Time domain double diffraction at a pair of coplanar skew edges

A time domain version of the uniform description of double diffraction at a pair of coplanar skew edges is presented, with source and observation at a finite distance. The time domain (TD) field description is obtained by direct Fourier inversion of the frequency domain (FD) doubly diffracted (DD) field. The TD-DD field response to a more general pulsed excitation is obtained via convolution. If the exciting signal has no low-frequency components and is thus dominated by high frequencies, the range of validity of the resulting pulsed response is enlarged to later observation times behind the wavefront. The present TD-DD field is limited to real time, and matches and compensates the spatial discontinuity of the TD singly diffracted field.


I. INTRODUCTION
W AVE propagation in terms of rays, obtained using highfrequency approximations, is an efficient tool in a variety of applications in the frequency domain (FD). Studies of ray propagation directly in the time domain (TD) have also been carried out in the past, since they offer a compact description of the field propagation when excited by short-pulses. In particular, previous studies of TD Geometrical Optics (GO) and TD diffracted fields have been presented in [1]- [3]. In [4]- [6], a spectral theory of transients was presented to represent TD fields in terms of their spectral components, also systematizing the various approximations that can be obtained from that representation. Successively, TD diffraction by wedges have been studied in [7] in a Geometrical Theory of Diffraction (GTD) framework, and in a systematic way in [8]- [10], by using ana- plane, e.g., the GO field from B exists only above its SB , and therefore cannot excite the field^ diffracted at edge 2. Equivalently, the field^ is excited only when the source point B is above the SB plane. The DD field mechanism 12 is shown in Fig. 2. lytic signals, thus allowing the time to be a complex variable. TD wedge diffraction in a uniform theory of diffraction (UTD) framework using analytic signals has been presented in [11]. In this paper a TD doubly diffracted (DD) field at a pair of coplanar skew edges is presented, with impulsive spherical source and observation at a finite distance. This is developed to augment the GO and singly diffracted (SD) fields in a UTD framework. Therefore, the total field is represented here as (1) where is the GO term, are the SD fields arising from edges , that are depicted in Fig. 1, and is the DD field, i.e., the field diffracted at edge and subsequently diffracted at edge , as depicted in Fig. 2. As usual in a GTD description, each field contribution is defined only in its lit region; thus, it is bounded by shadow boundaries (SBs). For example, referring to Fig. 1, the GO field contribution from source exists only above its . Also, as long as the source lies above the plane containing the two edges, a SD field is excited at edge 2. Therefore, when the source moves from to and crosses the , the field abruptly disappears. Analogously, the SD field contribution abruptly disappears when the observer crosses the . Indeed, ray descriptions limited to GO and SD field contributions present abrupt discontinuities at SBs of SD rays, that occur at in Fig. 1. Furthermore, the field description is not satisfactory in shadow regions where neither GO nor UTD rays are traced, e.g., when both source and observer are below the plane; thus, predicting a vanishing field. The augmentation by DD fields solves these impairments. The TD-DD field formulation is constructed via spectral synthesis, as it was done for the FD-DD field recently developed in [12]- [14] for point source time-harmonic excitation, and in [15] for line source time-harmonic excitation. In Section II, the spectral synthesis is completely rephrased in the TD via a continuous superposition of elementary impulsive spherical sources. Next, in Section III, the resulting double spectral integral is evaluated under the wavefront approximation (here denoted by nonuniform), i.e., for space-time location at and immediately behind the DD wavefront. However, this result suffers of the usual impairments of the GTD, i.e., it is unbounded at the SBs of SD fields. Therefore, a uniform wavefront approximation for the TD-DD field is then carried out in terms of canonical spectral integrals that are evaluated in closed form defining two novel TD transition functions for TD-DD fields. These TD canonical functions have also a broader application and can be applied to other diffraction problems with pulse excitation (see [16]). Furthermore, the present result is cast in such a way that the uniform wavefront approximation reduces to the nonuniform one out of DD transition regions. In Section IV, it is shown that the present TD uniform result can be obtained also by inverse Fourier transforming the high-frequency uniform approximation for the FD-DD field in [12] which is conveniently expressed in terms of generalized Fresnel integrals [17]. For describing the electromagnetic DD fields at coplanar skew edges with perfectly conducting faces, in Section V the scalar DD coefficients determined in Section III are used to obtain a dyadic diffraction coefficient. Section VI illustrates how our uniform approximation for TD-DD fields compensates for the discontinuities of SD fields at their SBs, thus providing a continuous field. Particular attention is given to the physical interpretation in the TD of the FD transition regions, and to the definition of the TD transition regions. Furthermore, DD fields at a flat plate are obtained in Section VII as a particular case of the solution of Section III. Their nonuniform versions coincide with the second order diffraction mechanisms described in [18], [19] for a strip. In Section VIII, numerical results illustrate the various transitional behaviors that are shown analytically in Section VI.
In principle, the present TD-DD result is valid only for early times, on and close to (behind) the DD ray wavefronts for the following reasons. First, it has been obtained by a suitable wavefront approximation, as demonstrated in [4]- [6] in general terms. Second, it is shown that the present uniform wavefront approximation for TD-DD field is the exact Fourier transform of the uniform expression for FD-DD field obtained in [12], whose accuracy has been proved at high-frequency in [12], [15], [13] against Method of Moments (MoM) and measurement results in various geometric configurations. The TD-DD field response to a general pulsed excitation is obtained via convolution between the exciting pulse and the impulsive DD response, as shown in Section VIII. When the exciting signal has no low-frequency components and is dominated by high frequencies, the range of validity of the resulting pulsed response is extended to later observation times behind the wavefront.

II. DD FIELD SPECTRAL SYNTHESIS
Let us consider a pair of half planes with soft/hard boundary conditions (BC) and coplanar edges, illuminated by a spherical source. It is useful to define spherical ray fixed coordinate systems at each edge with their origins at the DD diffraction points , determined by standard ray tracing. Our description of the double diffraction mechanism is constructed directly in the TD as the superposition of two analogous mechanisms: a field diffracted from edge 2 when it is illuminated by the field diffracted from edge 1 (12), and that from 1 when it is illuminated by 2 (21). In the following, only the contribution 12 will be considered. The ray geometry for the field DD at and is depicted in Fig. 1 with the distance between the two diffraction points and , and the azimuthal coordinate of measured in the system at edge 1 (2). TD and FD quantities are related by the Fourier transform pair (2) (3) and a caret tags time-dependent quantities. In order to permit spectral integration path deformations, it is convenient to introduce also the analytic transform [4]- [6].
with (4) The transform is analytic in the upper complex time plane. The physical transient field for real is recovered by the limit on the real axis .

A. Spectral Synthesis
Consider the first half plane illuminated by an impulsive spherical excitation , with where and are the source and observation locations, respectively, and is the wave velocity in the medium. It is convenient to consider for the moment the analytic excitation with . We recall that for (6) whereas for real time The TD response from the first edge to an impulsive spherical wave is obtained by the inverse Fourier transform of its FD counterpart, i.e., the FD-SD field by a half plane excited by a time harmonic spherical source [see [12] and (2)] with the free space wavenumber. The contour of integration shown in Fig. 3(a) is defined along for , whereas it must be defined along for . The spectral weight is (9) with the upper/lower sign referring to hard/soft BC, and possesses "incidence" and "reflection" pole singularities as shown in Fig. 3(a). The integration path is indented as in Figure in order to include/exclude the GO incident field residue contribution when the observation point at is above/below the at . The TD-SD field from the first edge excited by the analytic waveform (6) is obtained from (4), and it is thus expressed as (10) The analytic delta (6) introduces a couple of time-dependent poles defined by the solutions of (11) It is important to note that the pole trajectories , with varying time , never cross the integration path (see Appendix A) so that no extra-field contribution due to the residue of a pole crossing the integration path must considered. The integral (10) is interpreted as a superposition of impulsive spectral sources , weighted by the spectral factor , as shown in [12, Fig. 2]. Each spherical spectral source provides a diffracted field contribution from edge 2 at the observation point , that is conveniently calculated using reciprocity, as the diffracted field from edge 2 at due to a point source at . Such contribution is represented as a summation of spectral spherical sources at weighted by the spectral , i.e. (12) in which , and the contour has the same definition as . Then, all the fields radiated by edge 2 (12) are superposed for each -spectral spherical source in (10), i.e., in (10) is replaced by , leading to the double integral representation (13) In (13), is the distance between the spectral source and observation whose explicit expression is (14) with (note that being the unit vectors and orthogonal to the plane containing the two edges). The introduction of the angles and , which automatically appear in our spectral synthesis, is convenient to parameterize the DD-field as a ray. In (13), and exhibit pole singularities that independently occur in the two variables of integration. Eventually, the change of variables and is performed, leading to (15) in which the integration contours and are now translated and thus defined along and , respectively. It is notable that our expression for the DD field (15) explicitly satisfies reciprocity. The term is now be interpreted as a "spectral delay function" and it will play a dominant role in the wavefront approximation. The integrand in (15) presents various spectral singular points such as branch points, poles and "stationary delay" points (i.e., stationary points of the spectral delay function in both variables that provide various localized wave mechanisms. For instance, residues in both variables at "optical" poles of the functions provide GO (direct, reflected or doubly reflected) fields; pole in one variable and saddle point in the other provide edge SD fields that may have also undergone reflection at the other half-plane. The time-dependent -poles introduced by the analytic delta are solution of the equation (16) and are discussed in Appendix A: it is shown that the trajectories of the poles in the plane for varying real and on , never cross the integration path . A two-dimensional (2-D) stationary-delay point [i.e., stationary points of the spectral delay function , defined by , at ] provides the DD ray. When the two edges are not parallel, other 2-D stationary-delay points are present that would provide spherical waves arising from the intersection of the two edges [20]. In the following, we will focus only on the wavefront approximation that will provide a uniform closed form result for the DD field.

III. WAVEFRONT APPROXIMATIONS
The DD field in (15) is evaluated using a "wavefront approximation" based on the 2-D stationary delay point evaluation at that provides a good accuracy when the observer is located near the wavefront of the DD ray. In order to retain also a higher-order term in the wavefront approximation, and to uniformly take into account for the closeness of the poles to the stationary delay point, it is useful to split each -spectral function in (15) in its even and odd parts (17) where the upper/lower sign applies to hard/soft BC, and Then, we introduce the change of variables , and the GO-pole singularities of the two functions in (17) are mapped onto  and  ,  where , and . The value (20) represents the arrival delay of the -spectral component. As shown in [5, sec. II-C.1], for the case of a single spectral integral, the wavefront approximations of the DD field is dominated by the value of the integrand at the stationary point of , defined as , that occurs at which is a minimum of the delay function . The spectral component there located dictates the DD field turn-on time (21) and the spectral components in its neighborhood build up constructively the early-time evolution of the DD field. As done in [12] for the FD case, a Taylor expansion of up to second order around the stationary point gives (22) with , and . The integration paths and , given by mapping the original contours and via the above change of variables, are now deformed onto the real axes. In these deformations, pole singularities , and , obtained by mapping the "incidence" and "reflection" poles of the functions (see Fig. 3), may be encountered. However their residue contributions are not considered here since they would give rise to standard GO or SD diffracted fields that are already accounted for a priori with their domains of existence (their SBs) determined by standard ray tracing.
The additional poles introduced by the analytic delta are defined by the solutions of (23) shown in Fig. 13 before and after the path deformations in both variables . Indeed, there the -trajectories of the -poles in the complex plane are shown for varying (a) along , and (b) along the real -axis integration path. The pole trajectories in the plane are analogous due to the symmetry of . In Appendix A it is illustrated how these TD-dependent poles are never captured in the path deformations in both variables. After all these manipulations, (15) is written as (24) with (25) and (26) The splitting of the original single spectral integral into (25) and (26) arises by multiplying the two even-even and odd-odd parts of the -spectral functions in (17), whereas the resulting odd part of the integral vanishes. The two even functions in (25) and (26) are defined as (27) (28) in which the upper/lower sign applies to hard/soft BC. As shown in Sections VI and VII, the terms are in general of higher order but render the final solution directly applicable to any double edge configuration, including the case when the two edges share a common face as in a strip or in a flat plate. Both and are regular slowly varying functions in a neighborhood of the stationary point . Moreover, their gradients vanish at and in this neighborhood , thus assuring that they can be evaluated at the stationary point and brought out of the integral with high accuracy.
In the sequel, a nonuniform evaluation for (25) and (26) is given first, that permits a ray-optical description of the TD-DD field but fails when both the SD and DD wavefronts arrive almost simultaneously at the observer location, i.e., when the pole singularities of the integrand occur close to or at the stationary point. Therefore, a uniform evaluation is also given that remains valid at any incidence/observation aspect describing the transitional behavior of the TD-DD field. The higher order terms in are of the same asymptotic (wavefront) order as the dominant term in when , but they are subdominant when . Therefore, they are neglected since only the dominant term in may undergo a double transition when (see Section VI-C). After the quadratic approximation in the neighborhood of , the spectral delay in the argument of the analytic delta is purely real for all on the real-axis integration paths. Therefore, in the following all the results are related to real time simply using and the property , and all the above integrals, in particular (24)-(26), will provide the physical transient field , as long as the analytic delta in (25), (26) is substituted by with real time. Therefore, the DD field will be evaluated directly in closed form (see Appendixes B and C) and not by residue evaluations at the time dependent poles, as done in [5] and [9].

A. Nonuniform Wavefront Approximations
The nonuniform wavefront approximations for (25) and (26)  are the incident spreading factor at and the DD spreading factor, respectively. The TD-DD coefficient is evaluated at the retarded time , where is the turn on time of the DD field defined in (21). As for the FD case [12], the splitting in (24) leads to a DD coefficient expressed by the sum

B. Uniform Wavefront Approximations
The nonuniform wavefront approximation (31)-(35) for the DD field is singular for those incidence or observation aspects such that or , respectively. The condition occurs when edge 2 is exactly either at the incidence or reflection SB due to edge 1 illuminated by the source at . Analogously, occurs when the observer at is either at the incidence or reflection SB due to edge 2 when is illuminated by edge 1. These peculiar conditions are depicted in Figs. 4-6 where they are analyzed in detail. A uniform wavefront approximation which is well behaved at those aspects and compensates for the discontinuity of the edge-1 and edge-2 SD fields at their SBs (see Figs. 1 and 4-6) is derived uniformly accounting for the poles at and when evaluating (25) and (26), that may occur near or at the stationary delay point Conditions t 0 t < and t 0 t < define regions, with parabolic shape near the edges 2 and 1, respectively, where wavefronts arrive "almost" simultaneously.  Overlapping TD transition regions: the observer is reached by a transitional field diffracted at edge 2 that is illuminated itself by the transitional SD field at edge 1. Observer A will be reached by direct, SD and DD fields, while B will be reached only by the DD field. The SB plane bounds the domain of existence of the direct field, while the SB plane bounds the domain of existence of the transitional SD field. The three almost simultaneous wavefronts, direct, SD and DD, arrive at the observer A at instants t ; t , and t , respectively, with t t t .
. To this end, the regular slowly varying parts in (25) and (26) are evaluated at (wavefront approximation) and the remaining integrals are expressed in a canonical form yielding , and the total time dependence is taken into account by the nondimensional parameters and .

1) Uniform TD-DD Coefficients:
In summary, the TD-DD field is given by (31)-(33) with the uniform DD coefficients that involve the TD-DD transition functions and parameterized in terms of the nondimensional parameters and . Furthermore, we have explicited the unit step functions to cast the uniform diffraction coefficients as a product between their nonuniform versions in (34) and (35), and the transition functions. Note that our wavefront approximations still preserve reciprocity.

IV. DIRECT TD INVERSION FROM HIGH-FREQUENCY DD FIELDS
We show here that the uniform wavefront approximation for TD-DD field (31)-(33) with (45) and (46) coincides with the direct Fourier inversion of the uniform high-frequency expressions for the FD-DD field in [12] for time harmonic excitation. The FD-DD field arising from edge 2 upon illumination of diffracted field from edge 1, is [12] (47) in which is the incident field at on edge 1, is the DD field spreading factor given in (32), and is the ambient wavenumber. The FD-DD coefficient is represented as , where and (49) with FD-DD transition functions In [12], and have been conveniently represented as combination of generalized Fresnel integrals [17], and computed easily as in [21]. However here, to determine the results for the inverse Fourier transform of we only need to transform and . The parameters of and in (48), (49) are all independent of and the normalization with respect to has been made to prepare and for Fourier inversion. Performing the change of variables , the exponential function becomes , all the disappear in the remaining part of the righthand side of (50), whereas in (51) one remains at the denominator, leading to (52) in which denotes inverse Fourier transform as in (3), and and are defined in (38) and (39), respectively. Since all the steps from (38), (39) to (43), (44) are exact, the inverse Fourier transform of the uniform high-frequency result is exactly the uniform wavefront approximation (31) with (45) and (46). Though this result would be straightforward for a nonuniform asymptotic/wavefront expansion, i.e., without transition functions, it is not for a uniform asymptotic/wavefront approximation. The key point is that the FD transition function is exactly related to the TD transition function as stated by (52) and (53), since the same decomposition of the weight spectral functions (15)- (28), and the same quadratic expansion (22) of the spectral delay in the neighborhood of the stationary point , have been adopted for both the FD and TD cases in [12] and here, respectively.

V. ELECTROMAGNETIC DD FIELD
The solution for the scalar problem of skew edges with hard/soft BCs can be directly used to construct a solution for the vectorial problem of an electromagnetic arbitrarily polarized spherical wave illuminating a pair of skew edges with perfectly conducting faces. As done in [12], [14], [22]- [24] for the double diffraction problem and in [25] for the single diffraction problem, the scalar solutions for hard and soft BCs are used to construct a compact diagonal dyadic DD coefficient in the ray fixed reference systems. To this end, at the two diffraction points and (see Fig. 2) the ray fixed coordinate systems are defined through the unit vectors from the source to , and from to the observation point . Ray fixed coordinate systems connecting edge 1 and 2 are not necessary since the two edges are coplanar and the interaction between the two edges does not cause depolarization for the leading asymptotic terms treated in this paper. The arbitrarily polarized impulsive incident wave at is denoted as , where the arbitrary polarization factor expressed in the incident ray fixed coordinate system is denoted by . The DD electric field mechanism 12 evaluated at the observation point is denoted by (54) and is given by (55) where is the incident spreading factor in (32), is the DD spreading factor in (32), is the retarded time in (21) and the dyadic TD-DD coefficient is represented as (56) where are the scalar soft and hard TD-DD coefficients in (33) with (45) and (46). The dyad includes a leading order term (45), and a higher order term (46) that however is important in single and double transition regions to recover the main phenomenology of electromagnetic scattered fields.

VI. PHENOMENOLOGY AND DISCUSSION
The uniform wavefront approximation for the DD field in (31)-(33) consists of two terms associated to and in (45) and (46). Accordingly, it will be convenient to split the DD field as (57) Their phenomenology is governed by the nondimensional parameters and . In the following, we highlight wave phenomenologies related to various locations of the observation point . Similar concepts (not shown for space limitation) also apply for incident aspects, i.e., to , since the TD-DD field here derived fully satisfies reciprocity.

A. TD-DD Field at Its Wavefront
At the wavefront of a DD ray, shown in Fig. 4, the uniform DD coefficient in (31) is evaluated for . Thus, and , whence the transition functions and in (45) and (46) are approximated for large first two parameters as (58) and the uniform DD coefficients (45), (46) reduce to their nonuniform versions (34) and (35). It is worth noting that at the DD wavefront, (58) still holds in the proximity of the SB of SD rays, e.g., in Fig. 4. In the FD, the transition region around is bounded by a parabola when (the shape is obtained imposing the parameter , being an assigned small quantity) as shown by the grey region in Fig. 4 for . In the TD, the FD transition region collects all the observation points where SD and DD wavefronts arrive almost simultaneously, i.e., where , with the turn-on time of the SD ray diffracted at edge 1, being the distance between and the diffraction point of the SD ray. Conversely, TD-DD transition regions are defined as the regions where the transition functions and differ significantly from unity, i.e., where their nondimensional argument , being a sufficiently small number. As shown in Fig. 5 for , when the TD transition region has an elliptical shape whose size grows with time . The evolving TD-DD transition regions are tangent to the moving DD wavefront.

B. TD-DD Field Behind Its Wavefront
Behind the wavefront, the arguments of the transition functions in (45) and (46) decrease, and the TD-DD transition region in Fig. 5 becomes larger and larger remaining tangent the the moving DD wavefront. Eventually, for and , and the transition functions and vanish providing a smooth transition from the early time to a late-time decaying response.
The uniform wavefront approximation for the TD-DD field is not expected to be accurate for late time instants since it is obtained expanding the argument of the delta Dirac function around the spectral components which define the ray field (usually referred to as wavefront approximation). Equivalently, the TD-DD field is also obtained inverting from the FD the high-frequency expressions for DD field (see Section IV); hence a convolution with an excitation waveform dominated by high frequencies, whose frequency spectrum has no low-frequency components, may enlarge the range of validity of the resulting pulsed response to later observation times.

C. Shadow Boundary Limits
It is well-known that the GO field is spatially discontinuous at its incidence and reflection SBs (see Figs. 4 and 5 where the incidence is shown) and the SD field compensates for those discontinuities undergoing a transitional behavior [7]- [9], [11], [25]. In our double edge configuration, the SD field at edge 1 is optically shadowed by edge 2 at its (see Fig. 5). We analyze here the transitional behavior of the DD field in two distinct configurations, when the SD field experience a ray optical shadowing at its (Fig. 5), and when the GO and SD fields experience optical shadowing at their almost overlapped and (Fig. 6). In the latter case the shadowed SD field is in its transition requiring a double transition of the DD field.
1) Observer or Source in Transition Region: We first consider the case when the observation point crosses the plane containing edges 1 and 2, depicted in Figs. 4 and 5. There, the SD field from edge 1 is spatially discontinuous at and behind its wavefront (i.e., for ), due to the shadowing by edge 2. At this aspect, , so that and consequently (see [12]), implying that , i.e., the SD and DD wavefronts arrive simultaneously. Therefore, approaching is the TD-UTD SD field [11] singular at the wavefront . The other field contribution in (61), as well as , are higher order terms that behave like a step function , and are spatially continuous across the line at and behind the wavefront. Thus, compensates for the discontinuity of at the , at and behind the wavefront, recovering the well-known time dependence of the SD field.
Conversely, when the source crosses the plane containing the edges causing edge 2 to be in the shadow (lit) region of the incident field, the SD field abruptly disappears (appears). At this aspect, , so that and consequently (see [12]). Again, reciprocally to the previous case, the DD field is in transition and restores the continuity of the total field for varying incident aspect. Similar considerations apply to discontinuities for and , caused by shadowed reflected-diffracted and diffracted-reflected mechanisms, respectively.
2) Double Transition Region: When both source and observer lie on the plane containing the two edges, all the arrival times of the incident, SD and DD wavefronts coincide, i.e., , where is the arrival time of the incident field. At this peculiar incident-observation aspect , hence immediately behind the DD wavefront and the TD-DD transition functions vanish as (63) As can be inferred substituting (63) and (64) in (45) and (46), the terms in and vanish for every . In order to understand the double transition behavior at the wavefront, we consider an observer tied to the DD wavefront, with both source and observer in proximity of (and not at) the plane containing the two edges. In this double transition, is close to , consequently the arrival times of the incident, SD and DD fields almost coincide and the parameters and are small but not vanishing. Exactly at the DD wavefront , the two nondimensional parameters , and the two transition functions equal one as shown in (58). As a consequence, at the DD wavefront, when both source and observer approach the plane containing the two edges, both , and the terms in and experience a singularity. In summary, in this double transition limit the terms have a delta Dirac impulsive behavior at the DD wavefront. Their weights are determined by time-integrating the terms of and prior to the limit . Using a decomposition of (43) and (44) in simple terms, and the formula , one obtains (65) and (66) Finally, performing the limit , leads to the double transition limit of the DD fields (67) The field reduces to a quarter of the GO field (i.e., the free-space spherical illuminating impulse). The sign function permits to restore the continuity of the total field on the at and after the wavefronts. The higher order DD field , at this double transition limit, is a delta Dirac impulse like and the incident field. Though it does not contribute to repair the discontinuity introduced by the appearance/disappearance of the incident and SD fields, its contribution is fundamental to achieve an accurate prediction of the DD response in this double transition region (see [26]- [28] for a FD description). Furthermore, it may be the only existing contribution in particular cases such as the one in the sequel. Similar considerations apply when reflected or doubly reflected GO SBs and reflected-diffracted or diffracted-reflected SBs overlap.

VII. DD FIELD AT A FLAT PLATE
The DD field scattered by two edges of a flat plate shown in Fig. 7 can be seen as a particular case of the general solution (31) when the two edges share a common face. In this case The phenomenology outlined in Sections VI-A and VI-B still applies to this particular case. At the wavefront , or out of transition regions, and , and the uniform wavefront approximations (71), (72) coincide with the nonuniform ones. Our nonuniform TD-DD result [(31) with (71) and (72), assuming ] applied to the conducting strip problem (two parallel edges) is coincident with the wavefront approximation of the second order diffraction mechanisms obtained in ( [18], (5)-(6)), and [19] for both hard and soft cases and plane wave excitation. This has been obtained reducing our result (31) to unit amplitude plane wave excitation via the limit and , and considering orthogonal incidence in (71) and (72).

VIII. NUMERICAL EXAMPLES
The compensation and transitional behavior discussed in Section VI-C is here investigated numerically for a particular geometry determined by cm; cm; cm; (see Fig. 2). The source at radiates a spherical pulse . If the incident field pulse is band limited, the DD band limited response is obtained by the time convolution (73) with the impulsive response in (31). In Fig. 8 we analyze a scalar case with hard BC. In Fig. 8(a) the excitation waveform is a Dirac delta function , whereas in Fig. 8(b) the excitation waveform is a normalized Rayleigh pulse , whose spectrum has a central frequency GHz, corresponding to a central wavelength cm. In both cases and , three distinct wave regimes are analyzed as shown in the insets. . For delta excitation the transitional behavior is not so apparent in Fig. 8(a), but it is clearly shown in Fig. 8  Along the observation scan in Fig. 9, the space-time behavior of the spatially discontinuous standard and DD fields, and of the continuous total field , are described for an observer that undergoes a single (Fig. 10) and double (Fig. 11) transition, respectively. In both cases, the same geometry and the same band limited source excitation as in the previous case are considered, though we have dropped the superscript BL for simplicty. Field contributions are plotted against time and observation angle . Existence of the GO field is limited by the , whereas the SD field is bounded by the at , causing abrupt spatial discontinuities.
In Fig. 10, the source is located at implying that the and are far apart as in Fig. 5. Therefore, the SD field is out of transition when abruptly disappears at the [ Fig. 10(a)]. The DD field exhibits an opposite discontinuity [ Fig. 10(b)], to restore the continuity of the total field in Fig. 10(c), at any point and any time.
In Fig. 11, the same space-time aspects are analyzed as in Fig. 10. The only difference is that the source is at , and the SD field is in its transition (ready to compensate for the GO field discontinuity at the ) at its . Indeed, as explained in Fig. 6, at and at almost overlap. Therefore, the strong discontinuity of the transitional SD field at its (note the different scales in Figs. 10 and 11) is compensated by an opposite discontinuity of the DD field [ Fig. 11(b)], that there experiences a double transition. The total field in Fig. 11(c) is continuous at any point and any time.
Note that for increasing observation angle , passing from the lit to the shadow region, the total TD field waveform in Figs. 10(c) and 11(c) gradually weakens and transforms, ac-  Fig. 9 varying the observation angle . The source dipole is at = 310 (single transition case) and oriented along u , all the geometry and source parameters are shown in Fig. 9. In (a), the standard GO + SD transient field abruptly vanishes beyond the SB at = 280 . In (b), the TD-DD field contribution experiences a transition and an opposite discontinuity at the SB [see (61)]. In (c), the total field (GO + SD + DD) is continuous across the SB , at any time, and gradually changes its TD waveform passing from the lit to the shadow region.
cording to the presence of the only higher order mechanism in the shadow region.

IX. CONCLUSION
A closed form solution for DD fields at a pair of coplanar skew edges has been obtained in Sections II-V. The formulation starts by using analytic signals to handle complex time and  Fig. 9 varying the observation angle . The source dipole is at = 281 (double transition case) and oriented along u , all the geometry and source parameters are shown in Fig. 9. The phenomenology is similar to that in Fig. 10, with the only difference that now the SB is close to the SB implying that in (a) the abruptly vanishing SD field is also in its transition. Accordingly, in (b) the DD field has to compensate for that angular discontinuity by experiencing a double transition [see (67)]. In (c), the total field (GO+SD+ DD) is continuous across the SB , at any time.
permit path deformation to prepare the radiation integral for "uniform" and "nonuniform" wavefront approximations. Novel transition functions for TD-DD fields have been determined permitting the uniform description of DD fields near and at the SB of SD rays. It is worth noting that our solution satisfies reciprocity. TD-DD field expressions for the electromagnetic case are obtained by using the scalar solutions to build a dyadic diffraction coefficient defined in the ray fixed coordinate systems. The present TD-DD field is limited to real time and analytic extension of the DD mechanisms to complex time, as in [8], [9], [11] for the SD field, is currently under study. This first investigation lies the foundations to treat subsequent problems involving more general double wedge geometries. The reduction of these TD results for the DD field to the particular case of a flat plate is considered in Section VII. Numerical results have shown the transitional behavior of TD-DD fields, for various source and observation configurations. It has been also shown, analytically and numerically, how the TD-DD field restores the continuity of the total field ( fields) across the SB of a SD field. Space limitations have limited our phenomenology discussion to a few representative cases. Closed form results for TD-DD fields excited by particular band limited waveforms can also be obtained and this is left to future investigations. Here we have restricted the analysis to the numerical convolution of the impulse-excited TD-DD with an arbitrary excitation waveform with no low frequency components.

APPENDIX A COMPLEX TRAJECTORIES OF TD POLES
In this Appendix, using an example, we clarify how the change of variables from the to the complex planes and the subsequent integration path deformations are performed. Due to the perfect symmetry between and , and between and , all the following considerations conducted in the and complex planes apply also to and but are thus not rephrased.
The analytic delta function in (15) introduces time dependent -poles that satisfy (16). The DD field is dominated by the spectral contributions near and at the 2-D stationary point , therefore the following calculations are performed using the quadratic approximation that permits a closed form evaluation of the two -poles symmetric to the origin, as a function of and . In Fig. 12 the trajectories of the two time-varying poles in the plane are plotted for various belonging to the contour [see Fig. 3(b)]. To better separate the trajectories, the time has a small imaginary part . The dotted and continuous curves show the noncausal and causal portions of the trajectories, respectively. Note that the trajectories of the -poles never cross the integration path (thick gray line). Note also that for , the two trajectories of the two -poles tend to meet at for (indeed, this happens exactly only for real time, i.e., when ) showing that at the wavefront , the two time dependent pole singularities coincide with the 2-D stationary point thus providing the main contribution to the DD field.
Once the mapping specified after (19) of the planes onto the planes is performed, the time dependent poles are solution of (23), and the trajectories in the complex plane are shown in Fig. 13(a) for varying , with . There, for each time , for fixed , there are two complex poles symmetric to the origin. The thick gray curve represents the contour , image of through the introduced mapping. Again, the dotted and continuous curves show the noncausal and causal portions of the trajectories, respectively. Since the topology is preserved by the introduced mapping, the trajectories of the -poles never cross the integration path , for every value of the variable that belongs to the integration path.
After the deformation of both integration paths and onto the real and axes, respectively, the trajectories of the time dependent -poles, for fixed belonging to the real axis, are shown in Fig. 13(b). Again, for each time and fixed , there are two complex -poles symmetric to the origin, and the trajectories of the -poles never cross the integration path (the -real axis), for every value of the variable that belongs to the -real axis. In summary, the complex trajectories of the -poles always lie on the same side of the integration path, before and after the deformation of the -and -integration paths in the -and -planes, respectively. Hence, when both integration paths are deformed, time dependent poles are never captured by these deformations.

APPENDIX B
The integrals where we have introduced the nondimensional parameters (recall that ) which completely account for the -dependence, and is a polynomial. Due to the vanishing behavior of the integrand when , the integration path in both (83) and (84) is closed at infinity in the upper half plane. The integrals (83), (84) are evaluated as the sum of the residue contributions at the two enclosed poles and (see Fig. 14), leading to the closed form TD transition functions (43) and (44). Note that at the DD wavefront, the time-dependent poles and coalesce at , whereas for they move apart as depicted in Fig. 14, and when and