### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Electromagnetic Field Formulation
- 3. State Space Method
- 4. Results and Discussion
- 5. Conclusions
- Appendix A: Geometrical Optics Reflected Field
- Appendix B: Asymptotic Approximations of the Reflection Coefficients
- References

[1] Creeping waves play an important role in the analysis of electromagnetic scattering by large objects with curved boundaries. In this paper, an asymptotic model is presented for the back-scattered field on a dielectric-coated cylinder at aspect angles near broadside incidence, a region where the creeping wave has maximum intensity. The monostatic radar cross section is analytically derived from creeping wave poles and their residues, and is validated with data extracted from rigorous method of moments computation of the scattered field by utilizing a novel state space spectral estimation algorithm. Detailed computations reveal a stark contrast between dielectric-coated objects and uncoated metallic ones with regard to creeping wave propagation. Because of smaller curvature-induced leakage and coherent interaction between incident wave and the dielectric coating, creeping waves, strongly attenuated by metallic objects, become quite pronounced for coated objects. As the frequency increases, the creeping waves are partially trapped inside the dielectric layer and the scattered field becomes quite small. Therefore, in contrast to leakage on a metallic cylinder, which displays smooth monotonic reduction in amplitude with increasing frequency, creeping waves on a coated cylinder exhibit a strong cutoff akin to a guided wave.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Electromagnetic Field Formulation
- 3. State Space Method
- 4. Results and Discussion
- 5. Conclusions
- Appendix A: Geometrical Optics Reflected Field
- Appendix B: Asymptotic Approximations of the Reflection Coefficients
- References

[2] Analysis of electromagnetic (EM) scattering by dielectric-coated curved objects is of significant practical interest, such as the radar cross section (RCS) of a missile structure or the installed radiation pattern of patch antennas conformal to rocket bodies. The underlying canonical configuration is a perfect electrically conducting (PEC) cylinder coated with one or more thin dielectric layers, the cylinder radius and length being much larger than the operating wavelength. In this paper, an asymptotic model is developed for the back-scattered field from a dielectric-coated cylinder, focusing on rigorous analytical characterization and validation of creeping waves. Detailed computations on a coated cylinder reveal stark contrast in propagation characteristics between the surface-guided field on a dielectric-coated cylinder and that on an uncoated metallic cylinder. The asymptotic approximations are facilitated by well-established representation of the source-excited field for a PEC cylinder in terms of an integral over the angular wave number spectrum using the theory of characteristic Green's functions [*Felsen and Marcuvitz*, 1973, pp. 685–697].

[3] The theoretical formulation describing the scattering by dielectric or dielectric-coated cylinders in terms of azimuthally propagating waves has existed for many years [*Elliott*, 1955; *Helstrom*, 1963; *Streifer*, 1964; *Tang*, 1957; *Wait*, 1960]. More recently, the advancement of numerical methods has enabled the computation of surface wave propagation constants and spectral integrals in the complex plane, resulting in the possibility of extending the asymptotic solution for a cylinder to more complex geometries. Several researchers have addressed the problem of asymptotic approximation of the EM field radiated by a line source or an infinitesimal dipole in the presence of an infinitely long coated cylinder, primarily focusing on numerical evaluation of azimuthally propagating waves derived from so-called creeping wave poles and their residues [*Albertsen*, 1989; *Erturk and Rojas*, 2000; *Felsen and Naishadham*, 1991; *Logan and Yee*, 1962; *Kim and Wang*, 1989; *Krasnojen*, 1998; *Naishadham and Felsen*, 1993; *Paknys and Wang*, 1986; *Paknys and Jackson*, 2005; *Pearson*, 1986, 1987; *Wang*, 1985]. Computation of the EM field itself, its verification with other methods, and its validation with either measured or computed data have received considerably less attention.

[4] In this paper, we evaluate asymptotically the monostatic RCS for a coated cylinder at normal incidence in terms of its geometrical optics (GO) field in the illuminated region and the creeping waves. Although the GO formulation is based on a standard approach [*Felsen and Marcuvitz*, 1973, pp. 693–697], the focus of this paper is on analytical treatment of creeping waves and evaluation of their contribution to the back-scattered field. We do not dwell on creeping wave propagation constants, as these are covered elsewhere [cf. *Paknys and Wang*, 1986]. *Kim and Wang* [1989] obtained the uniform theory of diffraction solution of the field exterior to a 2-D cylinder with a thin lossy coating by utilizing line source excitation at a fixed frequency. As the frequency increases, the coating becomes thicker relative to a wavelength in the dielectric, and the properties of wave propagation and the supported modes change significantly. To our best knowledge, the frequency behavior of the creeping wave modal field and its transition to leaky and trapped modes have not been investigated. We present new results on wideband characterization of creeping waves utilizing practical plane wave excitation, which show that creeping waves cannot be neglected for the coated cylinder. As the frequency is increased, the creeping wave exhibits a sharp modal cutoff, transitioning from an azimuthally propagating mode to a trapped mode. This should be contrasted with the relatively large monotonic decay of creeping waves with respect to frequency on a conducting cylinder.

[5] A second contribution of this paper is the rigorous validation of creeping wave scattering using the state space method (SSM) [*Piou*, 2005; *Naishadham and Piou*, 2005; *Naishadham and Piou*, 2008]. SSM is applied to extract spectral content and synthesize the wideband frequency response of creeping waves from a method of moments (MoM) solution for the scattering by the coated cylinder. It is shown that the extracted creeping wave contribution compares quite favorably with the analytical result for the creeping waves, on the basis of a 2-D formulation over a wide frequency band for a thick lossy dielectric coating.

[6] Mathematical details on the formulation of the creeping wave field, starting from line source excitation of an infinite coated cylinder and developing plane wave synthesis as a subsequent limiting case, are presented in section 2. SSM is summarized in section 3 for completeness. Representative computed results parameterized in terms of dielectric constant, polarization, and frequency are presented in section 4, emphasizing corroboration with wave phenomena extracted using the MoM/SSM spectral estimation approach. Important conclusions are summarized in section 5.

### 2. Electromagnetic Field Formulation

- Top of page
- Abstract
- 1. Introduction
- 2. Electromagnetic Field Formulation
- 3. State Space Method
- 4. Results and Discussion
- 5. Conclusions
- Appendix A: Geometrical Optics Reflected Field
- Appendix B: Asymptotic Approximations of the Reflection Coefficients
- References

[7] We begin with the line source-excited configuration in cylindrical coordinates for formulating the EM field in terms of transverse magnetic (TM) and transverse electric (TE) potentials [*Harrington*, 1961], and compute the scattered response for plane wave incidence in the limit the source point approaches infinity. In general, for oblique incidence on a dielectric cylinder, the fields are TE-TM coupled. However, for angles approaching broadside incidence, such coupling can be neglected and the scattered field can be constructed asymptotically from the line source potentials. For angles off broadside, one should employ the dipole Green's functions formulation [*Pearson*, 1986; *Naishadham and Felsen*, 1993] as the starting point for plane wave synthesis of creeping waves.

[8] Invoking the line source formulation of TE and TM potentials for a coated cylinder [*Paknys and Wang*, 1986] and using completeness relations of the characteristic Green's functions in cylindrical coordinates [*Felsen and Marcuvitz*, 1973, sections 3.3, 6.6], each potential is represented as a spectral integral in the angular wave number (*ν*) domain, with the azimuthal (ϕ) domain extended to span the infinite range −∞ < ϕ < ∞, thereby removing the 2*π*-periodicity constraint and allowing for propagation of creeping waves. The required physical periodicity of the field at ϕ-ϕ′ = *π* is restored by summing over image sources located at ϕ′ ± 2*nπ*, *n* = 1,2,⋯, in the infinite angular domain. The spectral amplitudes depend only on the radial coordinates owing to separation of variables in the Helmholtz equation for the potentials. This radial domain problem is solved by a superposition of inward and outward propagating waves, with their amplitudes determined by invoking the required continuity conditions across the layer boundaries. The (source-free) potentials for plane wave incidence are obtained by moving the source point to infinity and evaluating the corresponding spectral integrals using the stationary phase asymptotic method. Standard far-field approximations are then invoked to compute the scattered field.

[10] A time dependence exp(*jωt*), *ω* = 2*πf* being the angular frequency, is assumed, and *k*_{0} is the wave number in free space. The polarization angle *γ*_{p} = 0 corresponds to horizontal (or TM) polarization and *γ*_{p} = *π*/2 denotes vertical (or TE) polarization. The observation point is located at (*r*, θ, ϕ) in spherical coordinates. For the coated cylinder, the PEC core is covered with a dielectric layer of relative permittivity ɛ_{r} and thickness *d* = *b* − *a* where *b* is the outer radius. We are primarily interested in the field at normal incidence obtained by making θ′ = *π*/2.

#### 2.1. Spectral Integral Representation

[11] For a line source positioned at *r*′ = (*ρ*′, ϕ′), the *z*-directed TM (magnetic) and TE (electric) vector potentials can be expressed as spectral integrals in the complex angular wave number:

The EM field can be readily derived from the potentials [cf. *Harrington*, 1961, equations (5–18), (5–19)]. The *ν*-integral above is recognized as the continuous angular spectrum that follows from the application of the Watson transformation to the conventional angular harmonic series representation [*Watson*, 1919]. However, we may obtain the integral representation in equation (2) directly by invoking a completeness relation for the Dirac *δ* function in the infinite angular domain [*Felsen and Marcuvitz*, 1973, pp. 306–313]. Henceforth, carets denote the spectral amplitudes in the transform domain. The vector *r* = (*ρ*, ϕ) locates the observation point, and *i* = 0, 1 specifies free space and dielectric, respectively. The radial variation of the fields is obtained by solving Bessel's differential equation as a superposition of cylindrical waves traveling in opposite directions. The spectral amplitudes _{i}, _{i} are determined by imposition of boundary conditions on the tangential field at the interfaces *ρ* = *a* and *ρ* = *b*.

[12] The source-excited field propagates on the surface of the coated cylinder along circumferential trajectories formed by varying *ν*. The integration path *C*_{ν} in the complex *ν*-plane is parallel to and just below the real axis. For an observer shadowed by the cylinder, we close this path in the lower half plane around the poles of the spectral amplitudes in equation (2) and evaluate the *ν*-integral as a summation of the corresponding residues. Modal amplitudes of the creeping waves are proportional to the residue at a given pole location multiplied by the appropriate excitation coefficient. The index *n* accounts for multiple circumnavigations of the creeping waves, and the infinite *n*-summation can be expressed in closed form (see equation (12) below). The *n* = 0 term is evaluated asymptotically by integrating along the real axis in the *ν*-plane and accounts for the GO field in the illuminated region [*Felsen and Marcuvitz*, 1973, pp. 693–697].

[13] A few points are in order regarding the computational advantages of the spectral representation in equation (2). Although this representation is valid for any cylinder radius, it is especially useful for large-radii cylinders. In this case, higher-order creeping waves decay rapidly for both PEC and coated cylinders, and the residue series converges efficiently, so that only the dominant residue (usually *p* = 1 term) would suffice. However, as we quantify in section 4, even after traversing partially around the cylinder and launching into the lit region, the dominant creeping wave on a coated cylinder has significant amplitude. Therefore, besides evaluation of the field in the deep shadow region, the asymptotic spectral representation of the creeping waves provides an efficient method to characterize their contribution to the back-scattered field.

[14] Next, we present expressions for the spectral amplitudes _{i}, _{i} in a form suitable for direct numerical implementation. The radial wave functions involve Hankel functions *H*_{ν}^{(1,2)}(*k*_{i}ρ) of complex order ν, where *k*_{i} = *ω*. In particular, spectral amplitudes in the free space region are given by the sum of incident and scattered fields:

where the incident field is derived from the potentials of a line source with amplitude *J*_{0} or *M*_{0}:

The amplitudes of the scattered field potentials are given by the following equation:

The TM and TE reflection coefficients, *R*_{ee} and *R*_{hh}, respectively, follow from the tangential field boundary conditions at *ρ* = *a* and *ρ* = *b*. Only the relevant equations are summarized below, with the reader referred to *Paknys and Wang* [1986] for details. The PEC boundary condition at *ρ* = *a* leads to the following calculations:

for the TM and TE reflection coefficients, respectively, at the conductor interface. Reflection coefficients at the outermost (dielectric) interface are computed as follows [*Paknys and Wang*, 1986; *Kim and Wang*, 1989]:

[15] It follows from equations (8) and (9) that the dispersion equation, whose zeros yield the poles *ν*_{p}(*β*) of the reflection coefficient in the complex *ν*-plane, is given by the following:

The solution to equation (10) has been discussed in *Naishadham and Felsen* [1993], classifying the poles as creeping wave type, trapped wave type, or leaky wave type, depending on whether Re(*ν*_{p}) ≈ *k*_{0}*b*, Re(*ν*_{p}) > *k*_{0}*b*, or Re(*ν*_{p}) < *k*_{0}*b*. When *b* = *a*, ɛ_{r} = 1 and *μ*_{r} = 1, we observe that *C*_{e} ∞, *C*_{h} 0, with the result that *R*_{ee} Γ_{e} and *R*_{hh} Γ_{h}. In other words, when the coating vanishes, the reflection coefficients in equations (8) and (9) correctly reduce to the TM and TE reflection coefficients of equation (7) for a PEC cylinder.

#### 2.2. Plane Wave Incidence

##### 2.2.1. Incident Field

[16] The line source field is now specialized to plane wave incidence by asymptotic evaluation of the incident and scattered field potentials in equations (5) and (6) for *ρ*′ ∞. Using equations (2), (3), and (5) with *ρ* < *ρ*′, and setting *J*_{0} = *M*_{0} = 1 henceforth, the incident TM and TE line source Green's function is given by the following equation:

where we have replaced the infinite summation in equation (2) with the closed-form result

Clearly, the integrand has simple poles at *ν*_{m} = *m*, *m* = 0, ±1, ±2, ⋯ in the *ν*-plane. Because the integrand is odd-symmetric about the origin, the integration contour *C*_{ν} may be deformed around the positive real axis (see Figure 2a), and by residue evaluation, equation (11) results in the 2D Green's function

Equation (13) is valid for observation angles in the *z* = 0 plane. The 3-D Green's function, appropriate to oblique plane wave incidence depicted in Figure 1, follows from the inverse Fourier transformation

where _{ρ} = *G*_{ρ} with *k*_{0} replaced by _{0} = , and *β* is the axial wave number. Substitution of equation (13) in equation (14) yields

The second equality in equation (15) follows from the addition theorem

Stationary phase evaluation of the second integral in equation (15) yields the 3-D Green's function

Using the standard far-field approximations in spherical coordinates for *r*′ ∞ along the ray defined by (θ′,ϕ′), equation (17) yields the incident TM and TE potentials

valid for illumination angle in the lit region 0 ≤ *ψ*_{i} ≤ *ψ*_{sb}, where *ψ*_{sb} denotes the shadow boundary. Appropriate polarization is effected by deriving the fields from these potentials using standard expressions [cf. *Harrington*, 1961]. With ϕ′ = 0, the propagation phase in equations (18) and (1) become identical. For the 2-D formulation relevant to this paper, we set θ = θ′ = *π*/2.

##### 2.2.2. Reflected Field

[17] Next, we approximate the scattered field in equation (6) for *ρ*′ ∞. In the lit region, the scattered field consists of the reflected and diffracted fields. The reflected field is obtained from the *n* = 0 term in equation (2) by asymptotic evaluation of the *ν*-integral along the real axis for Re(*ν*) < *k*_{0}*b* [*Felsen and Marcuvitz*, 1973, pp. 693–697; *Kim and Wang*, 1989]. For Re(*ν*) > *k*_{0}*b*, this integral does not contribute to the reflected field. For the diffracted field, however, the contour *C*_{ν} is closed around poles *ν*_{p} of the reflection coefficient in the lower half plane (see Figure 2b) or zeroes of the dispersion equation (10). Residue evaluation results in surface-guided modes such as creeping waves. Unlike the GO field, surface diffraction occurs in the shadow region behind the object.

[18] A general treatment of plane wave scattering requires variation of elevation angle θ′ in Figure 1, whereas the 2-D reflected field derived in *Kim and Wang* [1989] is valid only in the *z* = 0 plane. Therefore, instead of equation (6), we start from the 3-D dipole-excited reflected field for a coated cylinder [*Pearson*, 1986; *Naishadham and Felsen*, 1993] and let the source point *r*′ ∞ for plane wave synthesis. However, since we are interested in aspect angles near broadside, we neglect TE-TM coupling in the dipole reflection coefficient expressions. The TM and TE plane wave potentials of the reflected field for large *k*_{0}*b* are given by the following equation (see Appendix A for details):

where _{ee}, _{hh} are locally planar approximations of the reflection coefficients in equations (8)–(9) for large *k*_{0}*b* (see Appendix B), *ν*_{s} is the saddle point in the angular spectral integral, and the cylindrical coordinates of the specular point of reflection are given by (*b*,ϕ_{b},*z*_{b}). With θ and θ′ set to *π*/2 and *z*_{b} = 0, equation (19) reduces to the 2-D-reflected field for plane wave incidence. The term within the radical denotes the spreading factor.

##### 2.2.3. Surface-Diffracted Field

[19] The surface-diffracted field for creeping waves launched by an incident plane wave is obtained from the *ν*−spectral integration in equation (2) using the scattered line source field in equation (6) with the reflection coefficients given in equations (8) and (9). Closing the *ν-*integration path around the zeroes of equation (10) (see Figure 2b) and evaluating the integral in terms of residues _{ee} and _{hh} leads to the following equations:

where *N*_{ii} and *D*_{ii} are the numerator and denominator, respectively, of the corresponding reflection coefficient in equations (8) and (9). The *n-*summation may be accomplished in closed form as indicated in equation (12) and large-argument asymptotic approximation may be utilized for the source Hankel function, yielding the following equation:

Equation (22), derived as a limiting case of the line source Green's function, does not capture the TE-TM mode coupling that occurs for arbitrary oblique aspect angles. Hence, unlike the incident field in equation (18) and the reflected field in equation (19), the asymptotic formulation in equation (22) for the creeping waves is valid only at normal incidence. The creeping waves in equation (22) implicitly assume multiple clockwise and counterclockwise circumnavigations through the closed-form sum in equation (12). The scattered field potentials are now given by the sum of reflected and diffracted components in equations (19) and (22), respectively. The reader is referred to *Naishadham and Felsen* [1993] for implementation details of various quantities in equation (22), in particular the Hankel functions with complex argument and order. Derivatives with respect to order in the residues in equation (21) are computed numerically using finite differences, and Davidenko's method, a complex-zero search procedure with exponential convergence, efficiently computes the *ν-*plane poles. We have observed that only the leading order term in the summation would suffice for coated cylinders (see section 4).

### 4. Results and Discussion

- Top of page
- Abstract
- 1. Introduction
- 2. Electromagnetic Field Formulation
- 3. State Space Method
- 4. Results and Discussion
- 5. Conclusions
- Appendix A: Geometrical Optics Reflected Field
- Appendix B: Asymptotic Approximations of the Reflection Coefficients
- References

[23] We examine the monostatic RCS of a coated cylindrical target with core radius *a* = 12.13 cm, length *h* = 0.85 m, and a dielectric coating of thickness 4.57 mm and relative permittivity ɛ_{r} = 3.5–*j*0.55. The reference solution to validate the analytical model is computed over a frequency band of 2–12 GHz and angular space 0 ≤ θ ≤ *π*, using the body-of-revolution MoM [*Mautz and Harrington*, 1977]. This computation yields monostatic RCS as a function of incident angle and frequency. The MoM data for specific angles is then processed with the state space spectral estimation algorithm, summarized in section 3, to extract the specular and creeping wave components. We assume that edge diffraction, both single edge as well as multiple creeping wave/edge diffraction, can be neglected at aspect angles near broadside. Therefore, the coherent addition of the two main wave constituents, the specular and creeping waves, generates the composite signal.

[24] Broadside incidence perpendicular to the cylinder axis corresponds to the maximum of the creeping wave radiation. In this direction, the back-scattered creeping wave is essentially a cylindrical wave, with all the creeping rays adding coherently at the receiver. The frequency-domain vertically copolarized (VV or TE/TE) RCS for θ = 90° computed by MoM is shown by the solid line in Figure 3b. To locate the creeping wave peak, the frequency response is Fourier transformed with a Hamming window to suppress the sidelobes, obtaining the composite range response shown in Figure 3a. The first peak denotes the specularly reflected field, and the next one denotes the creeping wave. The phase reference (zero range) is on the cylinder axis. Since the excitation of the broadside creeping wave is essentially a 2-D problem, using ray tracing, one may verify that the creeping wave should occur at *πb/*2 down-range from the zero-phase reference, where *b* is the outer cylinder radius. Thus, the range to the creeping wave peak should be 0.25 m as confirmed in Figure 3a. The specular return, on the other hand, appears to emanate from a line source located on the front side of the cylinder (−0.12 m range), as physical optics would predict [*Ufimtsev*, 2007].

[25] Figure 3b depicts comparison between the total response computed by the analytical solution using equations (19) and (22) with the corresponding MoM-derived RCS at broadside. The interference pattern between the specular components of the GO return and the creeping wave is clearly evident at frequencies below the creeping wave cutoff around 7.5 GHz. Favorable comparison with the MoM data reveals that the dominant wave phenomena at broadside are the specular reflection and the creeping wave. Other diffraction effects, such as edge diffraction and multiple interactions (e.g., edge-diffracted creeping waves), which are included in the MoM but not considered in the analytical model, seem to be secondary. These secondary effects contribute to some error in the analytical solution for the total field at frequencies beyond the creeping wave cutoff. However, since the work reported herein focuses on creeping waves, we have not addressed the augmentation of the analytical model to include edge diffraction effects.

[26] Next, a 20th-order state-space model is employed to process the MoM-simulated frequency response for the broadside aspect, and using only three range-classified poles for each wave species, the specular and creeping wave constituents are extracted. Figure 4 depicts excellent corroboration between the analytical solution from Section 2 for these two wave components and the SSM-extracted reference data. The creeping wave response is plotted only until 8 GHz because it cuts off at 7.5 GHz. It is emphasized again that the discrepancy between MoM and analytical data at high frequencies is caused by ignoring secondary diffraction effects in the analytical solution. *To the best of our knowledge, this work represents the first wideband validation of creeping waves on a coated cylinder using a rigorous MoM reference solution*. The state space method allowed us to accurately extract creeping wave and specular constituents from the MoM data and facilitated such validation. Later in this section, we establish such validation even for perfectly conducting cylinders.

[27] It is seen that the creeping wave cuts off to −40 dB at ∼7.5 GHz. Above the cutoff, the creeping wave transitions into a trapped mode, which bounces back and forth internal to the layer, similarly to a surface wave on a planar slab, and leaks outside tangentially on the surface due to curvature [*Naishadham and Felsen*, 1993]. The transition of a creeping wave to trapped mode can be illustrated by examining its propagation constant as a function of frequency. Figure 5 displays the dominant TE mode creeping wave propagation constant, *ν*_{1}/*b*, computed by solving equation (10) for each frequency. As the creeping wave propagates, it attenuates by shedding energy tangentially because of curvature. The attenuation remains relatively constant until ∼7.5 GHz and then rapidly increases in magnitude with frequency, resulting in the trapped mode. Beyond the creeping wave cutoff, there is very little curvature-induced leakage. From the phase constant plot, it is observed that dispersion exists at higher frequencies. It has been observed that higher-order creeping waves (*p* > 1) suffer significantly larger attenuation than the dominant mode and are therefore neglected.

[28] Next, we examine the TM or HH polarized response at broadside incidence. In this case, analogous to the PEC cylinder, the creeping wave should be weaker than its VV polarized counterpart. The range response in Figure 6a shows the specular at the down-range of −0.12 m, the same as in the TE case (Figure 3a). However, the next peak, about −40 dB in amplitude, occurs at 0.12 m and does not appear to be the creeping wave specular (expected at 0.25 m). To examine this further, we plot in Figure 6b the total response extracted from the MoM solution for broadside incidence along with the sum of analytically evaluated GO specular and creeping wave returns. If the creeping wave were present, it would beat with the GO specular until it is cut off, as we have observed for VV polarization in Figure 3b. Absence of such interference alludes to a weak creeping wave for HH polarization. On the basis of the ray path analysis sketched in Figure 7, we infer that the peak at 0.12 m is due to an edge-diffracted wave, which becomes dominant at high frequencies. Since the edge diffraction is lumped in the MoM solution and ignored in the analytical solution, there exists a significant discrepancy beyond 9 GHz between these two sets of data in Figure 6b.

[29] Figure 7 shows the ray path associated with an edge-diffracted wave, which attaches broadside to the cylinder, travels diametrically along the base to the detachment point, and returns broadside to the receiver. Since the electric field is tangential to the cylinder at the two diffraction points for horizontal polarization, this edge wave is stronger for HH than for VV polarization [*Ufimtsev*, 2007]. In contrast, the creeping wave along the curved surface of the electrically long cylinder is not affected by edge diffraction.

[30] The MoM-extracted specular and edge-diffracted components are plotted in Figure 8 along with the analytical solution for the specular and creeping wave responses for HH polarization. Again, a 20th-order state-space model is employed to process the MoM-simulated frequency response, and using only three range-classified poles for each, the specular and edge wave constituents are extracted. This plot reiterates that the creeping wave is negligibly small in comparison with the edge-diffracted wave. The creeping wave could not be extracted from MoM data as its amplitude is within the range of numerical “noise” (−100 to −80 dB). Also, the creeping wave for HH polarization does not exhibit the frequency rolloff observed in the VV polarization. Good agreement is observed between the analytical and MoM constituents of the specular field for frequencies less than 9 GHz.

[31] Next, we present salient results on broadside RCS for a PEC cylinder of the same dimensions as the conducting core of the coated cylinder. First, the vertical polarization is considered. The range plot in Figure 9a for VV polarization shows the occurrence of specular and creeping waves at the anticipated locations. The frequency response in Figure 9b displays good corroboration of the analytical (specular + creeping wave) solution with the composite MoM-extracted data. Although the ripple caused by the interference between specular and creeping wave components is much smaller than that of the coated cylinder (Figure 3), weak oscillations persist throughout the 10 GHz bandwidth. The absence of a standing wave pattern in Figure 9b indicates that the PEC cylinder creeping wave does not have a steep high-frequency cutoff like the coated cylinder, as we show next.

[32] Figure 10 depicts the analytically evaluated specular response and the creeping wave components in comparison with MoM-extracted wave objects for the PEC cylinder. SSM with a model order of 10 has been used in the extraction process, with only two range-classified poles to represent each wave species. At high frequencies, the broadside specular RCS increases as square root of the frequency, as can be shown easily from equation (A2) [cf. *Kim and Wang*, 1989], and does not depict the resonant standing wave pattern of its dielectric counterpart in Figure 4. Creeping wave for the PEC cylinder decreases monotonically with frequency and does not have the high-frequency rolloff associated with trapped waves on a coated cylinder. Figure 10 quantitatively demonstrates the strong attenuation of the creeping wave on a large conducting curved surface.

[33] For completeness, horizontal polarization for the PEC cylinder is considered next. The range response in Figure 11a shows the specular at the down range of −0.12 m, the same as in the VV case (Figure 9a). However, the next peak, about −36 dB in amplitude, does not correspond to the creeping wave specular, but a manifestation of the edge diffraction along the path depicted in Figure 7. The frequency response in Figure 11b displays good corroboration of the analytical (specular + creeping wave) solution with the composite MoM-extracted data. The absence of ripple in the composite frequency response indicates that the creeping wave amplitude is very small, as one would expect for horizontal polarization. Indeed, this is corroborated in Figure 12, which compares the analytically evaluated specular response and the creeping wave components with MoM-extracted wave objects for the HH case. SSM with a model order of 10 has been used in the extraction process, with only two range-classified poles to represent each wave species. It is noted that the specular response is identical between the VV and HH cases for the PEC cylinder. However, the creeping wave is 50 to 80 dB smaller in the HH case and practically negligible. In comparison, edge diffraction, extracted from the MoM solution, is significantly larger in amplitude. It is interesting to note that edge diffraction for the PEC cylinder is similar in amplitude to that for the HH-polarized coated cylinder (see Figure 8b), whereas the creeping wave for the PEC cylinder is smaller than that for the coated cylinder, especially at high frequencies. This demonstrates conclusively that the creeping waves are strongly dependent on polarization and enhanced by the dielectric constant of the coating. At the higher frequencies, although creeping waves can be neglected for both coated and PEC cylinders in horizontal polarization, edge diffracted waves, relatively insignificant for the PEC cylinder, cannot be neglected for the coated cylinder. As is evident from Figure 6, ignoring edge diffraction causes considerable discrepancy between the MoM-extracted composite response for the coated cylinder and an analytical solution comprising the coherent addition of only the creeping waves and the specular response. In contrast, there is no such discrepancy for the PEC cylinder (see Figure 11).