This section contains examples of PO estimations for RCS of simple objects. Whenever possible, these estimations are accompanied by more precise PTD counterparts that include the contributions of primary edge waves generated by the nonuniform edge currents. Only objects with symmetry of revolution are considered. All given data are taken from (15) and (16).
Exact, numerical solutions of scattering problems for bodies of revolution can be found, for example, in (6), (18), and (25).
The geometry of this scattering problem is shown in Fig. 8. The backscattering direction is determined by the spherical coordinates ft = n — 7, p = —n/2. The disk radius is denoted by the letter a. The incident wave can have either E — or H — polarization. In the first case, the electric vector is perpendicular to the incidence plane and parallel to the disk face. The disk properties are described by the reflection and transmission coefficients, re(7), te(7), with respect to the electric vector. In the case of H-polarization, the magnetic vector of the incident wave is perpendicular to the incident plane and parallel to the disk face. The reflection and transmission coefficients, rh(7), th(7), determine the magnetic vector on the front (z = —0) and rear (z = +0) faces of the disk, respectively. According to Eq. (67) in (23), the backscattering RCS is given by
af°(y ) — re(У )2пa2[Jj(2ka siny)]2cot2 у
(24)
ак°(У) — rh (y)2na2[J1 (2ka sin y)]2cot2 у
where J1(x) is the Bessel function and the incidence angle is restricted by the values 0 < 7 < n/2. For perfectly conducting disks, one should put re(y) = — 1 and rh(y) = +1. Then, in the case of the normal incidence (7 = 0), Eq. (24) reduces to
aePO — of0 — n a2 (ka)2 (25)
Figure 7-24 on p. 514 of (2) shows that this equation is in good agreement with the exact results when ka > 5. Note also that Eq. (18), with A = na2 cos 7, determines the PO bistatic RCS of this disk for the forward direction (ft = 7). PTD estimations for RCS of a perfectly conducting disk are presented in Chapters 2 and 5 of (14). See also pp. 514-521 of (2). Some important corrections in the PTD expressions for bistatic scattering from a disk are given in (24). Contributions of multiple edge waves to forward scattering are presented in (15) (pp. 149-151).
Geometrical parameters of a perfectly conducting cone are shown in Fig. 14. The incident wave direction is parallel to the symmetry axis of the cone. The PO backscattering RCS is
|
Figure 14. Backscattering from a truncated cone. The base diameter of the cone (2a) is large compared to the wavelength. The length of the cone (l) can be arbitrary. In the limiting case l = 0, the cone transforms into a disk. |
perfectly conducting paraboloid equals
|
|
|
Figure 15. Backscattering from a truncated paraboloid. The base diameter of the paraboloid (2a) is large compared to the wavelength. The length of the paraboloid (l) can be arbitrary. In the limiting case l = 0, the paraboloid transforms into a disk. |
|
a |
|
(30) |
aPO = 4na2 tan2 a sin2 kl This equation can be written in another form as aPO = na2 tan2 a • e-lkl — elkl |2
(31)
which is more convenient for the physical analysis. The term with the exponential e~ikl gives the correct contribution of the specular reflection from the paraboloid tip. The term with the exponential eikl represents the edge wave contribution and is wrong. PTD includes the additional contribution from the nonuniform edge currents and provides the correct result, given by Eq. (18.04) in (14):
|
2 . n — sm — tan a + 11 11 |
|
a = n a2 |
|
(32) |
|
e |
|
n 2a cos cos — n n |
|
given by Eqs. (17.06) and (17.09) in (15), |
2
|
aPO = na2 ■ |
|
(26) |
|
aPO = n a2 |
|
(27) |
|
tan ae — |
|
2ka |
|
(33) |
|
e |
|
a P = n a2 |
|
(28) |
|
(34) |
|
— tan ae |
|
n 2a cos cos — n n |
|
The first term (with exponential e~ikl) is related to the wave scattered by the cone tip. Comparison with the exact solution [Fig. 18.15 on p. 691 of (3)] shows that this PO approximation is quite satisfactory for all cone angles (0 < a < n/2). The second term (with the exponential eikl) describes the edge wave contribution. This PO approximation is incorrect. PTD takes into account the additional contribution from the nonuniform (diffraction) currents located near the cone edge and provides a more accurate result, given by Eqs. (17.06) and (17.08) in (14), |
|
where the cone length equals l = a cot a. To clarify the physics in this equation, we rewrite it as |
|
-— tan2 a sin kl — tanae1′ ka |
|
tan a + —— tan a ) e 2ka |
|
-— tan a sin kl + ka |
|
2 . n — sm — n n |
|
2 |
|
2 |
|
where n = 3/2 + a/n. When the paraboloid transforms into the disk (a ^ n/2 and l ^ 0), these expressions reduce to Eq. (29). Truncated Sphere The geometry of this scattering problem is shown in Fig. 16. The angle a is formed by the tangent to the sphere generatrix and the symmetry axis. The sphere radius equals p = a/cos a, where a is the base radius. The length of the truncated sphere equals l = p ■ (1 — sin a). It is assumed that l < p. The PO backscattering RCS of a perfectly conducting sphere equals [Eq. (19.05) in (14)] |
|
In this equation, the first two terms represent the specular reflection from the sphere, and both are correct. The third term (with the exponential e2ikl) gives the contribution from the edge and it is wrong. With ka > 1, Eq. (33) simplifies to |
|
1 i U i ———- ——— tana — —— cos a 2ka 2ka |
|
aPO = n a2 |
|
1 |
|
2 |
|
2 |
|
cos a |
|
(29) |
where n = 3/2 + a/n. When the cone transforms into the disk (a ^ n/2, l ^ 0) the previous expressions reduce to
aPO = aPTD = n a2 (ka)2
which coincides with Eq. (25).
Paraboloid
The directrix of a paraboloid is given by the equation r = 2pz where p = a tan a (Fig. 15). The length of the paraboloid equals l = a2/(2p) = (a/2)cot a. The angle a is formed by the symmetry axis z and the tangent to the directrix at the point z = l. The radius of the paraboloid base equals a. The incident wave propagates in the positive direction of the z-axis. According to Eq. (18.02) in (14), the PO backscattering RCS of a
|
|
a
Figure 16. Backscattering from a truncated sphere. The base diameter of the sphere (2a) is large compared to the wavelength. The length of the sphere (l) can be arbitrary. In the limiting case l = 0, the sphere transforms into a disk.
|
2 . n — sin — |
|
J2ikl |
|
n |
|
n |
|
(35) |
|
+ ■ |
where n = 3/2 + a/n. When the sphere transforms into the disk (a ^ n/2, p ^ oo, l ^ 0), Eqs. (34) and (35) reduce exactly to Eq. (29).
Circular Cylinder with Flat Ends
The diameter and length of a perfectly conducting cylinder are assumed to be large as compared with the wavelength of the incident wave. PO and PTD estimations for backscatter — ing RCS are developed in Chapter 3 of (14). They are also presented in (2) (pp. 308-312). PTD asymptotic expressions for bistatic RCS are given in (15) (pp. 152-154).