THE IMPEDANCE SCATTERING PROBLEM FOR A POINT-SOURCE FIELD. THE SMALL RESISTIVE SPHERE

Transcription

1 THE IMPEDANCE SCATTERING PROBLEM FOR A POINT-SOURCE FIELD. THE SMALL RESISTIVE SPHERE By GEORGE DASSIOS (Division of Applied Mathematics, Department of Chemical Engineering, University of Patras, Greece) and GREGORY KAMVYSSAS (Institute of Chemical Engineering and High Temperature Chemical Processes, GR Patras, Greece) [Received 13 October Revise 1 May 1996] SUMMARY A small resistive scatterer disturbs a spherical time-harmonic field emanating from a point source. The incident point-source field is normalized in such a way as to be able to recover the corresponding results for plane-wave incidence. The full low-frequency expansion for the corresponding total field is reduced to an exterior boundary-value problem for the Laplace equation, which has to be solved repeatedly. Exact results for the case of a small resistive sphere are obtained. It is shown that the leading low-frequency approximations for the scattering as well as for the absorption cross-section are increasing functions of the impedance parameter and decreasing functions of the distance of the source from the scatterer. It is also shown that a small sphere scatters and absorbs more energy when it is illuminated by a point rather than by a plane-wave field establishing the fact that the closer the source of illumination to the scatterer, the stronger the interaction. The leading approximation of.the absorption cross-section is independent of the wavenumber, while the leading approximation of the scattering cross-section is proportional to the second power of the wavenumber. Hence, in the low-frequency realm, absorption is by two orders of magnitude stronger than scattering. Finally, a comparison between point- and plane-wave incidence, based on multipole expansions, is included. 1. Introduction THE PROBLEM of scattering of a point-generated spherical wave, by an object whose characteristic dimension is much less than the wavelength of the incident wave, has been given much less attention than the corresponding problem for plane-wave incidence (1). Nevertheless, most applications of scattering theory, such as underwater acoustics, medical imaging, nondestructive testing, and in general laboratory experiments involving waveobstacle interaction processes, pertaining to point-source radiation of the scatterer. Jones (2) was the first to calculate exact low-frequency results for the case of a circular disc in the presence of a point-source field. The [Q. Jl Mech. appl. Math., Vol. SO, PL 2, 1997] Oxford University Press 1997

2 322 GEORGE DASSIOS AND GREGORY KAMVYSSAS corresponding theory, and exact results for a small soft, or hard, sphere were presented in (3). The incident spherical wave in (3) was normalized in such a way as to preserve the energy flux at the origin, no matter where the source is located along a ray emanating from the origin. This normalization was chosen in such a way as to be able to recover the plane-wave incidence as the point source recedes to infinity. In this work we develop the low-frequency method for a resistive scatterer, where Robin boundary conditions hold on its surface, adopting the same kind of normalization for the incident wave as the one used in (3). All low-frequency approximations are reduced to the solution of a simple exterior boundary-value problem for the Laplace equation. Exact low-frequency approximations are derived for the case of a resistive sphere. These involve the leading three terms for the near-field and the leading two non-vanishing terms for the far-field. It is shown that the absorption cross-section is by two orders of magnitude (second power of the wavenumber) larger than the scattering cross-section and that the leading terms of both cross-sections are increasing functions of the coefficient of impedance. Furthermore, they are decreasing functions of the distance between the point source and the centre of the sphere. It is observed that the near field is constructed from infinitely many multipole terms, while the far field is expressible in terms of only a finite number of multipoles. This is not true in the case of plane-wave incidence where both near- and far-field approximations are expressible in terms of only a finite number of multipoles. The formulation of the problem is presented in section 2. Section 3 involves the decomposition of the scattering problem into low-frequency approximations and the reduction of the initial wave problem to an exterior problem for special harmonic functions. Then the method is applied to the case of a small resistive sphere and the exact results are furnished in section 4. The explicit results provided in section 4 demand very long calculations which are not included since they are, to a large extent, tedious and straightforward. A final section 5 recovers the corresponding results for the case of plane-wave incidence as limiting values of the results obtained in section Formulation of the problem The scatterer V c is specified as a bounded smooth and closed subset of R 3 and S is its boundary. The complement V of the scatterer forms the medium of propagation, within which a point-source field u', emanating from r 0, is propagating. We work in the frequency domain established by the harmonic dependence exp{ iwt}, where <o denotes the angular frequency. The source point r 0 of the incident field is located outside the smallest sphere that circumscribes the scatterer V c. Furthermore, we consider the case of a

3 SCATTERING BY A SMALL RESISTIVE SPHERE 323 modified point-source field which conforms with the demands of being able to recover the plane-wave incidence in the direction -f 0, and to preserve the normal acoustic intensity at the origin, as the source point r 0 recedes radially to infinity. As was shown in (3) these requirements lead to an incident field of the form u'(r) = ^ e*<*-*-i0, (1) l r ~ r ol where k denotes the wavenumber. Since, both the incident field u' and the scattered field u s have their sources within a bounded region, it follows that the total field «(r) = u'(r) + u'(r), rev-{r 0 } (2) satisfies the Sommerfeld radiation condition uniformly over directions. The total field u solves the Helmholtz equation OT )0 (3) (A + * 2 )u(r) = 0, rev-fo} (4) and satisfies the Robin boundary condition ^ ^ + jjtvu(r) = 0, re 5, (5) dn where dldn indicates exterior normal differentiation on 5 and v is a real dimensionless quantity defined as the ratio of the mass density times the phase velocity divided by the acoustic impedance of the medium occupying the exterior region V. In view of the boundary condition (5) the total field assumes the following integral representation: f (^ ) r - r' ) «fa(r') (6) where the zeroth-order spherical Hankel function of the first kind h(x) = h$\x) = e ix lix (7) specifies the fundamental solution of the Helmholtz operator and the representation (6) holds true for every exterior point r with the exception of the source point r 0. Standard asymptotic analysis furnishes the form «'«= g(i)kkr) + (l/r 2 ), r -»», (8) where the scattering amplitude g(r) is given by the surface intergral (9)

4 324 GEORGE DASSIOS AND GREGORY KAMVYSSAS Note that, in contrast to the case of plane-wave excitation, where g is dependent upon five quantities (the two orientational angles of incidence, the two orientational angles of observation and the wavenumber), in the case of point-source excitation the scattering amplitude is dependent upon six quantities (the two orientational angles of observation, the two orientational angles of the point source, the distance between the source and the scatterer, and the wavenumber). This extra quantity measuring the distance of the source point from the origin, which is located in the interior of the scatterer, could be of use in inverse problems connected to target location (3). Once the scattering amplitude is obtained the scattering cross-section is evaluated via the formula a, = -\ImI u*(r) ^^<fc(r) = ^<f g(r) 2 ds(j), (10) K J$ an K J52 where S 2 stands for the unit sphere in R 3, and the absorption cross-section is given by a a = \ Im <f «(r) ^ W ) = v I u(r) 2 ds(r), (11) k J s an J s where the integration is over the surface of the scatterer. Then a s, a a as well as the extinction cross-section = a s + a a = - J are functions of the three coordinates specifying the location of the point source, and the wavenumber of the excitation field. Formula (12) is obtained from (10), (11) and the fact that the incident waves u' and u'* solve the Helmholtz equation in the interior of 5, which implies that Im <j> u'(ir) du *^ ds(t) = 0. (13) Js dn 3. Hie low-frequency expansion For scatterers with characteristic dimension much smaller than the wavelength of the incident field, the analyticity of the total field at k = 0 can be used to expand it in powers of the wavenumber (4,5). In other words, if we define the characteristic dimension as the radius a of the smallest sphere that circumscribes the scatterer, then the assumption that ka «1 implies that f n (r), (14)

5 SCATTERING BY A SMALL RESISTIVE SPHERE 325 where the field u, which is dependent on k, is replaced by the sequence {u n } of the low-frequency approximations, which are independent of k. Substituting (14) into the equation (4) and the boundary condition (5) and equating equal powers of k, we arrive, for every n = 0,1,2,..., at the differential equation and the boundary condition Au n (r) = n(n - l)u n _ 2 (r), r E V - {r 0 } (15) du n (r) dn nvm n _i(r)=0, res (16) that the low-frequency approximations M O» "I, u 2,--, have to satisfy. In order to establish the behaviour of the low-frequency approximations in the far field we substitute (14), as well as the expansion for the incident field and the expansion 00 (iky ikh(k\r-r'\)= y Z L -r\r-*t l (18) for the fundamental solution, into the integral representation (6). Then, a simple asymptotic analysis yields the form ««(r)=/ n (r) + w n (r), tev-{r 0 }, (19) where, at the nth order of approximation, the function ~ r ol [ o o r + J~ "S (")(«- P) f 4rt p =i \p/ J s () r " rt 3 (r-r').n'u n _ p (r')ds(r') (20) is known, and w n (t) = Oillr), r-*oo. (21) Straightforward calculations show that A/ n (r) = n(n - IK-2W, rev- {r 0 }, (22) which implies that f n furnishes a particular solution of Poisson's equation (15). Therefore, the problem of obtaining the low-frequency approximations

6 326 GEORGE DASSIOS AND GREGORY KAMVYSSAS {u n } is reduced to that of solving the following exterior boundary-value problems: Aw n (r) = 0, rsv, (23) res, (24), r-^cc (25) for the Laplace equation. Once the low-frequency approximations {u n } have been obtained, similar arguments to those above, lead to the low-frequency expansion g<?) = f 2 ^7^ (^(-ir 1 1 u n - p (r')(t.rtkf.fi') - v] ds(r') (26) for the scattering amplitude. In a similar way we obtain low-frequency approximations for the scattering cross-section and the absorption cross-section via formulae (10) and (11), respectively. In what follows we shall apply the above technique explicitly for the case of a small resistive sphere that disturbs the propagation of a point-source field. 4. The resistive sphere If the scatterer is a sphere of radius a centred at the origin, then by symmetry we can assume that the point source, that generates the incident field (1), is located at the point r 0 with Cartesian coordinates (0,0, r 0 ) and a < r 0. Then the zeroth-order low-frequency approximation u 0, also known as Rayleigh approximation (5), is a harmonic function which assumes the asymptotic form «o(r) + r-r o and satisfies the Neumann boundary condition (27) du o (T)/dr = 0, r = a. (28) Therefore, the Rayleigh approximation of the resistive sphere coincides with the Rayleigh approximation of the hard sphere which, as was shown in (3), is given by where 8 is the spherical polar angle of the vector r.

7 SCATTERING BY A SMALL RESISTIVE SPHERE 327 The first-order approximation seeks also a harmonic function which confirms the asymptotic form and the boundary condition dn r^oo (30) VK O (r) = 0, r = a, (31) where «0 (r) is given by (29). Expanding all the known and the unknown fields in solid spherical harmonics and using the orthogonality of the Legendre polynomials, we are able to evaluate the expansion coefficients of the total field, and arrive at a spectral representation of the solution. This procedure yields the first-order approximation «,(r) = r o (l - «0 (r)) + av f -^-^ (-)" (-)" +I p B (cos d). (32) n=o(n + l) \ro/ \r/ Considerably more work is demanded in order to evaluate the second-order approximation of the total field. First, we evaluate and expand the non-harmonic particular solution f 2 (r) in terms of surface spherical harmonics. This leads to la r 2 \, A / 2/i 1 Af r\ n+i and the solution is obtained in the form \ro' \/v P n (cos 0) (33) M 2 (r) = fiir) + w 2 (r), (34) where w 2 (r) is a harmonic function, which is of order r~ l at infinity, and satisfies the boundary condition an on = a, (35) and where / 2 (r) is given by (33) and u^r) by (32). The expansions in (33) converge only for r < r 0. But this is enough, because all we want to use is

8 328 GEORGE DASSIOS AND GREGORY KAMVYSSAS the orthogonality of the spherical harmonics on 5 to evaluate the unknown coefficients of the expansion of w 2, and this is secured because the surface 5 is located within the region r < r 0. Long and tedious calculations involving spectral decompositions of harmonic functions yield the following representation of the second-order low-frequency approximation for the resistive sphere: u 2 (r) = -^- [ r - ro - r o f + 2va 2 (l + - r - r o \ r (n + l)(2n - 1 \ n - 2 Availability of the low-frequency approximations u 0, u^ and u 2 allows the calculation of the scattering amplitude up to order A: 4 through the expression (36) (? fi ' } " Vl[ " l(r ' ) " (? * r ' )Mo(r ' )]^^+ { [(f * fi<) " x [-\u 2 (t') + (f.r')u,(r') - (f «-') 2 «o(r')] dsfr') + O(k 5 ). (37) In order to arrive at a closed-form expression for the far-field quantities we need to evaluate a number of particular surface integrals. This can be done using the orthogonality of the spherical harmonics (6) over the unit sphere. The values of these integrals are provided in the Appendix. The low-frequency approximation for the scattering amplitude is then given by (38)

9 SCATTERING BY A SMALL RESISTIVE SPHERE 329 Based on the expression (38) one can easily obtain the following approximation for the scattering cross-section: 6 ), fca^o. (39) In view of formula (11), the approximations u Q, U\ and u 2 for the total field provide the following approximation for the absorption cross-section: Since = vl \u o (r')\ ds(i') + v* 2 < [ Ml (r') 2 - «o(r')u 2 (r')] ds(r') + O(k 4 ), I u o (r') 2 ds(f) = 4^ 2 k-+0+. (40) 2 - ^ 5 f^), (41) («+ l) ^V and f + 2v + 2v 2 it follows that 2n-1 (2«- l)(n +1) f (n + l) 2 rg n + lr o ]\r o (43) 2n-2 Note that the leading asymptotic terms of the absorption cross-section involve terms of order Ac 0 and k 2, while the corresponding terms for the scattering cross-section are of order k 2 and k 4. Therefore, a resistive body within a low-frequency field absorbs, by two orders of magnitude, more energy than it scatters.

10 330 GEORGE DASSIOS AND GREGORY KAMVYSSAS The extinction cross-section is obtained from (12), (39) and (44) if we confine ourselves to terms up to order k 2, as follows:,=,2/7 + 1 l 12 rl =,(«+ l) 2 (2n - 1) \r 0 Obviously, every result for the resistive scatter recovers the corresponding result for the hard body, obtained in (3), in the limit as v» Recovering plane-wave incidence As the point source r 0 in (1) approaches infinity the incident field reduces to the plane wave u'jw) = lim u'(r) = «"** ' (46) ro-h.00 propagating in the -f 0 -direction. As was shown in (3), the plane wave ul generates the same normal energy flux at the origin as the energy flux genrated by the point field «', no matter where the point r 0 is located along the ray in the f 0 direction. Taking the limit as r 0 tends to infinity for every result obtained in section 4 we end up with the following expressions which correspond to plane-wave incidence in the negative z-direction. The zeroth-, first- and second-order approximations are given by «o(r) = 1, (47) «,(r) = vy - (l + ^)r/>, (cos 6), (48) (49) respectively. Then the scattering amplitude yields g(-?o,f) = -v(*a) 2 - /(&a) v 2 + JP,(cos 6)) + v(to) 4 (l + v + v 2 + ^.(cos 0)) + O((ka) 5 ), ka -* 0. (50) Finally, the scattering cross-section a s and the absorption cross-section a a assume the form a s = Ana 2 v 2 {kaf + Ajia 2 {ka) A [^ - v 2-2v 3 - v 4 ] + O((ka) 6 ), ka -»0, (51) <r a = 4;ra 2 v - 4na 2 v(jtfl) 2 [(l + v) 2 - ] + O((ka) A ), ka -+ 0, (52)

11 SCATTERING BY A SMALL RESISTIVE SPHERE 331 which then imply the extinction cross-section a e = 4TO 2 V - m 2 v(2v + l)\kaf + O{{kaf), ka (53) Comparing the corresponding results for point- and plane-wave excitations, we observe a significant difference in the near-field approximations u 0, «i and u 2, which are given as complete multipole expansions in the case of point-source excitation, while for plane-wave excitation these expansions are restricted up to monopoles for u 0, up to dipoles for Uj and up to quadrupoles for u 2 - In other words, the differences between point- and plane-wave incidence are not as prominent in the far-field region, where every low-frequency approximation, for the case of point-source excitation, involves multipole terms of one or two orders higher than the corresponding term for the case of plane-wave excitation. This difference, between plane- and point-wave excitation, reflects the fact that in the near field every plane-wave component has an appreciable contribution to the representation of the spherical wave (7) and therefore it generates an appreciable interaction with the scatterer. On the other hand, in the far field, only a few multipole contributions survive, namely those that are needed for the multipole representation of the corresponding contribution of the incident wave. In other words, only the multipole components of the incident wave that interact with the scatterer manage to survive all the way to the far field, while the other terms that represent the geometrical transformation from the spherical system, centred at the scatterer, to the one centred at the point source, are annihilated as the observation point recedes from the scatterer as well as from the point source. A comparison with the plane-wave incidence case reveals that in the case of point-source excitation, every far-field approximation involves more multipoles than the corresponding approximation for the plane-wave incidence. Furthermore, as is observed by comparing the Rayleigh approximations in (39) and (51), as well as (44) and (52), a small sphere scatters and absorbs more energy when it is radiated from a point source than from a plane-wave field. The closer the point source is located to the sphere, the more intense the scattering interaction appears, even in the case where the intensity of the point source is so normalized as to have the same amount of energy flux arriving at the sphere, independently of where the source is located. Finally, we observe that the leading-term approximation for both the scattering cross-section and the absorption cross-section are increasing functions of the impedance parameter v. Acknowledgment The first author acknowledges partial support from the Greek General Secretariat for Research and Technology during the preparation of the present work.

12 332 GEORGE DASSIOS AND GREGORY KAMVYSSAS REFERENCES 1. J. J. BOWMAN, T. B. A. SENIOR and P. L. E. UNSLEIGH, Electromagnetic and Acoustic Scattering by Simple Shapes (North-Holland, New York 1969). 2. D. S. JONES, Communs pure appl. Math. 9 (1956) G. DASSIOS and G. KAMVYSSAS, IMA J. appl. Math. 55 (1995) G. DASSIOS and R. E. KLEINMANN, SIAM Rev. 31 (1989) R. E. KLEINMANN, Proc. IEEE 53 (1965) E. D. RAINVILLE, Special Functions (Chelsea, New York 1960). 7. M. TYGEL and P. HUBRAL, Transient Waves in Layered Media (Elsevier, Amsterdam 1987). APPENDIX The surface integrals needed to obtain the expression (38) for the scattering amplitude assume the following values: (f. n>o(r') ds(j') = 2/r- P,(cos 6), (A.I) = 4TO 2, (A.2) (f. n>,(r') ds(j') = 2na 3 (~- 1 Wcos 9), \2r 0 I (A.3) = 4na 3 v, (A.4) (f. n')(i. r> 0 (r') <fc(r') = ^f- (l +1 ^ /> 2 (cos 0)), (A.5) <f (f. r>o(r') ds(r') =2n- /'.(cos 6), (A.6) r o Js (f.n')u 2 (r')ds(r') = 2na A \^--- v]p,(cos0), (A.7) L 2 r 0 J ' l [2v 2 + 2v + l], (A.8) Ana* f 2a (v a \ 1 n= [ 3v +- (3 - -!)^( cos 0 )} ( A - 9 > I (f. r>,(r') ds(r') = 2na\~ - \\P,(zos d), J s \2 r 0 I (r.fi'kf.r') 2 Mo T J r( 0 \ > ( ) ^ r o (A.10) j (P. r') 2 «o(?') A(r') = ^ (l + Ipr/Wcos 0)). (A.12)