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WIDE-BAND APPROXIMATION OF THE SOUND FIELD SCATTERED BY AN IMPENETRABLE BODY OF ARBITRARY SHAPE
Journal of Sound and Vibration (1996) 194(4), 537–572
WIDE-BAND APPROXIMATION OF THE SOUND FIELD SCATTERED BY AN IMPENETRABLE BODY OF ARBITRARY SHAPE A. W T. S Laboratoire de Me´canique et d’Acoustique, U.P.R. 7051 du C.N.R.S., 31 chemin Joseph Aiguier, 13402 Marseille ce´dex 20, France (Received 26 July 1995, and in final form 27 November 1995) The scattered field on the boundary of an insonified hard or soft cylinder of arbitrary shape (CAS) is obtained in approximate manner by using explicit forward scattering solutions of insonified impenetrable circular cylinders (CC) whose radii vary with location on the boundary of the CAS. This so-called intersecting canonical body approximation (ICBA) is employed in the boundary integral representation in order to determine field functions at arbitrary points on or outside the scattering body. The approximation is evaluated in the near and farfield zone by comparison with rigorous solutions obtained from a boundary-integral equation scheme. The ICBA is shown to give results that are of better quality in the low frequency and resonance regions and of equivalent quality in the high frequency region as compared with the approximate solution deriving from the oft employed Kirchhoff ansatz of the surface field.
7 1996 Academic Press Limited
1. INTRODUCTION
Exact solutions of scattering problems in acoustics and electromagnetics are available [1] only for isolated bodies of simple shape (e.g., impenetrable, penetrable, or coated circular, elliptic and parabolic cylinders, spheres, ellipsoids, layered half-space bodies with flat boundaries and interfaces etc.). The need for approximate methods to describe scattering from arbitrarily shaped obstacles is still acute, particularly in the context of 3D forward scattering (for which the number of unknowns can be huge) and 2D or 3D inverse scattering problems (i.e., determine the shape of a body from the scattered field, a task [2, 3] which also can require an enormous amount of computations if rigorous field representations are employed). At low frequencies, one can appeal to the Rayleigh method [4] and its variants [5–7], but this requires that exact solutions of the Laplace equation be available, which is not usually the case when the shape of the given body is arbitrary (i.e., non-canonical). At high frequencies, it is appropriate to make use of (ordinary) geometric optics [8], the geometrical theory of diffraction [8, 9] or different variants of the so-called physical optics approximation (e.g., the Kirchhoff approximation [10], the Adachi approximation [11], the Fock approximation [12–15] etc.), the Lynch approximation [16]. However, a reliable approximate method still remains to be found for scattering by arbitrarily shaped bodies in the medium frequency (resonance) region which, hopefully, could also apply at high and low frequencies, although there is some evidence [17] that the recently proposed OSRC method [18] may partially fulfill these requirements (at least for circular cylinders, as concerns the backscattering cross-section, at all frequencies, and for cylinders with other 537 0022–460X/96/290537 + 36 $18.00/0
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. .
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shapes, as concerns the bistatic cross-section, in the medium to high frequency range). Another promising method [19], based on the so-called highpass scattering approximation (HPSA), applies to scatterers of a variety of shapes and at all frequencies from low to high, but concerns only the backscattering cross-section and involves heuristic (curve-fitting) functions to account for the fine-scale features of the response. The present work addresses this issue by considering a canonical boundary-value problem which is in some sense close to the given one and which can be solved, in closed form, in both the near-field and farfield zones as well as at all frequencies. Herein (i.e., for 2-D configurations) the canonical boundary-value problem is that of scattering from an impenetrable circular cylinder (CC). The impenetrable CC is replaced by a penetrable CC when the scattering is provoked by a penetrable cylinder of arbitrary shape (CAS) or y
Ω0 Γ
r
θ
Ω1 O
θi
x
Figure 1. The 2-D scattering configuration.
Figure 2. Definition of ‘‘inner’’ and ‘‘outer’’ zones in the vicinity of the scattering body.
Figure 3. The canonical bodies passing through points a and b of the given scattering body.
Figure 4. Graphs (in the cross-section plane) of the scattering bodies considered in the numerical study. (a), Three-leaf clover cylinder; (b) elongated elliptical cylinder; (c), nearly circular elliptic cylinder.
soft soft hard hard hard soft soft hard
7 8 9 10 11 12
Sound hard/soft
5 6
Figure number
ellip. ellip. clover
ellip. ellip. ellip.
ellip. ellip.
Type cylinder
1·5 1·5 1·0
1·5 1·5 1·5
1·2 1·2
a
1·0 1·0 0·3
1·0 1·0 1·0
1.0 1·0
b
0 p/2 0
0 0 0
0 0
ui
1·0 1·0 1·0
0·1 1·0 20·0
0·1 20·0
k
15 15 18
6 15 30
4 20
N ICBA ICBA, KA ICBA ICBA ICBA, KA ICBA ICBA ICBA
Best appr. on G
ICBA ICBA ICBA
ICBA ICBA, KA ICBA ICBA
Best appr. on G'
T 1 Input and output parameters for the numerical examples
ICBA ICBA, KA ICBA ICBA ICBA KA ICBA ICBA ICBA
Best appr. on G a
excel. good good
excel. good good
excel. excel.
Overall quality
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540
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by an impenetrable or penetrable sphere when the scatterer is a 3-D body of arbitrary shape (work to be published elsewhere). There exists a host of examples in which canonical problem solutions have been used in the past in a local setting. The hypothesis underlying the use of canonical problems to approximate the field of more complicated problems is that there is a one-to-one correspondence between the field scattered by an appropriate canonical body with that scattered by the complicated body at each point of the boundary of the latter. A recent example of a local technique employing canonical body solutions for complicated cylinder-like scatterers can be found in the work of Stanton [20]. His idea (which is called the deformed cylinder approximation (DCA)), for instance, for a cylinder with varying (although always circular) cross-section area along its axis, is to apply the known solution to the canonical problem of scattering by the infinite cylinder passing through successive points of the boundary as one moves along the axis of the actual cylinder to obtain the overall response outside the given cylinder. As shown further on, the DCA is somewhat similar to the intersecting canonical body approximation (ICBA) proposed herein, although it should be borne in mind that the ICBA concerns (here) a cylinder with generally non-circular cross-section which is the same and of the same area all along the cylinder axis as opposed to the DCA which concerns a cylinder with usually circular cross-section (it could also be elliptical but this makes the DCA much more unyieldy) which varies in area along the cylinder axis. An older, more well-known example of the use of canonical body solutions, the so-called impedance boundary condition (also known as the Leontovich boundary condition), which is often used to simplify transmission problems [10, 21], derives from the closed-form solution of the canonical problem of a plane wave impinging on a flat interface between two media and is thought to be valid even at a given point of a curved boundary provided the product of the radius of curvature at that point with the wavenumber in the medium of higher refractive index is large. The Kirchhoff or tangent plane approximation [10, 22] is another example in which the solution to the canonical problem of a plane wave impinging on a flat interface between two media (one of which may be impenetrable) is employed locally to provide an approximation of the field on the illuminated portion of the surface of a curved body, it being assumed that the field vanishes on the portion of the boundary within the geometric shadow when the body is impenetrable. Another approximation for impenetrable bodies, called the Adachi approximation [11], assumes that the Kirchhoff approximation continues as such onto the shadowed portion of the surface. As shown by Bass and Fuks [10], the (ordinary) geometrical optics approximation [8] of the field outside a scattering body can be obtained by a stationary phase approximation of the Kirchhoff–Helmholtz integral (rigorous boundary integral representation of the field) subsequent to use of the Kirchhoff approximation of the surface field within the integral. In this respect, geometric optics is also an outcome of using canonical solutions in a local context. An improvement of the Kirchhoff approximation has been obtained by Lynch [16] via a phase modification dependent on the curvature of the scattering body at each point of its surface. A more exact description of the fields at high frequencies, particularly near caustics [23], in the transition region between light and shadow, and deeper into the shadow region of a curved obstacle, requires examination of a canonical problem that is closer to the original problem, notably by the fact of incorporating curvature of the scatterer. For this purpose, partial-wave solutions of the circular cylinder and sphere scattering problems, recast into forms more suitable for high frequency analysis by means of the Watson transform [24], have been used by Fock [12, 13, 15], Franz and Depperman [25], Keller
541
[8, 9] and others [26–28] to develop the so-called Fock, geometrical theory of diffraction and creeping wave approximations of the field. A different and original line of attack to approximation of scattered fields at high frequencies was taken by Ursell [29, 30]. Instead of introducing a suitable approximation of the field on the surface into the Kirchhoff–Helmholtz integral representation (which employs the free space Green function), Ursell suggested generating approximations to the field represented by an integral involving a Green function satisfying either a Neumann or Dirichlet boundary condition on the boundry of the scatterer. Owing to the fact that finding this Green function is as difficult as determining the scattered field itself, Ursell assumes locality (justifiable at high frequencies) which fact makes it possible to replace the
Figure 5(a). Scattering response (surface layer potential 1v u) on the boundary G of a sound-soft almost-circular elliptic cylinder to an end-fire plane wave for wavenumber k = 0·1. q, BIEM reference response; W, ICBA; r, KA.
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. .
Figure 5(b). Same as Figure 5(a) except that the response now concerns the total field u on G' (measurement circle of radius = 2 outside of the body).
actual surface of the scatterer at a given point by its circle or spheres of curvature and thereby reduce the task of finding the Green function to that of employing the (known) Green function of a local canonical body (circular cylinder or sphere), this Green’s function varying from point to point on the boundary of the actual scatterer due to the fact that the radii of curvature of the latter vary from point to point (unless, of course, the scatterer is a circular cylinder or sphere). By employing the boundary integral representation involving this spatially varying canonical Green’s function on the boundary, Ursell was able to show that the kernel of the resulting integral equation of the second kind is small enough for it to be feasible to generate a convergent series solution by a process of iteration. The different orders of approximation of the field, obtained in this manner, correspond to different choices of the number of terms in the series. It has been shown recently by Jones [31] that the Ursell analysis is tantamount to the demonstration that, at high frequencies, the field and its normal derivative are related locally on the surface of a body of arbitrary (convex) shape in the same manner as on the local circle or sphere of curvature. Jones also demonstrated that a useful consequence of
543
this fact is the so-called on-surface radiation condition (OSRC), discovered in another manner, by Kriegsmann et al. [18], which provides yet another local approximation to the field on the surface of a perfectly reflecting body (or impedance boundary body [17]) of arbitrary shape. At lower frequencies, it is questionable, to say the least, whether the local character of the field can be retained. It seems obvious that the field at a point on a hemisphere is generally different if the latter is backed by a disc (of equal radius) rather than by another identical hemisphere (in which case the object is a sphere). Thus, at other than very high frequencies, it would appear that the field at an arbitrary point of a scattering object is related not only to the physical environment in the immediate vicinity of this point, but also to that of the configuration at distant points of the scatterer. In other words, using the tangent plane or local circle of curvature approximation might be justified at high frequencies, but not at medium-range or low frequencies. The question is then whether the notion of canonical body approximations can be extended to non-local phenomena (in the sense we have just described). Our approach to this problem is somewhat similar to that of Ursell [29, 30] in that we first adopt a boundary integral representation of the field and then concentrate our attention on the Green function, rather than on the single and double layer potentials that enter into this representation. This enables us to obtain different partial-wave representations outside different concentric circular disc-like domains that more or less cover the cross-section domain of the generally non-circular cylindrical object. Invoking a type of outgoing wave hypothesis and noticing the similarity of the actual scatterer with a circular cylinder in the circular disc-like domains suggest that a good approximation of the field in the neighborhood of the point of intersection of each circular domain with the
Figure 5(c). Same as Figure 5(a) except that the response now concerns the farfield scattering pattern B(u).
544
. .
Figure 6(a). Scattering response (surface layer potential 1v u) on the boundary G of a sound-soft almost-circular elliptic cylinder to an end-fire plane wave for wavenumber k = 20. q, BIEM reference response; W, ICBA; r, KA.
545
non-circular domain is that which would be obtained by scattering by the cylinder associated with the circular domain. An approximate expression is obtained for the field at other points of space (i.e., not on the boundary) by introducing the approximate expression of the surface field into the boundary integral representation in the same manner as with the Kirchhoff and OSRC approaches.
Figure 6(b). Same as Figure 6(a) except that the response now concerns the total field u on G' (measurement circle of radius = 2 outside of the body).
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. .
Figure 6(c). Same as (a) except that the response now concerns the farfield scattering pattern B(u).
To evaluate the new method (termed hereafter the intersecting canonical body approximation (ICBA) method because the boundaries of the canonical bodies generally intersect (i.e., are not osculating with respect to) the boundary of the given body at successive points of the latter) of computing the field scattered by a (n impenetrable) body, we shall compare the near and far-zone fields obtained in this manner with those computed
547
by a rigorous boundary integral equation and approximate Kirchhoff methods. This will be done for isolated elliptic and clover-leaf cylinders, insonified by a plane wave for two incident angles and three frequencies (low, medium and high). We find, using the rigorous boundary integral equation method (BIEM) results as reference solutions (i.e., reality) that
Figure 7(a). Scattering response (surface layer potential u) on the boundary G of a sound-hard (relatively) elongated elliptic cylinder to an end-fire plane wave for wavenumber k = 0·1. q, BIEM reference response; W, ICBA; r, KA.
548
. .
Figure 7(b). Same as Figure 7(a) except that the response now concerns the total field u on G' (measurement circle of radius = 2 outside of the body).
549
Figure 7(c). Same as Figure 7(a) except that the response now concerns the farfield scattering pattern B(u).
550
. .
Figure 8(a). Scattering response (surface layer potential u) on the boundary G of a sound-hard elongated elliptic cylinder to an end-fire plane wave for wavenumber k = 1·0. q, BIEM reference response; W, ICBA; r, KA.
the ICBA method gives results that are in generally better agreement with reality than the Kirchhoff method results, this being particularly true at low and medium frequencies. 2. PROBLEM INGREDIENTS
A sound-soft or sound-hard cylinder of known arbitrary shape is located within unbounded space such that its axis is the z-axis of the Oxyz Cartesian co-ordinate system.
551
The origin O is located within the cylinder. The exp (−ivt) time dependence (v the angular frequency and t the time) of the incident plane wave is imparted to the entire wave-field outside the cylinder and is omitted in the sequel. The orientation of the incident wave-vector (assumed to lie in the Oxy plane) and the cylindrical nature of the body imply
Figure 8(b). Same as Figure 8(a) except that the response now concerns the total field u on G' (measurement circle of radius = 2 outside of the body).
. .
552
that the total wave-field does not depend on z. The body occupies the simply-connected domain V1 of the cross-section plane Oxy, the remainder of the cross-section plane (not including the bounding curve G = 1V1 ) being denoted by V0 (see Figure 1). The object is to determine the field scattered in V0 by the cylinder, with G assumed to be representable by the polar parametric equation r = r(u), when r(u) is a continuous, twice differentiable, single valued function of u. This is done on the assumption that r(u), the frequency and angle of incidence of the incident wave are known. Let x denote the position vector in the cross-section plane and u i(x), u s(x) and u(x) = u i(x) + u s(x), the incident, scattered and total velocity potentials, respectively, in V0 , such that u i(x) = exp [−ikr cos (u − u i)],
(1)
with u i(u) the incident (current) angle measured counterclockwise from the positive x-axis, k = v/c the wavenumber and c the celerity in V0 . u(x) is locally square-integrable in V0 , governed by the Helmholtz equation [D + k 2 ]u(x) = 0, satisfies the outgoing wave condition at infinity, and obeys either the Dirichlet boundary condition u(r, u) = 0 (sound-soft cylinder) or Neumann boundary condition 1v u(r, u) = 0 (sound-hard cylinder, with 1v the normal derivative) for r = r(u), 0 E u Q 2p. It is required to obtain a good approximation of u(x) in V0 , as well as of 1v u(r, u) (for the sound-soft case) or u(r, u) (for the sound-hard case) on G.
3. BOUNDARY INTEGRAL FIELD REPRESENTATIONS USING THE FREE SPACE GREEN FUCNTION
The point of departure is Green’s theorem in V0 , with the free space Green function G(x'; x) = (i/4)H(1) 0 (k =x' − x=),
(2)
which leads [1] to
HV 0 (x')u(x') = u i(x') +
g
[G(x', x)1v u(x) − u(x)1v G(x', x)] dg(x),
(3)
G
wherein H(1) 0 is the first kind Hankel function of order 0, dg is the infinitesimal arc length along G, the normal derivative is taken for the normal to G directed towards V1 and the Heaviside function HV 0 equals 1 for x' $ V0 , and 0 for x' ( G + V0 . Next we make use of the well-known expression, in polar co-ordinates, of the free space Green’s function [1] a
(1) G(x'; x) = (i/4) s ein(u' − u) [H(r − r')H(1) n (kr)Jn (kr') + H(r' − r)Hn (kr')Jn (kr)],
(4)
n = −a
(where H(1) n and Jn are the first kind Hankel function and Bessel function of order n, H(z)
553
Figure 8(c). Same as Figure 8(a) except that the response now concerns the far-field scattering pattern B(u).
. .
554
is the Heaviside function equal to 1 for z q 0 and to 0 for z Q 0) and of its normal derivative 1v G(x'; r, u) = s−1 (r−1r˙ 1u − r1r )G(x'; r, u) a
=(i/4s) s ein(u' − u){(−inr−1r˙ )[H(r − r')H(1) n (kr)Jn (kr') n = −a
+ H(r' − r)H(1) n (kr')Jn (kr)] (1) −r[d(r − r')H(1) n (kr)Jn (kr') − d(r' − r)Hn (kr')Jn (kr)] (1) − kr[H(r − r')H (1) n (kr)Jn (kr') + H(r' − r)Hn (kr')J n (kr)]},
(5)
where s = s(u) = (r˙ + r ) , r = r(u), r˙ = dr/du, J n (z) = dJn (z)/dz, H (z) = dH (z)/ dz, d(z) is the Dirac delta function and the second bracket [] above vanishes in the sense of distributions. Thus, 2
(1) n
2 1/2
(1) n
a
1v G(x'; r, u) = (i/4s) s ein(u' − u){H(r − r')Jn (kr')[( − inr−1r˙ )H(1) (1) n (kr) − krH n (kr)] n = −a −1 +H(r' − r)H(1) r˙ )Jn (kr) − krJ n (kr)]}, n (kr')[(−inr
(6)
so that a
inu' HV 0 (x')u(x') = u i(x') + s [an (x')H(1) , n (kr') + bn (x')Jn (kr')]e
(7)
n = −a
Figure 8(d). Same as Figure 8(a) except that the response now concerns the function =B(u)=2 proportional to the bistatic scattering cross-section.
555
Figure 9(a). Scattering response (surface layer potential u) on the boundary G of a sound-hard elongated elliptic cylinder to an end-fire plane wave for wavenumber k = 20. q, BIEM reference response; W, ICBA; r, KA.
where
an (x') =
g
(i/4s)e−inuH(r' − r){Jn (kr)1v u(x)
G
− [(−inr−1r˙ )Jn (kr) − krJ n (kr)]u(x)} dg(x),
(8)
. .
556 bn (x') =
g
(i/4s)e−inuH(r − r'){H(1) n (kr)1v u(x)
G
(1) − [(−inr−1r˙ )H(1) n (kr) − krH n (kr)]u(x)} dg(x).
(9)
4. THE INTERSECTING CANONICAL BODY APPROXIMATION
For x' $ V0outM{r' q rmax ; 0 E u' Q 2p}, where rmax M max [r(u)], we find (see Figure 2, 0 E u Q 2p from equation (3), a
inu' , u(x') = u i(x') = s An H(1) n (kr') e
(10)
n = −a
where An =
g
(i/4s)e−inu{Jn (kr)1v u(x) − [(−inr−1r˙ )Jn (kr) − krJ n (kr)]u(x)} dg(x).
(11)
G
Equation (10) indicates that the scattered field in the ‘‘outer zone’’ V0out is expressible in terms of outgoing wave cylindrical harmonics with constant coefficients (i.e., that do not depend on the observation point x'). For x' $ V0inM{r(u') Q r' Q rmax ; 0 E u' Q 2p}, we find (see Figure 2), from equation (3), a
inu' , u(x') = u i(x') + s [an (x')H(1) n (kr') + bn (x')Jn (kr')] e
(12)
n = −a
where the coefficients are given by equations (8) and (9). This expression demonstrates the important property that the field in the ‘‘inner zone’’ neighborhood (V0in ) of a cylindrical body of arbitrary shape cannot be expressed in terms of cylindrical harmonics with constant coefficients as it can in the ‘‘outer zone’’ (see equation (10)). The spatially varying character of these coefficients in the ‘‘inner zone’’ (reminiscent of what is obtained in the DCA [20], which is an outcome of the rigorous Green theorem representation of the field, will be conserved in the intersecting canonical body approximation described below. Consider a point a (see Figure 3) on the scattering boundary G through which we draw a circle B centered at the origin O of the laboratory system. This circle B will generally intersect G at least at one other point (at present, there are three other intersection points d, e and f in Figure 3). Unless O is very far from the centroid of the body and/or the given body is very different (i.e., cigar-shaped) from a centered circular cylinder, there exists, as regards an observer in the neighborhood of points a, d, f and e, somewhat of a similarity between the region (shaded region in Figure 3) included between the actual body and that subtended by the circular cylinder B passing through these points (interior of circle B in Figure 3). This suggests that it is not unreasonable to assume that the scattered field, in, for instance, the neighborhood of point a on the actual body, can be approximated by the scattered field in the same neighborhood of the impenetrable circular cylinder B passing through this point. In order to make this hypothesis tangible, we first have to assume (assumption 1) that the second term in the brackets [] in equation (12) vanishes since no such term involving incoming cylindrical waves, appears in the field diffracted outside an impenetrable circular cylinder illuminated by a plane wave. Secondly, we have to assume (assumption 2) that
557
the an (x') in the first term of the brackets depend only on the product kh, where h is the radius of the intersecting circular cylinder (at present cylinder B). This is so because the exact solution of diffraction of a plane wave by an impenetrable circular cylinder of radius h is well-known and given by i an (kh) = −{Jn (kh)/H(1) n (kh)} exp ( − in(u + p/2)),
(13)
Figure 9(b). Same as Figure 9(a) except that the response now concerns the far-field scattering pattern B(u) relative to u s on G a.
. .
558
Figure 9(c). Same as Figure 9(a) except that the response now concerns the function =B(u)=2 proportional to the bistatic scattering cross-section.
for the sound-soft boundary and i an (kh) = −{J n (kh)/H (1) n (kh)} exp ( − in(u + p/2))
(14)
for the sound-hard boundary. Now if we apply this method of reasoning to the neighborhood of another point b on the actual body through which we draw the intersecting cylinder C, then we must once again make assumptions 1 and 2, except that, at present, h is not the radius of cylinder B, but of cylinder C. The generalization of this approach leads to none other than the intersecting canonical body approximation, expressed by a
u(r', u') = u i(r', u') + s an (kr(u'))H(1) n (kr') exp (inu'),
(r', u') $ V0in,
(15)
n = −a
wherein an is given by equations (13) or (14) with h being replaced by r(u'). We will evaluate the ICBA by introducing approximate single and double layer potentials associated with equation (15) into the boundary integral representation obtained from equation (3): i.e., u(x') = u i(x') +
g
[G(x', x)1v u(x') − u(x)1v G(x', x)] dg(x'), x' $ V0out
(16)
G
or, more specifically, u(x') = u i(x') +
g
G
G(x', x)1v u(x) dg(x),
x' $ V0out
(17)
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Figure 10(a). Scattering response of the surface layer potential 1v u on the boundary G of a sound-soft elongated elliptic cylinder to an end-fire plane wave for wavenumber k = 1·0. q, BIEM reference response; W, ICBA; r, KA.
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. .
Figure 10(b). Same as Figure 10(a) except that the response now concerns the total field u on G' (measurement circle of radius = 2 outside of the body).
561
for the sound-soft body, and u(x') = u i(x') −
g
u(x)1v G(x', x) dg(x),
x' $ V0out
(18)
G
for the sound-hard body. These ICBA layer potentials are (on the basis of equation (15)) N
1v u(x)=x$G = 1v u i(x)=x$G + s an (kr(u))1v [H(1) n (kr) exp (inu)]=x$G
(19)
n = −N
for the sound-soft body, and N
u(x)=x$G = u i(x)=x$G + s an (kr(u))[H(1) n (kr) exp (inu)]=x$G
(20)
n = −N
for the sound-hard body, wherein the series are limited to 2N + 1 (i.e., a finite number of) terms in order to anticipate the numerical applications. 5. THE KIRCHHOFF APPROXIMATION
The basis of the Kirchhoff approximation is a local description of the field on a scattering body, with an impenetrable medium of half-infinite extent bounded by a flat surface, tangent to the body at the given point at which the approximation is sought, playing the role of the canonical body [3, 10, 22, 32–34]. The Kirchhoff approximation (KA) distinguishes between the lit and shadow zones, so that 1v u(x)=x$G = 21v u i(x)=x$G 1v u(x)=x$G = 0
on the lit portion of G,
(21)
on the shadow portion of G,
(22)
on the lit portion of G,
(23)
for the sound-soft body, and u(x)=x$G = 2u i(x)=x$G
on the shadow portion of G,
u(x)=x$G = 0
(24)
for the sound-hard body. We will compare the KA with the ICBA by introducing the approximate single and double layer potentials of equations (21)–(26) into the boundary integral representations given by equations (17) and (18). 6. THE RIGOROUS BOUNDARY INTEGRAL SCHEME
The rigorous boundary integral method (BIEM) is now classical [3, 35] and the only reason we evoke it here is to indicate just which variant we employ to obtain the reference results which are compared in the next section to the ICBA and KA approximations. From equation (3) and the boundary conditions we obtain the integral equations (wherein PV denotes a Cauchy principal value) 0 = u i(x') +
g
G
G(x', x)1v u(x) dg(x), x' $ G,
(25)
. .
562 for the sound-soft body, and 1 2
u(x') = u i(x') − PV
g
u(x)1v G(x', x) dg(x),
x' $ G,
(26)
G
for the sound-hard body. These integral equations are solved numerically for the layer
Figure 10(c). Same as Figure 10(a) except that the response now concerns the farfield scattering pattern B(u).
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Figure 11. (a) Scattering response (surface layer potential 1v u) on the boundary G of a sound-soft elongated elliptic cylinder to a broadside plane wave for wavenumber k = 1·0. q, BIEM reference response; W, ICBA; r, KA.
potentials by the PS method described by Bolomey and Wirgin [36]. The layer potentials are then introduced into equations (17) and (18) to compute the field outside the scattering body. Thus, the approximate methods (ICBA and KA) and the rigorous method rely on the same computation (via equations (17) and (18)) for determining the field outside the scattering body.
. .
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7. EVALUATION OF THE ICBA AND KA BY COMPARISON WITH RIGOROUS BOUNDARY INTEGRAL METHOD REFERENCE RESULTS
The computations were done for elliptic cylinders whose boundary is defined by the parametric equation r = r(u) = ab/(a 2 sin2 u + b 2 cos2 u)1/2.
(27)
The chosen elliptic cylinders were almost circular (half axes a = 1·2, b = 1·0 arbitrary units) or relatively elongated (half axes a = 1·5, b = 1·0 arbitrary units). Computations were also done for the (non-convex) three-leaf clover cylinder r = r(u) = a − b sin (3u),
(28)
for a = 1, b = 0·3. Each cylinder is such that its center line coincides with the z-axis of the laboratory system (see Figure 4). The cylinders are illuminated by end-fire (u i = 0) or broadside (u i = p/2) incident plane waves at three frequencies corresponding to k = 0·1, 1 and 20 (same arbitrary units as the axes lengths), which are referred to hereafter as low, medium and high frequencies. Surface layer potentials (1v u or u on G) are computed as well as the total field u in the near zone on the circle G' of radius r = 2 (arbitrary units) and the diffracted field u s(x) in the far zone on the circle G a of radius r:a. More precisely, the field has the asymptotic behavior u s(x') 0 B(u')
0 1 2 pkr'
1/2
exp [i(kr' − p/4)], kr':a (i.e., x':G a),
(29)
e−ikrcos(u' − u)1v u(r, u)(r˙ 2 + r 2 )1/2 du
(30)
where B(u') =
i 4
g
2p
0
for the sound-soft body and B(u') =
−k 4
g
2p
e−ikrcos(u' − u)u(r, u)[r˙ sin (u' − u) − r cos (u' − u)] du
(31)
0
for the sound-hard body. We compute the real and imaginary parts of the farfield scattering function B(u) in order to appreciate to what extent both the magnitude and phase of this oft-measured farfield quantity, both of which are critical in inverse problem applications, are well-accounted for by the ICBA. In addition, we compute the quantity =B(u)=2 which is proportional to the widely used bistatic scattering cross-section in practical forward scattering problems. Table 1 gives a synoptic picture of the input (columns 2–8) and output (columns 9–12) parameters of the different numerical results. The values of N (recall that 2N + 1 terms are retained in the ICBA surface layer potential series) in column 8 are only indicative; tests were made to ensure that the results did not change significantly for larger values of N. Column 9 indicates which type of approximation (ICBA or KA) gives the closest agreement with the reference (BIEM) results on the boundary curve G. Columns 10 and 11 indicate the same sort of thing on measurement circles G' and G a, respectively. Column
565
12 gives an overall (on G, G' and G ) indication of quality of the best approximation to the scattering response. In all cases, the ICBA gives the best approximation of the response, which, in some exceptional cases is attained also by the KA (actually, there are not very large differences between the KA and ICBA at high frequencies), and the quality of this approximation is either good or excellent. a
Figure 11(b). Same as Figure 11(a) except that the response now concerns the total field u on G' (measurement circle of radius = 2 outside of the body).
566
. .
Figure 11(c). Same as Figure 11(a) except that the response now concerns the farfield scattering pattern B(u).
Let us consider Figures 5–12 in some detail. We have included results for the almost circular cylinder because the ICBA is exact for the circular cylinder (with center at the origin of the laboratory system) and should be all the better the closer the shape of the given cylinder is to circular. The KA, on the other hand, is not, on the whole, more valid for a circular cylinder than for an elliptical cylinder and should not be accurate at low
567
Figure 11(d). Same as Figure 11(a) except that the response now concerns the function =B(q=2 proportional to the bistatic scattering cross-section.
frequencies. This is borne out by the results in Figures 5 and 7 where it is also seen that the ICBA is far superior to the KA at all sampling points. Another feature of Figures 5 and 7 is that the ICBA approximation of the field on G' and G a is generally better than on the scattering surface G itself. Comparing Figures 6 and 9 shows, at high frequencies, that the KA and ICBA give almost equivalent results that are in excellent or good agreement with reality at all sampling points, including on the scatterer itself. However, Figure 8 shows that only the ICBA correctly accounts for reality, although with some difficulty on G a (note that the large differences in Figure 8(d) as compared to Figure 8(c) are due to the squaring operation). Figures 10 and 11 (as compared to Figure 8) show that the ICBA generally fares better for sound-soft than for sound-hard cylinders of the same shape, at least at intermediate frequencies. Once again, these figures show that the ICBA is a much better approximation than the KA in the resonance region, be it for one or the other incident angles. Finally, Figure 12 shows that the ICBA provides a decent approximation of the field at all sampling points even for moderately non-convex scattering bodies, the same being much less true for the KA. 8. CONCLUSIONS
The proposed (ICBA) approximation (embodied by equations (13)–(15)) is extremely simple to use and shares with the widely used Kirchhoff approximation the attractive feature of eliminating the arduous numerical task of solving systems of algebraic or integral equations in order to determine the surface layer potentials. This is of particular interest for solving the inverse scattering problem, i.e., determining the shape of the body from measurements of the scattered field, as has been shown by Lewis [2] and Tabbara [36] for the KA, and by Scotti and Wirgin [37–39] for the ICBA.
568
. .
Figure 12(a). Scattering response (surface layer potential u) on the boundary G of a sound-hard three-leaf clover cylinder to an end-fire plane wave for wavenumber k = 1. q, BIEM reference response; W, ICBA; r, KA.
569
Figure 12(b). Same as Figure 12(a) except that the response now concerns the total field u on G' (measurement circle of radius = 2 outside of the body).
570
. .
Figure 12(c). Same as Figure 12(a) except that the response now concerns the farfield scattering pattern B(u).
571
The ICBA has another unique feature which is that it is applicable at all frequencies (recall that the Kirchhoff approximation furnishes essentially a high-frequency description of the field); herein we have shown this to be true over a range of frequencies for which the highest is 200 times the lowest frequency. This makes the ICBA very useful (and more suitable than the Kirchhoff approximation) for determining scattering response in the time-domain as well as for pulse-echo target identification. The ICBA, shares with the Kirchhoff approximation, the feature of being less-well adapted to the description of the response of highly non-convex bodies for which multiple scattering can play a major role [33]. For the same reason, it is not appropriate to use the ICBA, in its present form, for scattering configurations involving more than one body (unless multiple scattering can be neglected). However, suitably-modified versions of the ICBA can be (and are being) developed for isolated penetrable 2-D [38–40] and 3-D bodies, isolated impenetrable 3-D bodies and isolated coated 2-D and 3-D bodies. A final feature of the ICBA is that it can be cast into even simpler form at both low and high frequencies by appropriate asymptotic manipulation (e.g., Watson transformation) of the partial wave solution of the canonical body scattering solution. This feature is common to the DCA and HPSA techniques of Stanton [19, 20]. For all these reasons it would appear that the ICBA provides a useful new alternative to the well-known Rayleigh, Kirchhoff, geometrical optics, Fock etc., approximations of the scattered field. ACKNOWLEDGMENTS
The authors would like to thank Walid Tabbara for calling our attention to the Adachi approximation and to other related references. We also thank Ralph Kleinman for underlining a possible link between the ICBA and Ursell’s work on local Green’s functions. Two anonymous reviewers have greatly helped to improve the presentation of this work and called our attention to the recent contributions of D. S. Jones to the OSRC issue and T. K. Stanton to the DCA and HPSA techniques. REFERENCES 1. P. M. M and H. F 1953 Methods of Theoretical Physics. New York: McGraw-Hill. 2. R. M. L 1969 IEEE Transations on Antennas and Propagation AP-17. 308–314. Physical optics inverse diffraction. 3. D. C and R. K 1992 Inverse acoustic and Electromagnetic Scattering Theory, Berlin: Springer (applied mathematical sciences No 93). 4. L R 1897 Philosophical Magazine 44, 28–52. On the incidence of aerial and electric waves upon small obstacles in the form of ellipsoids or elliptic cylinders and on the passage of electric waves through a circular aperture in a conducting screen. 5. A. F. S 1953 Journal of Applied Physics 24, 1134–1142. Solutions of electromagnetic scattering problems as power series in the ratio (dimension of scatterer/wavelength). 6. R. E. K 1978 in Electromagnetic Scattering (P. L. E. Uslenghi, editor) 1–28. Low frequency electromagnetic scattering. New York: Academic Press. 7. R. E. K and T. B. A. S 1986 in Low and High Frequency Asymptotics (V. K. Varadan and V. V. Varadan, editors) 1–70. Rayleigh scattering. Amsterdam: Elsevier. 8. J. B. K 1962 Journal of the Optical Society of America 52, 116–321. Geometrical theory of diffraction. 9. J. B. K 1956 IRE Transactions, AP-4, 312–321. Diffraction by a convex cylinder. 10. F. G. B and I. M. F 1979 Wave Scattering from Statistically Rough Surfaces. Oxford: Pergamon. 11. S. A 1965 Proc. IEEE 53, 1067–1068. The nose-on echo area of axially symmetric thin bodies having sharp apices. 12. V. A. F 1945 Journal of Physics (USSR) 9, 255–266. Diffraction of radio waves around the earth’s surface.
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13. V. A. F 1946 Journal of Physics (USSR) 10, 130–136. The distribution of currents induced by a plane wave on the surface of a conductor. 14. V. A. F 1946 Journal of Physics (USSR) 10, 399–409. The field of a plane wave near the surface of a conducting body. 15. V. A. F and A. A. F 1958 Zhurnal Tekhicheskoi Fiziki 28, 2548–2566. Diffraction of a plane electromagnetic wave by a perfectly conducting paraboloid of revolution. 16. P. J. L 1970 Journal of the Acoustical Society of America 47, 804–815. Curvature corrections to rough-surface scattering at high frequencies. 17. J.-M. J, J. L. V and V. V. L 1989 IEEE Transactions on Antennas and Propagation 37, 118–124. A comparative study of the OSRC approach in electromagnetic scattering. 18. G. A. K, A. T and K. R. U 1987 IEEE Transactions on Antennas and Propagation AP-35, 153–161. A new formulation of electromagnetic wave scattering using an on-surface radiation condition approach. 19. T. K. S 1989 Journal of the Acoustical Society of America 86, 1499–1510. Simple approximate formulas for backscattering of sound by spherical and elongated objects. 20. T. K. S 1989 Journal of the Acoustical Society of America 86, 691–705. Sound scattering by cylinders of finite length, III: Deformed cylinders. 21. T. B. A. S 1960 Applied Scientific Research B8, 418–436. Impedance boundary conditions for imperfectly conducting surface. 22. P. B and A. S 1963 The Scattering of Electromagnetic Waves from Rough Surfaces. Oxford: Pergamon. 23. J. C. B and A. W 1976 Proceedings of the Institution of Electrical Engineers 123, 1201–1207. Theoretical determination of the electromagnetic field in the neighborhood of a parabolic cylinder antenna. 24. G. N. W 1918 Proceedings of the Royal Society of London A95, 83–99. The diffraction of electric waves by the earth. 25. W. F and K. D 1952 Annalen der Physik 10, 361–373. Theory of diffraction by a cylinder as affected by the surface wave. 26. D. S. J 1979 Methods in Electromagnetic Wave Propagation. Oxford: Clarendon. 27. V. H. W 1967 in Electromagnetic Wave Theory (J. Brown, editor). Oxford: Pergamon, 835–844. Recent highlights in diffraction theory. 28. F. U 1968 Proceedings of the Cambridge Philosophical Society 64, 171–191. Creeping modes in a shadow. 29. F. U 1957 Proceedings of the Cambridge Philosophical Society, 53, 115–133. On the short-wave asymptotic theory of the wave equation (9 2 + k 2 )f = 0. 30. F. U 1966 Proceedings of the Cambridge Philosophical Society 62, 227–244. On the rigourous foundation of short-wave asymptotics. 31. D. S. J 1994 Quarterly Journal of Mechanics and Applied Mathematics 47, 629–634. Physical optics and the surface radiation condition. 32. N. N. B 1982 IEEE Transactions on Antennas and Propagation AP-30, 980–989. A survey of the physical optics inverse scattering identity. 33. A. W 1983 Journal of the Optical Society of America 73, 1028–1041. Scattering from sinusoidal gratings: an evaluation of the Kirchhoff approximation. 34. L. B and S. S 1995 Acoustical Physics 41, 637–648. Use of the Kirchhoff approximation in scattering from elastic rough surfaces. 35. P. J. T. F 1977 Journal of Sound and Vibration 54, 473–500. Layer potentials and acoustic diffraction. 36. J. C. B and A. W 1974 Proceedings of the Institution of Electrical Engineers 121, 794–804. Numerical comparison of the Green’s function and the Waterman and Rayleigh theories of scattering from a cylinder with arbitrary cross-section. 37. W. T 1973 IEEE Transactions on Antennas and Propagation AP-21, 245–247. On an inverse scattering method. 38. T. S and A. W 1995 C.R. Acad. Sci. Paris II b, 330, 641–646. Location and shape reconstruction of a soft body by means of canonical solutions and measured scattered sound fields. 39. T. S and A. W 1995 Inverse Problems 11, 1097–1111. Shape reconstruction using diffracted waves and canonical solutions. 40. A. W and T. S 1995 in Nondestructive Testing of Materials (R. Collins, W. D. Dover, J. R. Bowler and K. Miya, editors). Oxford: IOS Press, 271–279. Location and shape reconstruction of a penetrable body using solutions of a canonical forward wave interaction problem and measurements of scattered electromagnetic waves.
WIDE-BAND APPROXIMATION OF THE SOUND FIELD SCATTERED BY AN IMPENETRABLE BODY OF ARBITRARY SHAPE A. W T. S Laboratoire de Me´canique et d’Acoustique, U.P.R. 7051 du C.N.R.S., 31 chemin Joseph Aiguier, 13402 Marseille ce´dex 20, France (Received 26 July 1995, and in final form 27 November 1995) The scattered field on the boundary of an insonified hard or soft cylinder of arbitrary shape (CAS) is obtained in approximate manner by using explicit forward scattering solutions of insonified impenetrable circular cylinders (CC) whose radii vary with location on the boundary of the CAS. This so-called intersecting canonical body approximation (ICBA) is employed in the boundary integral representation in order to determine field functions at arbitrary points on or outside the scattering body. The approximation is evaluated in the near and farfield zone by comparison with rigorous solutions obtained from a boundary-integral equation scheme. The ICBA is shown to give results that are of better quality in the low frequency and resonance regions and of equivalent quality in the high frequency region as compared with the approximate solution deriving from the oft employed Kirchhoff ansatz of the surface field.
7 1996 Academic Press Limited
1. INTRODUCTION
Exact solutions of scattering problems in acoustics and electromagnetics are available [1] only for isolated bodies of simple shape (e.g., impenetrable, penetrable, or coated circular, elliptic and parabolic cylinders, spheres, ellipsoids, layered half-space bodies with flat boundaries and interfaces etc.). The need for approximate methods to describe scattering from arbitrarily shaped obstacles is still acute, particularly in the context of 3D forward scattering (for which the number of unknowns can be huge) and 2D or 3D inverse scattering problems (i.e., determine the shape of a body from the scattered field, a task [2, 3] which also can require an enormous amount of computations if rigorous field representations are employed). At low frequencies, one can appeal to the Rayleigh method [4] and its variants [5–7], but this requires that exact solutions of the Laplace equation be available, which is not usually the case when the shape of the given body is arbitrary (i.e., non-canonical). At high frequencies, it is appropriate to make use of (ordinary) geometric optics [8], the geometrical theory of diffraction [8, 9] or different variants of the so-called physical optics approximation (e.g., the Kirchhoff approximation [10], the Adachi approximation [11], the Fock approximation [12–15] etc.), the Lynch approximation [16]. However, a reliable approximate method still remains to be found for scattering by arbitrarily shaped bodies in the medium frequency (resonance) region which, hopefully, could also apply at high and low frequencies, although there is some evidence [17] that the recently proposed OSRC method [18] may partially fulfill these requirements (at least for circular cylinders, as concerns the backscattering cross-section, at all frequencies, and for cylinders with other 537 0022–460X/96/290537 + 36 $18.00/0
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. .
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shapes, as concerns the bistatic cross-section, in the medium to high frequency range). Another promising method [19], based on the so-called highpass scattering approximation (HPSA), applies to scatterers of a variety of shapes and at all frequencies from low to high, but concerns only the backscattering cross-section and involves heuristic (curve-fitting) functions to account for the fine-scale features of the response. The present work addresses this issue by considering a canonical boundary-value problem which is in some sense close to the given one and which can be solved, in closed form, in both the near-field and farfield zones as well as at all frequencies. Herein (i.e., for 2-D configurations) the canonical boundary-value problem is that of scattering from an impenetrable circular cylinder (CC). The impenetrable CC is replaced by a penetrable CC when the scattering is provoked by a penetrable cylinder of arbitrary shape (CAS) or y
Ω0 Γ
r
θ
Ω1 O
θi
x
Figure 1. The 2-D scattering configuration.
Figure 2. Definition of ‘‘inner’’ and ‘‘outer’’ zones in the vicinity of the scattering body.
Figure 3. The canonical bodies passing through points a and b of the given scattering body.
Figure 4. Graphs (in the cross-section plane) of the scattering bodies considered in the numerical study. (a), Three-leaf clover cylinder; (b) elongated elliptical cylinder; (c), nearly circular elliptic cylinder.
soft soft hard hard hard soft soft hard
7 8 9 10 11 12
Sound hard/soft
5 6
Figure number
ellip. ellip. clover
ellip. ellip. ellip.
ellip. ellip.
Type cylinder
1·5 1·5 1·0
1·5 1·5 1·5
1·2 1·2
a
1·0 1·0 0·3
1·0 1·0 1·0
1.0 1·0
b
0 p/2 0
0 0 0
0 0
ui
1·0 1·0 1·0
0·1 1·0 20·0
0·1 20·0
k
15 15 18
6 15 30
4 20
N ICBA ICBA, KA ICBA ICBA ICBA, KA ICBA ICBA ICBA
Best appr. on G
ICBA ICBA ICBA
ICBA ICBA, KA ICBA ICBA
Best appr. on G'
T 1 Input and output parameters for the numerical examples
ICBA ICBA, KA ICBA ICBA ICBA KA ICBA ICBA ICBA
Best appr. on G a
excel. good good
excel. good good
excel. excel.
Overall quality
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by an impenetrable or penetrable sphere when the scatterer is a 3-D body of arbitrary shape (work to be published elsewhere). There exists a host of examples in which canonical problem solutions have been used in the past in a local setting. The hypothesis underlying the use of canonical problems to approximate the field of more complicated problems is that there is a one-to-one correspondence between the field scattered by an appropriate canonical body with that scattered by the complicated body at each point of the boundary of the latter. A recent example of a local technique employing canonical body solutions for complicated cylinder-like scatterers can be found in the work of Stanton [20]. His idea (which is called the deformed cylinder approximation (DCA)), for instance, for a cylinder with varying (although always circular) cross-section area along its axis, is to apply the known solution to the canonical problem of scattering by the infinite cylinder passing through successive points of the boundary as one moves along the axis of the actual cylinder to obtain the overall response outside the given cylinder. As shown further on, the DCA is somewhat similar to the intersecting canonical body approximation (ICBA) proposed herein, although it should be borne in mind that the ICBA concerns (here) a cylinder with generally non-circular cross-section which is the same and of the same area all along the cylinder axis as opposed to the DCA which concerns a cylinder with usually circular cross-section (it could also be elliptical but this makes the DCA much more unyieldy) which varies in area along the cylinder axis. An older, more well-known example of the use of canonical body solutions, the so-called impedance boundary condition (also known as the Leontovich boundary condition), which is often used to simplify transmission problems [10, 21], derives from the closed-form solution of the canonical problem of a plane wave impinging on a flat interface between two media and is thought to be valid even at a given point of a curved boundary provided the product of the radius of curvature at that point with the wavenumber in the medium of higher refractive index is large. The Kirchhoff or tangent plane approximation [10, 22] is another example in which the solution to the canonical problem of a plane wave impinging on a flat interface between two media (one of which may be impenetrable) is employed locally to provide an approximation of the field on the illuminated portion of the surface of a curved body, it being assumed that the field vanishes on the portion of the boundary within the geometric shadow when the body is impenetrable. Another approximation for impenetrable bodies, called the Adachi approximation [11], assumes that the Kirchhoff approximation continues as such onto the shadowed portion of the surface. As shown by Bass and Fuks [10], the (ordinary) geometrical optics approximation [8] of the field outside a scattering body can be obtained by a stationary phase approximation of the Kirchhoff–Helmholtz integral (rigorous boundary integral representation of the field) subsequent to use of the Kirchhoff approximation of the surface field within the integral. In this respect, geometric optics is also an outcome of using canonical solutions in a local context. An improvement of the Kirchhoff approximation has been obtained by Lynch [16] via a phase modification dependent on the curvature of the scattering body at each point of its surface. A more exact description of the fields at high frequencies, particularly near caustics [23], in the transition region between light and shadow, and deeper into the shadow region of a curved obstacle, requires examination of a canonical problem that is closer to the original problem, notably by the fact of incorporating curvature of the scatterer. For this purpose, partial-wave solutions of the circular cylinder and sphere scattering problems, recast into forms more suitable for high frequency analysis by means of the Watson transform [24], have been used by Fock [12, 13, 15], Franz and Depperman [25], Keller
541
[8, 9] and others [26–28] to develop the so-called Fock, geometrical theory of diffraction and creeping wave approximations of the field. A different and original line of attack to approximation of scattered fields at high frequencies was taken by Ursell [29, 30]. Instead of introducing a suitable approximation of the field on the surface into the Kirchhoff–Helmholtz integral representation (which employs the free space Green function), Ursell suggested generating approximations to the field represented by an integral involving a Green function satisfying either a Neumann or Dirichlet boundary condition on the boundry of the scatterer. Owing to the fact that finding this Green function is as difficult as determining the scattered field itself, Ursell assumes locality (justifiable at high frequencies) which fact makes it possible to replace the
Figure 5(a). Scattering response (surface layer potential 1v u) on the boundary G of a sound-soft almost-circular elliptic cylinder to an end-fire plane wave for wavenumber k = 0·1. q, BIEM reference response; W, ICBA; r, KA.
542
. .
Figure 5(b). Same as Figure 5(a) except that the response now concerns the total field u on G' (measurement circle of radius = 2 outside of the body).
actual surface of the scatterer at a given point by its circle or spheres of curvature and thereby reduce the task of finding the Green function to that of employing the (known) Green function of a local canonical body (circular cylinder or sphere), this Green’s function varying from point to point on the boundary of the actual scatterer due to the fact that the radii of curvature of the latter vary from point to point (unless, of course, the scatterer is a circular cylinder or sphere). By employing the boundary integral representation involving this spatially varying canonical Green’s function on the boundary, Ursell was able to show that the kernel of the resulting integral equation of the second kind is small enough for it to be feasible to generate a convergent series solution by a process of iteration. The different orders of approximation of the field, obtained in this manner, correspond to different choices of the number of terms in the series. It has been shown recently by Jones [31] that the Ursell analysis is tantamount to the demonstration that, at high frequencies, the field and its normal derivative are related locally on the surface of a body of arbitrary (convex) shape in the same manner as on the local circle or sphere of curvature. Jones also demonstrated that a useful consequence of
543
this fact is the so-called on-surface radiation condition (OSRC), discovered in another manner, by Kriegsmann et al. [18], which provides yet another local approximation to the field on the surface of a perfectly reflecting body (or impedance boundary body [17]) of arbitrary shape. At lower frequencies, it is questionable, to say the least, whether the local character of the field can be retained. It seems obvious that the field at a point on a hemisphere is generally different if the latter is backed by a disc (of equal radius) rather than by another identical hemisphere (in which case the object is a sphere). Thus, at other than very high frequencies, it would appear that the field at an arbitrary point of a scattering object is related not only to the physical environment in the immediate vicinity of this point, but also to that of the configuration at distant points of the scatterer. In other words, using the tangent plane or local circle of curvature approximation might be justified at high frequencies, but not at medium-range or low frequencies. The question is then whether the notion of canonical body approximations can be extended to non-local phenomena (in the sense we have just described). Our approach to this problem is somewhat similar to that of Ursell [29, 30] in that we first adopt a boundary integral representation of the field and then concentrate our attention on the Green function, rather than on the single and double layer potentials that enter into this representation. This enables us to obtain different partial-wave representations outside different concentric circular disc-like domains that more or less cover the cross-section domain of the generally non-circular cylindrical object. Invoking a type of outgoing wave hypothesis and noticing the similarity of the actual scatterer with a circular cylinder in the circular disc-like domains suggest that a good approximation of the field in the neighborhood of the point of intersection of each circular domain with the
Figure 5(c). Same as Figure 5(a) except that the response now concerns the farfield scattering pattern B(u).
544
. .
Figure 6(a). Scattering response (surface layer potential 1v u) on the boundary G of a sound-soft almost-circular elliptic cylinder to an end-fire plane wave for wavenumber k = 20. q, BIEM reference response; W, ICBA; r, KA.
545
non-circular domain is that which would be obtained by scattering by the cylinder associated with the circular domain. An approximate expression is obtained for the field at other points of space (i.e., not on the boundary) by introducing the approximate expression of the surface field into the boundary integral representation in the same manner as with the Kirchhoff and OSRC approaches.
Figure 6(b). Same as Figure 6(a) except that the response now concerns the total field u on G' (measurement circle of radius = 2 outside of the body).
546
. .
Figure 6(c). Same as (a) except that the response now concerns the farfield scattering pattern B(u).
To evaluate the new method (termed hereafter the intersecting canonical body approximation (ICBA) method because the boundaries of the canonical bodies generally intersect (i.e., are not osculating with respect to) the boundary of the given body at successive points of the latter) of computing the field scattered by a (n impenetrable) body, we shall compare the near and far-zone fields obtained in this manner with those computed
547
by a rigorous boundary integral equation and approximate Kirchhoff methods. This will be done for isolated elliptic and clover-leaf cylinders, insonified by a plane wave for two incident angles and three frequencies (low, medium and high). We find, using the rigorous boundary integral equation method (BIEM) results as reference solutions (i.e., reality) that
Figure 7(a). Scattering response (surface layer potential u) on the boundary G of a sound-hard (relatively) elongated elliptic cylinder to an end-fire plane wave for wavenumber k = 0·1. q, BIEM reference response; W, ICBA; r, KA.
548
. .
Figure 7(b). Same as Figure 7(a) except that the response now concerns the total field u on G' (measurement circle of radius = 2 outside of the body).
549
Figure 7(c). Same as Figure 7(a) except that the response now concerns the farfield scattering pattern B(u).
550
. .
Figure 8(a). Scattering response (surface layer potential u) on the boundary G of a sound-hard elongated elliptic cylinder to an end-fire plane wave for wavenumber k = 1·0. q, BIEM reference response; W, ICBA; r, KA.
the ICBA method gives results that are in generally better agreement with reality than the Kirchhoff method results, this being particularly true at low and medium frequencies. 2. PROBLEM INGREDIENTS
A sound-soft or sound-hard cylinder of known arbitrary shape is located within unbounded space such that its axis is the z-axis of the Oxyz Cartesian co-ordinate system.
551
The origin O is located within the cylinder. The exp (−ivt) time dependence (v the angular frequency and t the time) of the incident plane wave is imparted to the entire wave-field outside the cylinder and is omitted in the sequel. The orientation of the incident wave-vector (assumed to lie in the Oxy plane) and the cylindrical nature of the body imply
Figure 8(b). Same as Figure 8(a) except that the response now concerns the total field u on G' (measurement circle of radius = 2 outside of the body).
. .
552
that the total wave-field does not depend on z. The body occupies the simply-connected domain V1 of the cross-section plane Oxy, the remainder of the cross-section plane (not including the bounding curve G = 1V1 ) being denoted by V0 (see Figure 1). The object is to determine the field scattered in V0 by the cylinder, with G assumed to be representable by the polar parametric equation r = r(u), when r(u) is a continuous, twice differentiable, single valued function of u. This is done on the assumption that r(u), the frequency and angle of incidence of the incident wave are known. Let x denote the position vector in the cross-section plane and u i(x), u s(x) and u(x) = u i(x) + u s(x), the incident, scattered and total velocity potentials, respectively, in V0 , such that u i(x) = exp [−ikr cos (u − u i)],
(1)
with u i(u) the incident (current) angle measured counterclockwise from the positive x-axis, k = v/c the wavenumber and c the celerity in V0 . u(x) is locally square-integrable in V0 , governed by the Helmholtz equation [D + k 2 ]u(x) = 0, satisfies the outgoing wave condition at infinity, and obeys either the Dirichlet boundary condition u(r, u) = 0 (sound-soft cylinder) or Neumann boundary condition 1v u(r, u) = 0 (sound-hard cylinder, with 1v the normal derivative) for r = r(u), 0 E u Q 2p. It is required to obtain a good approximation of u(x) in V0 , as well as of 1v u(r, u) (for the sound-soft case) or u(r, u) (for the sound-hard case) on G.
3. BOUNDARY INTEGRAL FIELD REPRESENTATIONS USING THE FREE SPACE GREEN FUCNTION
The point of departure is Green’s theorem in V0 , with the free space Green function G(x'; x) = (i/4)H(1) 0 (k =x' − x=),
(2)
which leads [1] to
HV 0 (x')u(x') = u i(x') +
g
[G(x', x)1v u(x) − u(x)1v G(x', x)] dg(x),
(3)
G
wherein H(1) 0 is the first kind Hankel function of order 0, dg is the infinitesimal arc length along G, the normal derivative is taken for the normal to G directed towards V1 and the Heaviside function HV 0 equals 1 for x' $ V0 , and 0 for x' ( G + V0 . Next we make use of the well-known expression, in polar co-ordinates, of the free space Green’s function [1] a
(1) G(x'; x) = (i/4) s ein(u' − u) [H(r − r')H(1) n (kr)Jn (kr') + H(r' − r)Hn (kr')Jn (kr)],
(4)
n = −a
(where H(1) n and Jn are the first kind Hankel function and Bessel function of order n, H(z)
553
Figure 8(c). Same as Figure 8(a) except that the response now concerns the far-field scattering pattern B(u).
. .
554
is the Heaviside function equal to 1 for z q 0 and to 0 for z Q 0) and of its normal derivative 1v G(x'; r, u) = s−1 (r−1r˙ 1u − r1r )G(x'; r, u) a
=(i/4s) s ein(u' − u){(−inr−1r˙ )[H(r − r')H(1) n (kr)Jn (kr') n = −a
+ H(r' − r)H(1) n (kr')Jn (kr)] (1) −r[d(r − r')H(1) n (kr)Jn (kr') − d(r' − r)Hn (kr')Jn (kr)] (1) − kr[H(r − r')H (1) n (kr)Jn (kr') + H(r' − r)Hn (kr')J n (kr)]},
(5)
where s = s(u) = (r˙ + r ) , r = r(u), r˙ = dr/du, J n (z) = dJn (z)/dz, H (z) = dH (z)/ dz, d(z) is the Dirac delta function and the second bracket [] above vanishes in the sense of distributions. Thus, 2
(1) n
2 1/2
(1) n
a
1v G(x'; r, u) = (i/4s) s ein(u' − u){H(r − r')Jn (kr')[( − inr−1r˙ )H(1) (1) n (kr) − krH n (kr)] n = −a −1 +H(r' − r)H(1) r˙ )Jn (kr) − krJ n (kr)]}, n (kr')[(−inr
(6)
so that a
inu' HV 0 (x')u(x') = u i(x') + s [an (x')H(1) , n (kr') + bn (x')Jn (kr')]e
(7)
n = −a
Figure 8(d). Same as Figure 8(a) except that the response now concerns the function =B(u)=2 proportional to the bistatic scattering cross-section.
555
Figure 9(a). Scattering response (surface layer potential u) on the boundary G of a sound-hard elongated elliptic cylinder to an end-fire plane wave for wavenumber k = 20. q, BIEM reference response; W, ICBA; r, KA.
where
an (x') =
g
(i/4s)e−inuH(r' − r){Jn (kr)1v u(x)
G
− [(−inr−1r˙ )Jn (kr) − krJ n (kr)]u(x)} dg(x),
(8)
. .
556 bn (x') =
g
(i/4s)e−inuH(r − r'){H(1) n (kr)1v u(x)
G
(1) − [(−inr−1r˙ )H(1) n (kr) − krH n (kr)]u(x)} dg(x).
(9)
4. THE INTERSECTING CANONICAL BODY APPROXIMATION
For x' $ V0outM{r' q rmax ; 0 E u' Q 2p}, where rmax M max [r(u)], we find (see Figure 2, 0 E u Q 2p from equation (3), a
inu' , u(x') = u i(x') = s An H(1) n (kr') e
(10)
n = −a
where An =
g
(i/4s)e−inu{Jn (kr)1v u(x) − [(−inr−1r˙ )Jn (kr) − krJ n (kr)]u(x)} dg(x).
(11)
G
Equation (10) indicates that the scattered field in the ‘‘outer zone’’ V0out is expressible in terms of outgoing wave cylindrical harmonics with constant coefficients (i.e., that do not depend on the observation point x'). For x' $ V0inM{r(u') Q r' Q rmax ; 0 E u' Q 2p}, we find (see Figure 2), from equation (3), a
inu' , u(x') = u i(x') + s [an (x')H(1) n (kr') + bn (x')Jn (kr')] e
(12)
n = −a
where the coefficients are given by equations (8) and (9). This expression demonstrates the important property that the field in the ‘‘inner zone’’ neighborhood (V0in ) of a cylindrical body of arbitrary shape cannot be expressed in terms of cylindrical harmonics with constant coefficients as it can in the ‘‘outer zone’’ (see equation (10)). The spatially varying character of these coefficients in the ‘‘inner zone’’ (reminiscent of what is obtained in the DCA [20], which is an outcome of the rigorous Green theorem representation of the field, will be conserved in the intersecting canonical body approximation described below. Consider a point a (see Figure 3) on the scattering boundary G through which we draw a circle B centered at the origin O of the laboratory system. This circle B will generally intersect G at least at one other point (at present, there are three other intersection points d, e and f in Figure 3). Unless O is very far from the centroid of the body and/or the given body is very different (i.e., cigar-shaped) from a centered circular cylinder, there exists, as regards an observer in the neighborhood of points a, d, f and e, somewhat of a similarity between the region (shaded region in Figure 3) included between the actual body and that subtended by the circular cylinder B passing through these points (interior of circle B in Figure 3). This suggests that it is not unreasonable to assume that the scattered field, in, for instance, the neighborhood of point a on the actual body, can be approximated by the scattered field in the same neighborhood of the impenetrable circular cylinder B passing through this point. In order to make this hypothesis tangible, we first have to assume (assumption 1) that the second term in the brackets [] in equation (12) vanishes since no such term involving incoming cylindrical waves, appears in the field diffracted outside an impenetrable circular cylinder illuminated by a plane wave. Secondly, we have to assume (assumption 2) that
557
the an (x') in the first term of the brackets depend only on the product kh, where h is the radius of the intersecting circular cylinder (at present cylinder B). This is so because the exact solution of diffraction of a plane wave by an impenetrable circular cylinder of radius h is well-known and given by i an (kh) = −{Jn (kh)/H(1) n (kh)} exp ( − in(u + p/2)),
(13)
Figure 9(b). Same as Figure 9(a) except that the response now concerns the far-field scattering pattern B(u) relative to u s on G a.
. .
558
Figure 9(c). Same as Figure 9(a) except that the response now concerns the function =B(u)=2 proportional to the bistatic scattering cross-section.
for the sound-soft boundary and i an (kh) = −{J n (kh)/H (1) n (kh)} exp ( − in(u + p/2))
(14)
for the sound-hard boundary. Now if we apply this method of reasoning to the neighborhood of another point b on the actual body through which we draw the intersecting cylinder C, then we must once again make assumptions 1 and 2, except that, at present, h is not the radius of cylinder B, but of cylinder C. The generalization of this approach leads to none other than the intersecting canonical body approximation, expressed by a
u(r', u') = u i(r', u') + s an (kr(u'))H(1) n (kr') exp (inu'),
(r', u') $ V0in,
(15)
n = −a
wherein an is given by equations (13) or (14) with h being replaced by r(u'). We will evaluate the ICBA by introducing approximate single and double layer potentials associated with equation (15) into the boundary integral representation obtained from equation (3): i.e., u(x') = u i(x') +
g
[G(x', x)1v u(x') − u(x)1v G(x', x)] dg(x'), x' $ V0out
(16)
G
or, more specifically, u(x') = u i(x') +
g
G
G(x', x)1v u(x) dg(x),
x' $ V0out
(17)
559
Figure 10(a). Scattering response of the surface layer potential 1v u on the boundary G of a sound-soft elongated elliptic cylinder to an end-fire plane wave for wavenumber k = 1·0. q, BIEM reference response; W, ICBA; r, KA.
560
. .
Figure 10(b). Same as Figure 10(a) except that the response now concerns the total field u on G' (measurement circle of radius = 2 outside of the body).
561
for the sound-soft body, and u(x') = u i(x') −
g
u(x)1v G(x', x) dg(x),
x' $ V0out
(18)
G
for the sound-hard body. These ICBA layer potentials are (on the basis of equation (15)) N
1v u(x)=x$G = 1v u i(x)=x$G + s an (kr(u))1v [H(1) n (kr) exp (inu)]=x$G
(19)
n = −N
for the sound-soft body, and N
u(x)=x$G = u i(x)=x$G + s an (kr(u))[H(1) n (kr) exp (inu)]=x$G
(20)
n = −N
for the sound-hard body, wherein the series are limited to 2N + 1 (i.e., a finite number of) terms in order to anticipate the numerical applications. 5. THE KIRCHHOFF APPROXIMATION
The basis of the Kirchhoff approximation is a local description of the field on a scattering body, with an impenetrable medium of half-infinite extent bounded by a flat surface, tangent to the body at the given point at which the approximation is sought, playing the role of the canonical body [3, 10, 22, 32–34]. The Kirchhoff approximation (KA) distinguishes between the lit and shadow zones, so that 1v u(x)=x$G = 21v u i(x)=x$G 1v u(x)=x$G = 0
on the lit portion of G,
(21)
on the shadow portion of G,
(22)
on the lit portion of G,
(23)
for the sound-soft body, and u(x)=x$G = 2u i(x)=x$G
on the shadow portion of G,
u(x)=x$G = 0
(24)
for the sound-hard body. We will compare the KA with the ICBA by introducing the approximate single and double layer potentials of equations (21)–(26) into the boundary integral representations given by equations (17) and (18). 6. THE RIGOROUS BOUNDARY INTEGRAL SCHEME
The rigorous boundary integral method (BIEM) is now classical [3, 35] and the only reason we evoke it here is to indicate just which variant we employ to obtain the reference results which are compared in the next section to the ICBA and KA approximations. From equation (3) and the boundary conditions we obtain the integral equations (wherein PV denotes a Cauchy principal value) 0 = u i(x') +
g
G
G(x', x)1v u(x) dg(x), x' $ G,
(25)
. .
562 for the sound-soft body, and 1 2
u(x') = u i(x') − PV
g
u(x)1v G(x', x) dg(x),
x' $ G,
(26)
G
for the sound-hard body. These integral equations are solved numerically for the layer
Figure 10(c). Same as Figure 10(a) except that the response now concerns the farfield scattering pattern B(u).
563
Figure 11. (a) Scattering response (surface layer potential 1v u) on the boundary G of a sound-soft elongated elliptic cylinder to a broadside plane wave for wavenumber k = 1·0. q, BIEM reference response; W, ICBA; r, KA.
potentials by the PS method described by Bolomey and Wirgin [36]. The layer potentials are then introduced into equations (17) and (18) to compute the field outside the scattering body. Thus, the approximate methods (ICBA and KA) and the rigorous method rely on the same computation (via equations (17) and (18)) for determining the field outside the scattering body.
. .
564
7. EVALUATION OF THE ICBA AND KA BY COMPARISON WITH RIGOROUS BOUNDARY INTEGRAL METHOD REFERENCE RESULTS
The computations were done for elliptic cylinders whose boundary is defined by the parametric equation r = r(u) = ab/(a 2 sin2 u + b 2 cos2 u)1/2.
(27)
The chosen elliptic cylinders were almost circular (half axes a = 1·2, b = 1·0 arbitrary units) or relatively elongated (half axes a = 1·5, b = 1·0 arbitrary units). Computations were also done for the (non-convex) three-leaf clover cylinder r = r(u) = a − b sin (3u),
(28)
for a = 1, b = 0·3. Each cylinder is such that its center line coincides with the z-axis of the laboratory system (see Figure 4). The cylinders are illuminated by end-fire (u i = 0) or broadside (u i = p/2) incident plane waves at three frequencies corresponding to k = 0·1, 1 and 20 (same arbitrary units as the axes lengths), which are referred to hereafter as low, medium and high frequencies. Surface layer potentials (1v u or u on G) are computed as well as the total field u in the near zone on the circle G' of radius r = 2 (arbitrary units) and the diffracted field u s(x) in the far zone on the circle G a of radius r:a. More precisely, the field has the asymptotic behavior u s(x') 0 B(u')
0 1 2 pkr'
1/2
exp [i(kr' − p/4)], kr':a (i.e., x':G a),
(29)
e−ikrcos(u' − u)1v u(r, u)(r˙ 2 + r 2 )1/2 du
(30)
where B(u') =
i 4
g
2p
0
for the sound-soft body and B(u') =
−k 4
g
2p
e−ikrcos(u' − u)u(r, u)[r˙ sin (u' − u) − r cos (u' − u)] du
(31)
0
for the sound-hard body. We compute the real and imaginary parts of the farfield scattering function B(u) in order to appreciate to what extent both the magnitude and phase of this oft-measured farfield quantity, both of which are critical in inverse problem applications, are well-accounted for by the ICBA. In addition, we compute the quantity =B(u)=2 which is proportional to the widely used bistatic scattering cross-section in practical forward scattering problems. Table 1 gives a synoptic picture of the input (columns 2–8) and output (columns 9–12) parameters of the different numerical results. The values of N (recall that 2N + 1 terms are retained in the ICBA surface layer potential series) in column 8 are only indicative; tests were made to ensure that the results did not change significantly for larger values of N. Column 9 indicates which type of approximation (ICBA or KA) gives the closest agreement with the reference (BIEM) results on the boundary curve G. Columns 10 and 11 indicate the same sort of thing on measurement circles G' and G a, respectively. Column
565
12 gives an overall (on G, G' and G ) indication of quality of the best approximation to the scattering response. In all cases, the ICBA gives the best approximation of the response, which, in some exceptional cases is attained also by the KA (actually, there are not very large differences between the KA and ICBA at high frequencies), and the quality of this approximation is either good or excellent. a
Figure 11(b). Same as Figure 11(a) except that the response now concerns the total field u on G' (measurement circle of radius = 2 outside of the body).
566
. .
Figure 11(c). Same as Figure 11(a) except that the response now concerns the farfield scattering pattern B(u).
Let us consider Figures 5–12 in some detail. We have included results for the almost circular cylinder because the ICBA is exact for the circular cylinder (with center at the origin of the laboratory system) and should be all the better the closer the shape of the given cylinder is to circular. The KA, on the other hand, is not, on the whole, more valid for a circular cylinder than for an elliptical cylinder and should not be accurate at low
567
Figure 11(d). Same as Figure 11(a) except that the response now concerns the function =B(q=2 proportional to the bistatic scattering cross-section.
frequencies. This is borne out by the results in Figures 5 and 7 where it is also seen that the ICBA is far superior to the KA at all sampling points. Another feature of Figures 5 and 7 is that the ICBA approximation of the field on G' and G a is generally better than on the scattering surface G itself. Comparing Figures 6 and 9 shows, at high frequencies, that the KA and ICBA give almost equivalent results that are in excellent or good agreement with reality at all sampling points, including on the scatterer itself. However, Figure 8 shows that only the ICBA correctly accounts for reality, although with some difficulty on G a (note that the large differences in Figure 8(d) as compared to Figure 8(c) are due to the squaring operation). Figures 10 and 11 (as compared to Figure 8) show that the ICBA generally fares better for sound-soft than for sound-hard cylinders of the same shape, at least at intermediate frequencies. Once again, these figures show that the ICBA is a much better approximation than the KA in the resonance region, be it for one or the other incident angles. Finally, Figure 12 shows that the ICBA provides a decent approximation of the field at all sampling points even for moderately non-convex scattering bodies, the same being much less true for the KA. 8. CONCLUSIONS
The proposed (ICBA) approximation (embodied by equations (13)–(15)) is extremely simple to use and shares with the widely used Kirchhoff approximation the attractive feature of eliminating the arduous numerical task of solving systems of algebraic or integral equations in order to determine the surface layer potentials. This is of particular interest for solving the inverse scattering problem, i.e., determining the shape of the body from measurements of the scattered field, as has been shown by Lewis [2] and Tabbara [36] for the KA, and by Scotti and Wirgin [37–39] for the ICBA.
568
. .
Figure 12(a). Scattering response (surface layer potential u) on the boundary G of a sound-hard three-leaf clover cylinder to an end-fire plane wave for wavenumber k = 1. q, BIEM reference response; W, ICBA; r, KA.
569
Figure 12(b). Same as Figure 12(a) except that the response now concerns the total field u on G' (measurement circle of radius = 2 outside of the body).
570
. .
Figure 12(c). Same as Figure 12(a) except that the response now concerns the farfield scattering pattern B(u).
571
The ICBA has another unique feature which is that it is applicable at all frequencies (recall that the Kirchhoff approximation furnishes essentially a high-frequency description of the field); herein we have shown this to be true over a range of frequencies for which the highest is 200 times the lowest frequency. This makes the ICBA very useful (and more suitable than the Kirchhoff approximation) for determining scattering response in the time-domain as well as for pulse-echo target identification. The ICBA, shares with the Kirchhoff approximation, the feature of being less-well adapted to the description of the response of highly non-convex bodies for which multiple scattering can play a major role [33]. For the same reason, it is not appropriate to use the ICBA, in its present form, for scattering configurations involving more than one body (unless multiple scattering can be neglected). However, suitably-modified versions of the ICBA can be (and are being) developed for isolated penetrable 2-D [38–40] and 3-D bodies, isolated impenetrable 3-D bodies and isolated coated 2-D and 3-D bodies. A final feature of the ICBA is that it can be cast into even simpler form at both low and high frequencies by appropriate asymptotic manipulation (e.g., Watson transformation) of the partial wave solution of the canonical body scattering solution. This feature is common to the DCA and HPSA techniques of Stanton [19, 20]. For all these reasons it would appear that the ICBA provides a useful new alternative to the well-known Rayleigh, Kirchhoff, geometrical optics, Fock etc., approximations of the scattered field. ACKNOWLEDGMENTS
The authors would like to thank Walid Tabbara for calling our attention to the Adachi approximation and to other related references. We also thank Ralph Kleinman for underlining a possible link between the ICBA and Ursell’s work on local Green’s functions. Two anonymous reviewers have greatly helped to improve the presentation of this work and called our attention to the recent contributions of D. S. Jones to the OSRC issue and T. K. Stanton to the DCA and HPSA techniques. REFERENCES 1. P. M. M and H. F 1953 Methods of Theoretical Physics. New York: McGraw-Hill. 2. R. M. L 1969 IEEE Transations on Antennas and Propagation AP-17. 308–314. Physical optics inverse diffraction. 3. D. C and R. K 1992 Inverse acoustic and Electromagnetic Scattering Theory, Berlin: Springer (applied mathematical sciences No 93). 4. L R 1897 Philosophical Magazine 44, 28–52. On the incidence of aerial and electric waves upon small obstacles in the form of ellipsoids or elliptic cylinders and on the passage of electric waves through a circular aperture in a conducting screen. 5. A. F. S 1953 Journal of Applied Physics 24, 1134–1142. Solutions of electromagnetic scattering problems as power series in the ratio (dimension of scatterer/wavelength). 6. R. E. K 1978 in Electromagnetic Scattering (P. L. E. Uslenghi, editor) 1–28. Low frequency electromagnetic scattering. New York: Academic Press. 7. R. E. K and T. B. A. S 1986 in Low and High Frequency Asymptotics (V. K. Varadan and V. V. Varadan, editors) 1–70. Rayleigh scattering. Amsterdam: Elsevier. 8. J. B. K 1962 Journal of the Optical Society of America 52, 116–321. Geometrical theory of diffraction. 9. J. B. K 1956 IRE Transactions, AP-4, 312–321. Diffraction by a convex cylinder. 10. F. G. B and I. M. F 1979 Wave Scattering from Statistically Rough Surfaces. Oxford: Pergamon. 11. S. A 1965 Proc. IEEE 53, 1067–1068. The nose-on echo area of axially symmetric thin bodies having sharp apices. 12. V. A. F 1945 Journal of Physics (USSR) 9, 255–266. Diffraction of radio waves around the earth’s surface.
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