Impulsive pyramid-vertex and double-wedge diffraction coefficients

A new time domain version of the uniform description of double diffraction at a pair of coplanar skew wedges and vertex diffraction at the tip of a pyramid is here presented. The diffracted fields are uniformly evaluated in closed form via wavefront approximations.


Impulsive Pyramid-Vertex and Double-Wedge
Diffraction Coefficients

Filippo Capolino and Matteo Albani
Dip. Ingegneria dell'InformaAione, Universici di Siena, Via Roma 56, 53100 Siena, Italy.e-mail: capolino@ing.unisi.it,albanim@ing.unisi.itA b s t r a c t A new time domain version of the uniform description of double diffraction at a pair of coplanar skew wedges and vertex diffraction at the tip of a pyramid is here presented.The diffracted fields are uniformly evaluated in closed form via wavefront approximations.

I N T R O D U C T I O N
Doubly diffracted (DD) fields at a pair of skew wedges illuminated by an impulsive point source are obtained directly in the TD using a TD spectral synthesis, while the vertex diffracted (VD) field at the tip of a pyramid is constructed via a TD version of the Incremental Theory of Diffraction (ITD) [l] representation of the field scattered at a wedge [2].The two diffraction mechanisms here considered complement, in a TD ray field framework, the field predicted by the TD-Uniform Theory of Diffraction (UTD) [3], leading t o new uniform field expressions for complex scattering problems involving vertexes and double edges.In both cases, closed form solutions are determined using wavefront approximation for the evaluation of the TD radiation integrals.TD-DD and TD-VD field responses to an impulsive spherical delta excitation ?, / I " ' ~ = d[t-r'/c]/(4~r) (r' being the distance from the source and c the ambient wavespeed), are valid only for early times, on and close to (behind) the wavefronts.The field response

G B L ( t )
to a more general Band Limited (BL) pulsed excitation with large-hut-finite bandwidth are found via convolution between the diffracted field due to an impulsive excitation G(t) and the illuminating waveform G(t).If the exciting signal G ( t ) has no low-frequency components and is thus dominated by high frequencies, the range of validity of the resulting pulsed response is enlarced to later observation times behind the wavefronts.The nresent TD-DD and TD-VD fie& are limited to real time, and matches and compensates the spatial discontinuity of the TD singly diffracted field developed in 131.Analytic extension of the DD mechanisms to complex time, as in [3],[4] for singly diffractkd field, is currently under investigation.

TIME D O M A I N DOUBLY D I F F R A C T E D F I E L D S
Let us consider a pair of wedges with soft/hard boundary conditions (BCs) and coplanar edges, illuminated by a spherical source.It is useful to define a a spherical ( T ; , 13,, 4;) ray fixed coordinate system at each edge with origin at the diffraction point Q; (i = 1,2) (see Fig.la).Our description of the DD mechanism is here constructed entirely in TD, following the procedure in [5] for the FD.In the following, only the TD field diffracted at edge 2 when it is illuminated by the field diffracted at edge 1 (12) will he considered.The DD mechanism (21) can be analogously obtained.The ray geometry for the DD field is depicted in Fig. 1 with e the distance between the two diffraction points QI and Q 2 , and c#q2 (&) the azimuthal coordinate of QZ (QI) measured in the system at edge 1 (2).The singly diffracted field at the first wedge illuminated by an impulsive spherical source at I"(@:) f (T; , T -pi, $ I ) , is expressed as superposition of impulsive a,-spectral spherical sources at P'(al weighted by G($I, $ ; ; a ~, n ~) [5], where n l x is the exterior wedge angle, and with the -/+ sign referring to soft/hard BC.Each impulsive al-spherical source provides a diffracted field contribution from edge 2 at the observation point P ( & ) = ( T ~, W , ~& ) , that is conveniently calculated using reciprocity, i.e., the diffracted field from edge 2 at P'(al + 412 +T) due to a point source at P(42) is represented as a summation of uz-spectral spherical sources at P(az + +i2 + T) weighted

TIME DOMAIN VERTEX DIFFRACTED FIELDS
using the hard and soft respective diffraction coefficients t o build dyadic diffraction coefficients in the ray fixed reference systems [5], [3], Df.$(t) = &BzD;$"(t) + &&D;',d."(t),DK(t) = B' fi fiSs(t) + &&,D$'(t).At those aspects in which a singly wedge diffracted ray abrGt1; dgappears, either because shadowed by a n other wedge or because of the truncation of the relevant edge at a vertex, the total field preserves its continuity t h a n k s t o the uniform description of these transitions provided by the transition functions T and TG, introduced for DD and VD fields, respectively.The smooth compensation still holds when the abruptly vanishing singly diffracted field is itself in its transition with a direct or reflected field.This emphasizes as both the DD and VD fields can exhibit a two level (simple or double) transitional behavior.To better explain this phenomenon we present a samplenumerical calculation (Fig. 2) for the DD field due to an electric dipole illuminating a couple of wedges.
The excitation field waveform is a normalized Rilyleigh pulse c(t) = Reb/(j + 2 ~f ~t / 4 ) ' ] , with spectrum with central frequency f~ = 3GHz (AM = c / f M = 10cm).In the first (from the left) figure both source and observation points lie far from their transition, i.e. they lie sufficiently below the edges plane.In these deep shadow region (Fig. Za), the only contribution arriving at the observer is the DD ray that is shaped as the primitive of the Rayleigb pulse.This is consistent with the l/w-frequency dependence of the FD-DD field.Shadow Boundary Limits.When the observation point crosses the plain containing edges 1 and 2 ( Fig. lb), the singly diffracted field from edge 1 is spatially discontinuous at any time, due to the shadowing by edge 2. At this aspect, & -i & + c (see [ 5 ] ) , it can he shown that &' = -$sgn(&,)&, with &' the TD-UTD singly aiffracted field [3], so that compensating for the discontinuity of 4;' at the SB $2 = 4iZ + n, at and after the wavefront.At this SB limit, the qfi time dependence recovers the well known 1/& behaviour of the singly diffracted field, as shown in Fig. 2b.When the source and the obsezver y e both close to the plane containing the edges, both dtr,bll -+ 0 and $$$l/4sgn(bllbll) $'".(P), !.e. the DD field reduces to one forth of the freespace direct contribution of the source, allowing the simultaneous compensation for the appearing/disappearing of the Geometrical Optics and of the two close-to-transition singly diffracted fields, restoring the continuity of the total field at and after the wavefront.At this double transition regime, the DD field time dependence is the same of the exciting pulse ( F i g 2 ).The VD field has an analogous behavior transitional behavior, not shown here for space limitation.