied by the incident field uI and the scattered field u, due linearity. For the total field uT we take the homogeneous impedance boundary condition − ∂uT ∂n + ZuT = 0 on Γ, (C.2) where Z is the impedance on the boundary. If Z = 0 or Z = ∞, then we retrieve respectively the classical Neumann or Dirichlet boundary conditions. In general, we consider a complex-valued impedance Z(x) that depends on the position x and that may depend also on the pulsation ω. The scattered field u satisfies the non-homogeneous impedance boundary condition − ∂u ∂n + Zu = fz on Γ, (C.3) where the impedance data function fz is given by fz = ∂uI ∂n − ZuI on Γ. (C.4) The solutions of the Helmholtz equation (C.1) in the full-plane R 2 are the so-called plane waves, which we take as the known incident field uI . Up to an arbitrary multiplicative factor, they are given by uI (x) = e ik·x , (k · k) = k 2 , (C.5) where the wave propagation vector k is taken such that k ∈ R 2 to obtain physically admissible waves which do not explode towards infinity. By considering a parametrization through the angle of incidence θI , for 0 ≤ θI < 2π, we can express the wave propagation vector as k = (−k cos θI , −k sin θI ). The plane waves can be thus also represented as uI (x) = e −ik(x1 cos θI+x2 sin θI ) . (C.6) An outgoing radiation condition is also imposed for the scattered field u, which specifies its decaying behavior at infinity and eliminates the non-physical solutions, e.g., plane waves and ingoing waves from infinity. It is known as the Sommerfeld radiation condition and receives its name from the German theoretical physicist Arnold Johannes Wilhelm Sommerfeld (1868–1951). This radiation condition allows only outgoing waves, i.e., waves moving away from the obstacle, and therefore characterizes an outward energy flux. It is also closely related with causality and fixes the positive sense of time (cf. Terrasse & Abboud 2006). The described outgoing waves have bounded energy and are thus physically admissible. 

Green’s function

The Green’s function represents the response of the unperturbed system (without an obstacle) to a Dirac mass. It corresponds to a function G, which depends on a fixed source point x ∈ R 2 and an observation point y ∈ R 2 . The Green’s function is computed in the sense of distributions for the variable y in the full-plane R 2 by placing at the right-hand side of the Helmholtz equation a Dirac mass δx, centered at the point x. It is therefore a solution G(x, ·) : R 2 → C for the radiation problem of a point source, namely ∆yG(x, y) + k 2G(x, y) = δx(y) in D ′ (R 2 ). (C.14) The solution of this equation is not unique, and therefore its behavior at infinity has to be specified. For this purpose we impose on the Green’s function also the outgoing radiation condition (C.8). Due to the radial symmetry of the problem (C.14), it is natural to look for solutions in the form G = G(r), where r = |y − x|. By considering only the radial component, the Helmholtz equation in R 2