Short-pulse radiation by a sequentially excited semi-infinite periodic planar array of dipoles

[ 1 ] This paper deals with the fourth in a sequence of canonical problems aimed toward an understanding of the time domain (TD) behavior of wideband-excited sequentially pulsed planar periodic finite arrays of dipoles, which play an important role in a variety of practical applications. The present investigation of sequentially pulsed semi-infinite planar dipole arrays extends our previous studies of sequentially pulsed infinite and semi-infinite line dipole arrays and of infinite planar dipole arrays. The discrete element-by-element radiations are converted collectively to radiations from a series of Floquet wave (FW)-modulated truncated smooth equivalent aperture distributions, and to corresponding FW-modulated edge diffraction. After a summary of necessary results from the earlier studies, emphasis is placed on the new truncation-induced TD results and interpretations, which are extracted via phenomenology-matched high-frequency asymptotics from rigorous frequency and time domain formulations parameterized in terms of the dispersive FW instantaneous frequencies and wave numbers. As in our previous studies, the outcome is a numerically efficient, physically incisive algorithm whose accuracy is verified preliminarily by application to a pulsed planar strip array of dipoles.


Introduction
[2] The prototype study of short-pulse radiation by a semi-infinite sequentially pulsed planar periodic array of dipoles ( Figure 1) plays an important role in the efficient modeling of time-dependent radiation from, or scattering by, actual rectangular phased array antennas, frequency selective surfaces and related applications. Impulsive (delta function) excitations of the array elements, leading to the time domain (TD) Green's functions (GF), are analyzed here, as well as band-limited short-pulse excitations to model more realistic signals.
[3] Our approach is based on the exact equivalence between summation over the contributions from individual sequentially pulsed elements in an array and their collective treatment (via Poisson summation) in terms of infinite series of time domain (TD) Floquet waves (FW). We have already investigated the basic canonical TD-GFs for infinite [Felsen and Capolino, 2000] and truncated [Capolino and Felsen, 2002] periodic line arrays, and that for an infinite periodic planar array [Capolino and Felsen, 2003]. These cases have been parameterized, respectively, in terms of nontruncated or truncated conical TD-FWs [Felsen and Capolino, 2000;Capolino and Felsen, 2002], truncation-induced TD FW-modulated tip diffractions [Capolino and Felsen, 2002], and nontruncated planar TD-FWs [Capolino and Felsen, 2003], which furnish understanding of the corresponding FW critical parameters and phenomenologies pertaining to time domain analysis of spatial periodicity. The present contribution extends the investigation of Capolino and Felsen [2003] to a semi-infinite periodic sequentially pulsed planar array, which introduces new truncationinduced TD phenomena.
[4] We proceed by accessing the time domain through asymptotic inversion of the frequency domain (FD) semi-infinite array results of Capolino et al. [2000bCapolino et al. [ , 2000c, and obtain thereby the instantaneous frequencies that parameterize the behavior of the constituent TD-FWs. The problem is formulated in section 2. Section 3 contains a summary of the relevant FD results from Capolino et al. [2000b], prepared so as to facilitate the inversion to the TD, which is carried out in sections 4 and 5. Section 4 summarizes the TD-FW behavior for the infinite planar array investigated by Capolino and Felsen [2003] because these FWs play an essential role in building up the behavior for the semi-infinite case. Here emphasis is on the distinction between the nondispersive lowest-order, and the dispersive higher-order TD-FWs. The TD-FW behavior for the semi-infinite array is developed in section 5, using the infinite array results in section 4 to parameterize and interpret truncation-induced phenomena affecting the bulk TD-FWs as well as giving rise to TD-FW-modulated diffractions from the array edge. Explaining these new TD results in section 5 in terms of band-limited (BL) problemmatched asymptotics implemented on the inversion integral (4) from the FD constitutes the main contribution in this paper. Preliminary numerical examples of radiation from a two-edged strip array with pulsed band-limited excitation in section 6 demonstrate the accuracy of the TD-FW algorithm and illustrate the rapid convergence of the (TD-FW)-based field representation since only a few terms are required for describing the off-surface field radiated by the truncated planar array. Conclusions are presented in section 7. Refinement and future calibration of these preliminary results through a systematic series of numerical experiments over broad ranges of parameters is reserved for a future publication.

Statement of the Problem
[5] The geometry of the array of parallel sequentially excited pulsed electric current dipole elements, oriented along the arbitrary vector direction J 0 and radiating into free space, is shown in Figure 1. The array is infinite in the z direction and truncated in the x direction. Both cartesian and cylindrical reference coordinate systems with their z-axes along the array edge are introduced such that the array occupies the region x > 0, y = 0. The interelement period is d x and d z along the x and z directions, respectively. The E field component is simply related to the J 0 -directed magnetic scalar potential A which shall be used throughout. A caret^tags timedependent quantities throughout and bold face symbols define vector quantities. With primed coordinates identifying generic source locations, any particular location is indexed by (x 0 , z 0 ) (nd x , md z ), and the corresponding transient current excitations are denoted byĵ(nd x , md z , t), with the frequency (w) and time (t) domains related via the Fourier transform pair The phased array FD and TD dipole currents J(w) and J (t), respectively, are given by where k = w/c denotes the ambient wave number and c denotes the ambient wave speed. In the m, n-dependent element current amplitudes in the last factor in (2), wh x /c and wh z /c in the FD portion account for an assumed (linear) phase difference between adjacent elements in the x and z directions, respectively, with h x /c and h z /c as the interelement phase gradients normalized with respect to w. The TD portion identifies sequentially pulsed dipole elements, with the element at (x 0 , z 0 ) = (nd x , md z ) turned on at time t mn = (h x nd x + h z md z )/c. The important Figure 1. Physical configuration and coordinates for a planar periodic semi-infinite array of sequentially pulsed dipoles which excite a wavefront progressing along the u 1 coordinate on the array. hk (with h = (h x 2 + h z 2 ) 1/2 ), phase gradient of the excitation along the direction u 1 ; v u1 ( p) = c/h, phase speed along u 1 ; k = w/c, c = ambient wave speed.
nondimensional single parameter, which is tied to the rotated coordinate system defined by u 1 (see Figure 1), combines both phasings h x and h z . In (3), is the normalized (with respect to w) phase gradient along u 1 , and v u1 ( p) = c/h is the corresponding impressed phase speed. The TD cutoff condition h = 1 (v u1 ( p) = c) separates two distinct wave dynamics. For h < 1 the excitation phase speed v u1 ( p) = c/h along u 1 is larger than the ambient wave speed c, and the projected phase speeds c/h x and c/h z along directions x and z are greater than c. Conversely, for h > 1, the excitation phase speed v u1 ( p) = c/h of the array along u 1 is slower than c and will not be considered in this paper [see Felsen and Capolino, 2000;Capolino and Felsen, 2003] for some details about the more intricate but practically less important h > 1 case).
[6] Let the total (m, n)-summed frequency domain field at r (x, y, z), due to an array whose individual elements are excited by unit amplitude harmonic currents with phasing exp[Àjk(h x nd x + h z nd z )] be represented by A tot (r, w). The total time-dependent electric field excited by a band-limited (BL) signal is then given bŷ in which G(w) is the weight assigned to each frequency component. In our applications (see section 6), the pulse spectrum G(w) is bounded away from w = 0 in such a manner as to render the high-frequency asymptotic expressions for A tot (r, w) by Capolino et al. [2000b] valid over the entire bandwidth. Therefore the integral in (4) has portions that can be regarded as amplitude functions which vary slowly with frequency and as phase functions which oscillate rapidly, denoted by F(w) andŷ(w), respectively, in (16), (32), and (36). This establishes the prerequisites for asymptotic evaluation in the w domain.
Since the composite phase function in (4) depends on space, time, and frequency, the asymptotic TD fields are parameterized by space-time-dependent saddle point frequencies found explicitly in the following sections.

Frequency-Domain Radiated FW Field: Preparation for TD Inversion
[7] It may be recalled that by Poisson summation, the sum of discretized element-by-element excitations in (2) can be converted into a sum of FW-modulated smoothly continuous equivalent source distributions covering the infinite (or semi-infinite) array aperture [see Capolino et al., 2000b]. For convenience, we summarize below the relevant results of the FD-FW asymptotics for the magnetic vector potential, including FW-modulated dif-fracted fields, that were presented by [Capolino et al., 2000b, Equations (14) -(16)] for the electric field. Also, we exhibit explicitly the w dependences in anticipation of the TD inversion from the FD. Without loss of generality, due to symmetry, we restrict our analysis to the upper half-space y > 0. The radiated field (with a suppressed time dependence exp( jwt)) is expressed as which contains infinite series of ( pq)-indexed FW contributions A pq FW (r, w), and q-indexed diffracted fields A q d (r, w) arising from the edge of the array. The pqth FW in (5) is given by and U(a) = 1 or 0 for a > 0 or a < 0, respectively. The spectral wave numbers and represent the FW propagation coefficients (wave numbers) along x and z, respectively. Furthermore, with the branch chosen to render =m(k ypq ) 0 on the top Riemann sheet; furthermore, <e(k ypq ) ! 0 or <e(k ypq ) 0 for w > 0 or < 0, respectively, consistent with the radiation condition at y = 1 for positive or negative frequencies. In (6), k xp 2 + k zq 2 < k 2 characterizes propagating FW while k xp 2 + k zq 2 > k 2 characterizes evanescent FWs. It is worth noting that the FW representation in (6) is the same as that for the infinite array, except for the Heaviside unit step function U in (5) that confines the domain of existence of the FWs to the region f < f pq SB [see Capolino et al., 2000c, Figures 1 and 2]. The angle f pq SB denotes the shadow boundary (SB) of the pqth FW and, for propagating FWs, is given by Capolino et al. [2000b, equation (24)]: which also specifies the direction of the azimuthal component (in the (x, y) plane) of the pqth FW vector. In (10), specifies the amplitude of the azimuthal (transverse-to-z) component of the pqth FW wave vector (see also the phase factor in (12)). As in (9), the branch is chosen to render =m(k rq ) 0 on the top Riemann sheet, while <e(k rq ) ! 0 or <e(k rq ) 0 for w > 0 or < 0, respectively. The asymptotic evaluation of the qth diffracted FW field in (5), carried out by [Capolino et al. [2000b, section III-B], leads to , and F is the transition function of the Uniform Theory of Diffraction (UTD) [Kouyoumjian and Pathak, 1974], whose arguments in (12) are given by In (12), P denotes the number of poles extracted in the Van der Waerden procedure [see Capolino et al., 2000b, equation (25)]. Owing to the exponential attenuation of the evanescent Floquet waves along y, the convergence of the ( p, q) sum of FWs is very rapid away from the array plane. Furthermore, owing to the exponential decay along r of the evanescent diffracted FWs, the convergence of the q-sum of diffracted FWs is rapid away from the edge. Thus, in practice, only a few terms have to be retained in (5) to provide excellent approximations of the FD radiated field [see Capolino et al., 2000c, section II-B].

Time Domain Floquet Waves for the Infinite Array
[8] Here we select from Capolino and Felsen [2003] those TD-FW pq results which bear directly on the trun-cation studies in section 5. Thus, for the present, we shall ignore the truncation SBs represented by the step function U in (5).

Nondispersive TD-FW 00
[9] The case p = q = 0 is nondispersive. This implies that the wave numbers in (7), (8), and (9) reduce for the p = q = 0 case to k x,0 = wh x /c, k z,0 = wh z /c, k y,00 = w(1 À h 2 ) 1/2 /c; i.e., they are all linearly dependent on w and therefore not amenable to saddle point evaluation. Inversion from the FD through the second relation in (1) with use of A 00 FW (r, w) from (6) yields the closed form (we only treat radiating FWs, for which h < 1, throughout this y)/c as has also been demonstrated after Capolino and Felsen [2003, equation (29)]. The TD-FW 00 is a planar step pulse turned on at time t = t 0 .

Dispersive TD-FW
in which F FW (w) = (4pjd x d z k ypq ) À1 , from (6), is considered slowly varying as a function of the radian frequency w. The exponential term witĥ in which k xp , k zq , and k ypq are functions of w as stated in (7), (8), and (9), is rapidly oscillatory. For these dispersive FWs with p 6 ¼ 0 or q 6 ¼ 0, the dominant contributions to the integral in (16) arise from the stationary (saddle) points w pq,i ofŷ (w) which satisfy (d/dw)ŷj w pq ,i = 0. For p or q 6 ¼ 0, the real w pq,i solutions are found to be [see Capolino and Felsen, 2003, equation (44)], )(1 À h 2 )] 1/2 , and Evaluation of the pqth TD-FW integral in (16) via the standard asymptotic formula [Felsen and Marcuvitz, 1993, pp. 382 yields explicitly, The unit step function U(t À t 0 ) arises because real saddle point frequencies w pq,i are restricted to t > t 0 (t > t 0 ). The range of validity of the FD-inverted asymptotic TD-FW in (23) is inferred from the analytic nondimensional estimator from Capolino and Felsen [2003, equation (55)] and plotted by Capolino and Felsen [2003, Figure 9]. Furthermore, (23) is the asymptotic version of the exact TD-FW evaluated by Capolino and Felsen [2003, equation (27)], and all interpretations relating to it apply here as well. The dispersive TD-FW pq has the same turn on time t 0 as the nondispersive TD-FW 00 .
[11] As shown by Capolino and Felsen [2003, equation (57)], the observation point r (x, y, z), together with the local instantaneous frequencies and their corresponding local instantaneous wave numbers in (7) -(9), k xp (w pq,i (t)), k zq (w pq,i (t)) and k ypq (w pq,i (t)), define localized points of ''emergence'' Q 0 pq,i (t) = [x 0 pq,i (t), z 0 pq,i (t)] [x À yk xp (w pq,i (t))/k ypq (w pq,i (t)), z À y k zq (w pq,i (t))/k ypq (w pq,i (t))] from the FW-modulated equivalent continuous source distribution on the array plane (a few such points are shown in Figure 2). Thus the first signal arrival at the observation point r (x, y, z) originates at the earlier point Q 0 (t 0 ) = [x 0 (t 0 ), z 0 (t 0 )] [x À yh x (1 À h 2 ) À1/2 , z À yh z (1 À h 2 ) À1/2 ]. Successively, for t > t 0 , these Q 0 pq,i (t) points all lie on the t-instantaneous ''equal delay'' ellipse defined by Capolino and Felsen [2003, equation (58)] and schematized in Figure 2. At each time t, all TD-FW pq propagate toward the observer along a t-dependent cone with the same group velocity c. For the semi-infinite array, the observer is reached only by those TD-FWs that were launched from points with x pq,i 0 (t) > 0 on the actual array, i.e., when the launch points lie on the solid part of the ellipse (Figure 2). The corresponding phenomenology is schematized in sections 5.2 and 5.3.

Time Domain Floquet Waves for the Semi-Infinite Array: Truncation of TD-FWs and Truncation-Induced TD Diffraction
[12] ''Bulk'' TD-FWs for the truncated planar array behave in the same way as those for the infinite planar array except for the presence of shadow boundaries that delimit their region of existence. Additionally, a diffracted field arising from the edge truncation at x = 0 exhibits FW-modulated q-dependent dispersive (q 6 ¼ 0) and nondispersive (q = 0) behavior. Inversion from the FD total radiated field in (5) leads tô Three distinct cases are distinguished, based on the p,qdependent FD dispersion relations for FWs and diffracted fields which are summarized in Table 1 and analyzed in sections 5.1, 5.2, and 5.3, respectively. In Table 1, the azimuthal angle f pq SB (w) defined in (10) denotes the SB and direction of propagation of the pqth propagating FW, and k rq (w) is the radial wave number, in (11), of the q-th diffracted field. We focus here only on propagating FWs since all the impulsively excited FD-FW pq in (6) are propagating when evaluated at their instantaneous frequencies w pq,i (t) in (18), as shown in (23).

Figure 2.
Propagating TD-FW phenomenology, based on radiation from the FW-modulated equivalent smoothly continuous source distribution on the infinite or semi-infinite array ''aperture.'' u 1 points in the direction u 1 (h x , 0,h z ) of the advancing excitation wavefront on the array (see Figure 1).
[13] For p = q = 0, the SB angle and direction of propagation of FD-FW 00 do not depend on w, and the radial wave number is linearly dependent on w. This case is therefore nondispersive. For q = 0 and p 6 ¼ 0, FWs are dispersive since the SB angle and direction of propagation vary with w as shown in Table 1. The q = 0 diffracted field is nondispersive because its radial wave number k r0 (w) still depends linearly on w (however, this field becomes ''weakly dispersive'' in the transition region surrounding the SB so as to match the phase speed there of the dispersive FD-FW p0 ). For q 6 ¼ 0 (and arbitrary p), both the FW and diffracted fields are dispersive since their directions of propagation vary with w (the FW and diffracted wave numbers depend nonlinearly on w).
[14] For frequency-dependent SBs (i.e., q 6 ¼ 0 and/or p 6 ¼ 0) and a fixed observer at f, there is a particular frequency w pq SB such that which is determined by squaring and rearranging the expression cos f pq SB (w) = cos f (see (10)). One obtains )cos 2 f À h x 2 ] À 2 w c (h x a p +h z a q cos 2 f) À (a p 2 +a q 2 cos 2 f) = 0, with solutions The illuminated region constraint f < f pq SB (w) on the FD-FWs in (24) implies that for p and/or q 6 ¼ 0, the FD-FWs and diffracted fields are individually w-discontinuous at w = w pq SB , although their sum A pq FW (r, w)U(f pq SB (w) À f) + A q d (r, w) is continuous there. The w pq SB radian frequencies will play an important role in the discontinuity-induced compensation mechanisms discussed in section 5.3.
[15] The three FW pq dispersion regimes in Table 1 parameterize subsequent high-frequency asymptotics for the FD-FW and diffracted fields in (24) away from the transition regions near the shadow boundaries where w % w pq SB . Nonuniform asymptotics leads to the FW-modulated ray optical interpretation of the relevant wave phenomenology. Near the SB pq one needs to employ uniform asymptotics for the diffracted fields, given in (12). Far from SBs, i.e., for w substantially different from w pq SB , the temporal dispersion of the diffracted field (12) is dominated by exp[Àjk rq (w)r], as noted in Table 1, since d pq 2 (w) ) 1 and thus F[d pq 2 (w)] % 1. However, in the pqth transition region, the diffracted field assumes a transitional behavior so as to match the phase speed of the FD-FW pq at the SB pq , thereby changing its dispersion properties. We recall that the expression for the qth diffracted field in (12) compensates for the SBdiscontinuity of all propagating pqth FD-FWs that have been regularized via the Van der Waerden procedure (i.e., those with extracted poles in ÀP p P). Approaching the SB pq , d pq This locally dispersive transitional behavior permits the qth FD diffracted field to compensate for the discontinuity of the pqth FD-FW at the SB pq , as explained by Capolino et al. [2000c, section III-A]. One finds that A q d (r,w) % 1 2 sgn[f À f pq SB (w)]A FW pq (r, w) / exp[Àj(k xp x + k ypq y + k zq z)], which matches the dispersive exponent of the FD-FW. This transitional phenomenology pertains to the cases treated in sections 5.2 and 5.3. 5.1. Nondispersive TD Diffracted Field q = 0 and p = 0 [16] As shown in Table 1, the SB f 00 SB is independent of w and the Fourier inversion (24) is carried out separately for the two terms A 00 FW (r, w)U(f 00 SB À f) and A 0 d (r, w). Fourier inversion of A 00 FW (r, w) produces the TD-FW in (15). A few steps have to be carried out before the inversion of A 0 d (r, w). Referring to (12) we write (with explicit insertion of the w-dependence), arising from the 1/2 and p = 0 terms in B 0 (w), and S p6 ¼0 A p0 d taking into account all higher-order p-terms as shown later on in section 5.2. In (27) , and recalling (20). The Fourier inversion of (27) can be carried out in closed form via an exact transform based on the formula (see expression before (23) and (50) of Capolino and Felsen [2002] followed by a convolution), which leads tô with Near the SB f = f 00 (where t d = t 0 ) we have ffiffiffiffiffiffiffiffiffiffiffiffi ffi sin((f À f 00 )/2) ! 0, which compensates for the discontinuity in the denominator of the second term in (29) (see Figure 3). The TD Fourier inversion of all the other p-terms (with q = 0) A p0 d (r, w) is treated next.

''Weakly Dispersive'' TD Diffracted Fields
With q = = 0 and p 6 ¼ 6 ¼ 6 ¼ 6 ¼ 0 w) for q = 0 and p 6 ¼ 0, we treat FW portion as in section 4.2 and use an approximate form of the diffracted field valid when the space-time position of the observer is in the vicinity of the SB. The relevant expression for the FD diffracted field arises from the higher-order p-terms in B 0 (w) (above (27)), combined with (12), in which k r0 (w), k z0 (w), f p0 (w) and d p0 2 (w) are all functions of w. The corresponding inverted TD-FW, valid away from and near the SB, is obtained as in section 4.2 for the infinite array, using the dominant instantaneous frequencies w p0,i (t) also in the shadow boundary truncation functions U[f p0 (w p0,i (t)) À f], i = 1, 2. A simple expression for the TD diffracted fields Â p0 d (r, t) can be derived when the observer is close to the moving SB p0 where w % w p0 SB . Therefore f % f p0 (w) in (31). and the argument of the transition function F becomes d p0 (w % w p0 SB ) % 0 (see (14)), yielding F % e Àu 2 du, with the upper/lower signs applying for <e(e jp/4 d) > < 0 [Capolino et al., 2000b;Rojas, 1987]. Noting that Àk r0 (w)r + d p0 2 (w) = À[k xp (w)x + k yp0 (w)y] (using (10)), the resulting inversion of A p0 T (r, w) near the SB becomesÂ wereŷ(w), defined in (17), takes into account the phase terms, and is the slowly varying part of A p0 T (r, w) near the SB.
[18] As in section 4.2, the dominant contributions to the integral in the high-frequency range arise from the stationary (saddle) points w p0,i (t) ofŷ (w) given in (18). FW and the conical diffracted field wavefront of Â 0 d reach the observer at t = t 0 and t = t d ! t 0 , respectively. Adding the TD-diffracted field to the truncated TD-FW restores continuity at the SB along and behind the wavefront.
In (33), erfc is considered as an amplitude function as did Capolino and Felsen [2002], [Carin and Felsen [1993], and Felsen and Carin [1994] because over its effective range f p0 (w) % f (in the proximity of the SB), its phase varies slowly. (Away from the SB, the phase varies rapidly; there, however, the FW field is not discontinuous and the diffracted field may either be neglected without appreciable loss of accuracy or evaluated by other techniques. Better approximations for the diffracted field away from the shadow boundaries are beyond the scope of the present investigation). The asymptotic evaluation of (32) is carried out via the formula in (21), with (33), and (d 2 /dw 2 )ŷj wp0,i = y(1 À h 2 ) 2 c À4w At later instants t > t 0 , they separate into SB pq,i (t), i = 1, 2, forming angles f pq,i (t) f pq SB (w pq,i (t)) with the x-axis, such that f pq,1 (t) > f 00 > f pq,2 (t). Observing that k rq = (k xp 2 + k zq 2 ) 1/2 and thus jcos f pq,i (t)j 1 (see (10)), implies that the angles f pq,i (t) are real for any p and any t > t 0 and that all the TD-FW pq are propagating toward the observer. According to (43), the FW pq,i are confined to the right side of SB pq,i (t) for i = 1, 2, respectively. Recalling Figure 2, the observer is reached only by those FWsÂ pq,i FW that originate at points x pq,i and t d defined in (30). For each FW order q, the two real solutions i = 1, 2 yield the two field contributions shown in Figure 5. These diffracted field local instantaneous frequency solutions w q,i d (r, t) are the same as those of Felsen and Capolino [2000, equation (28)] and they match the order of the w pq,i (r, t) solutions in (18) at a given point r and a given instant t in the causal domain t 0 > t d or t > t d = h z z/c + t d . At turn-on t = t d , jw q,i d j ! 1 for all q (see Figure 6). The frequencies increase with FW mode index q, decrease with time t, and approach their cutoff value w q,i d (t ! 1) = w q,i d,cutoff = c(1 À h z equality states that in the transition region, the dominant frequencies must satisfy dy/dw = 0, leading to w pq,i (t) in (18). This is not surprising since we have already encountered the same phenomenology in section 5.2 for q = 0 and p6 ¼ 0; therefore, the asymptotic evaluation for an observer in the space-time-dependent pqth transition region can be carried out in a similar fashion. Here, however, we note that dd pq 2 /dw = 2d pq Á(dd pq /dw) = 0 for w = w pq SB and thus from (41), dy d /dw = dy/dw for w = w pq SB . This demonstrates that if for a certain t = t pq SB , one has w pq,i (t pq SB ) = w pq SB , then also In other words, the asymptotic evaluation carried out above for w q,i d (t) substantially different from w pq SB automatically patches onto the asymptotic evaluation in the transition region w % w pq SB , when the observer is located in space-time at the SB pq (see section 5.2). Therefore the solution forÂ q d (r, t) in (40) can be used for all (r, t), inside and outside the transition regions.
[21] Behavior of local instantaneous frequencies: Space limitations prevent coverage of all possible combinations of phasing h x and h z , and FW indexes p, q<0. We only consider positive frequencies, i.e., those with i = 2. The behavior of negative frequencies (i = 1) is inferred by noting that w pq,1 (r, t) = Àw Àp,Àq,2 (r, t) and w q,1 d (r, t) = Àw d Àq,2 (r, t) (see (18) and (38), respectively). Since t d ! t 0 , the diffracted signal reaches the observer always later than (or simultaneously with, for f = f 00 ) the TD-FW signal, as is also confirmed observing the domain of the real instantaneous frequencies in Figure 6. There, the dynamics of both FW and diffracted field local instantaneous frequencies is compared for the simple but representative nonphased case h x = h z = 0. Width d z used for normalization, the x-domain interelement spacing is d x = 0.1 d z , the observer is placed at r (x, y, z) = (À0.05d z , 8d z , 0) and time and frequency are normalized through T = d z /c. As predicted above, for certain t = t pq SB , one has w q,i d (t SB ) = w pq,i (t SB ) = w pq SB , which means that at t = t pq SB the pqth moving SB intercepts the stationary observer at r. In that vicinity, theÂ q d behavior undergoes a dispersive transition that compensates for the truncation of the TD-FW pq , and restores total field continuity at time t = t pq SB . Except for the moving SBs, the compensation mechanism in the TD is equal to that in the FD [see Capolino et al., 2000c, section III-A], for q 6 ¼ 0.

Conclusions
[25] The motivation and methodologies for the present study having been summarized in the abstract and introduction, we re-affirm that our basic approach to parameterizing and understanding periodicity-induced FW-based TD dispersion can be extended to successively more complicated planar array configurations. The results are again appealingly expressed in terms of periodicity-modulated TD GTD-like wave phenomena, nonuniform outside and uniformized inside transition regions, with novel interpretations of the truncationinduced edge diffractions. Work is in progress to refine Figure 7. Radiated fields versus normalized time t/T, with T = d z /c. The initial spikes are due to the q = 0 diffracted field from x = 0. The second spike in the edgedominated response (a) is the q = 0 diffracted field arriving later from the edge at x = 139d x ; the chirped tail at the higher frequency in (b) is typical of a dispersive FW (here q = À1, 1). and extend the present preliminary asymptotic results beyond the parameter ranges included here, and to perform comprehensive numerical tests over broad parametric excursion, with emphasis on validation and error estimates. The outcomes will be submitted for separate publication. The final canonical problem to be addressed is the TD counterpart of the FD plane sectoral array of Capolino et al. [2000a] and Maci et al. [2001], which through inclusion of corner diffraction, lays the foundation for treating polygonal periodic planar array configurations.