Simplified closed‐form expressions for computing the generalized fresnel integral and their application to vertex diffraction

Abstract

The generalized Fresnel integral serves as a canonical function for the uniform ray field representation of several high‐frequency diffraction mechanisms. In this article, an example of application of this canonical function is examined, which is concerned with the diffraction at a plane angular sector with soft boundary conditions on its faces. Closed‐form expressions for computing this canonical function are presented, and it is shown that the approximate formulation adopted herein is accurate and very fast to calculate. © 1995 John Wiley & Sons, Inc.

Shielding effect of a thick screen with corrugations

“…The contribution is obtained from the formulation presented in [6], [7] that is relevant to a line source illumination, by introducing a suitable modification in the spreading factor and in the distance parameters. This leads to (8) where the diffraction dyad is defined as (9) In (9), denotes the DD coefficient for soft BC that applies to TM pol, while and denote the DD diffraction coefficients for the hard and the artificially soft cases, respectively; these latter apply to the TE pol for smooth and corrugated screen, respectively. They are expressed as (10) (11) and csc (12) in which and .…”

“…They are expressed as (10) (11) and csc (12) in which and . Equations (10)- (12) involve the transition functions (13) and (14) where is the generalized Fresnel integral (GFI) defined as in [8] where a very simple algorithm is suggested for its numerical computation.…”

High-frequency analysis of an array of line sources on a truncated ground plane

“…Thus, the same asymptotic evaluation as that in [6] is applied to yield (35) where the transition function (36) is the same as that introduced in [6] and [7]. A convenient expression for (36) which is suitable for numerical calculations is (37) where is the generalized Fresnel integral [14] (38) which may efficiently be calculated as suggested in [15].…”

Double diffraction at a pair of coplanar skew edges

“…Furthermore, in (12), are the diffraction coefficients for the hard and soft cases that are expressed as (14) in which (15) and (16) where is the same defined in (6) and the upper (lower) sign applies to the soft (hard) case. Equations (15) and (16) involve the transition functions (17) and (18) respectively, that are the same as those introduced in [16] for the problem of double diffraction at a thick screen; in (17) and (18), is the Generalized Fresnel Integral (GFI) (19) which is defined for negative arguments as .…”

“…Equations (15) and (16) involve the transition functions (17) and (18) respectively, that are the same as those introduced in [16] for the problem of double diffraction at a thick screen; in (17) and (18), is the Generalized Fresnel Integral (GFI) (19) which is defined for negative arguments as . A very simple algorithm for computing the GFI in (19) is suggested in [14]. The arguments of the transition functions in (15) and (16) are defined as (20) and (21) in which and are defined in (10) and (11).…”