Geometrical Shadow Boundaries Research Articles

Exploring the physical theory of diffraction for solutions of wedge fields in room-acoustic simulation

The need of developing accurate and cost-effective room-acoustic simulations has prompted recent investigations on edge diffractions with some progresses on solving multiple-order diffractions of finite wedges. This work explores an alternative approach based on the physical theory of diffraction [P. Ya. Ufimtsev, J. Acoust. Soc. Am. 120, 631-635 (2006)] which is well suited for solving acoustic scattering problems from reflecting objects for room-simulation purposes. Although the approaches based on the principle of geometrical acoustics (GA) are widely applied, they are less suitable for geometrical shadow boundaries. The physical theory of diffraction still relies on both geometrical and physical principles, yet emphasizes the physical one. One of the important features of this physical acoustics (PA) approach is the ability to calculate the sound field more accurately in shadow boundaries. To this end, exact and asymptotic solutions of wedge fields are discussed for secondary edge sources induced by the incident wave. This paper will discuss implemented results of several canonical cases with emphasis on Neumann boundary condition.

Diffraction imaging by uniform asymptotic theory and double exponential fitting

ABSTRACTSeismic diffracted waves carry valuable information for identifying geological discontinuities. Unfortunately, the diffraction energy is generally too weak, and standard seismic processing is biased to imaging reflection. In this paper, we present a dynamic diffraction imaging method with the aim of enhancing diffraction and increasing the signal‐to‐noise ratio. The correlation between diffraction amplitudes and their traveltimes generally exists in two forms, with one form based on the Kirchhoff integral formulation, and the other on the uniform asymptotic theory. However, the former will encounter singularities at geometrical shadow boundaries, and the latter requires the computation of a Fresnel integral. Therefore, neither of these methods is appropriate for practical applications. Noting the special form of the Fresnel integral, we propose a least‐squares fitting method based on double exponential functions to study the amplitude function of diffracted waves. The simple form of the fitting function has no singularities and can accelerate the calculation of diffraction amplitude weakening coefficients. By considering both the fitting weakening function and the polarity reversal property of the diffracted waves, we modify the conventional Kirchhoff imaging conditions and formulate a diffraction imaging formula. The mechanism of the proposed diffraction imaging procedure is based on the edge diffractor, instead of the idealized point diffractor. The polarity reversal property can eliminate the background of strong reflection and enhance the diffraction by same‐phase summation. Moreover,the fitting weakening function of diffraction amplitudes behaves like an inherent window to optimize the diffraction imaging aperture by its decaying trend. Synthetic and field data examples reveal that the proposed diffraction imaging method can meet the requirement of high‐resolution imaging, with the edge diffraction fully reinforced and the strong reflection mostly eliminated.

Uniform surface-to-line integral reduction of physical optics for curved surfaces by Keller-type equivalent edge currents in MER: Behaviors of fields on reflection shadow boundary

The Modified Edge Representation (MER) is a procedure for defining the equivalent edge currents (EECs) to be used in the surface-to-line integral reduction for computing the physical optics radiation integrals in diffraction analysis. The physical optics surface integration is transformed into the line integration of MER-EECs rigorously for the planar surfaces. It works with the remarkable accuracy even for the curved surface. In contrast to conventional EECs, those in MER are defined without ambiguity not only at the diffraction points but also at arbitrary ones along the periphery. One remarkable advantage is that the MER-EECs, after integrated along periphery, provide uniform fields across the geometrical shadow boundaries though the EECs in the integrand consist of nonuniform Keller-type diffraction coefficients. As the starting point of the applicability check of MER for the curved surfaces, this paper investigates the MER diffraction field behaviors on and near the reflection shadow boundary (RSB). First, the stabilities or the robustness of the numerical line integration are interpreted. Second, the dependence of the RSB field errors upon the curvature of the surfaces is investigated. They become smaller in higher frequency and are related with the aberration of the geometrical reflected ray; for the dipole wave illumination, the error is generally smaller and larger for the concave and convex surfaces, respectively. This surface shape dependence of MER errors in diffraction is quite analogous to those in predicting geometrical optics contributions, the latter of which is the complement of the former and was previously reported by the authors.

Young's Diagnostics of Phase Singularities into Polychromatic Light Fields

Two original approaches for the diagnostics of phase singularities (such as optical vortices, screw dislocations of a wave front) based on the Young-Rubinowicz mOdel of diffraction phenomena are represented. Both techniques are implemented without using a separate reference wave, as in common interference techniques. That is why they are very convenient for analysis of spatially coherent polychromatic fields. The first technique is based on the use of an opaque strip as the diffraction device. Bending Young's interference fringes produced by the edge diffraction waves from two rims of the strip within the geometrical shadow region reflect helicity of a wave front, so that the direction and magnitude of bending correspond directly to the sign and the modulus of topological charge of the optical vortex, respectively. This technique is practicable for diagnostics of isolated polychromatic vortices, such as rainbow Laguerre-Gaussian mode, where the condition of mutual spectral purity is satisfied. Another technique is based on the use of knife-edge diffraction. The edge of an opaque screen serves as the source of a reference wave, which interferes with the tested field within the directly illuminated region. One observes typical interference forklets near the geometrical shadow boundary, which detect optical vortices. This technique is more applicable to diagnostics of phase singularities in polychromatic speckle fields due to the condition of mutual spectral purity is satisfied automatically.

Straight-edge diffraction of Planck radiation

The irradiance diffraction profile of a straight edge is given as a Taylor series in powers of the distance from the geometrical shadow boundary to any point in the profile for monochromatic radiation. The coefficients of the series, which are obtained as simple analytic expressions, are proportional to the real part of a complex number whose phase cycles through a complete period every eight terms in the series. Integration of this series over a Planck distribution of radiation yields the power series for the Planck profile; this derived series has a finite radius of convergence. The asymptotic series for the Planck profile far from the shadow boundary and beyond the radius of convergence of its power series is obtained by analytic continuation of the power series with the aid of a Barnes type of integral representation.

A uniform asymptotic expansion in the problem of half‐plane diffraction

Three terms in the uniform asymptotic expansion of the diffraction wave have been obtained directly from the nonuniform asymptotic expansion by an approximative method of formal summation. The accuracy of the order up to k−2 in the vicinity of geometrical shadow boundaries, and up to k−3 outside transition regions is ensured, k being the wave number. The first two terms coincide with the known UTD + MSD solution.

Uniform asymptotic theory of diffraction by apertures

Simple and explicit formulas that represent uniform asymptotic approximations to double integrals are derived and applied to obtain a uniform asymptotic theory of diffraction by apertures. The formulas are used to evaluate the diffracted field or its angular spectrum in regions in which the results of ordinary asymptotic theories, such as the theory of the boundary diffracted wave and the geometrical theory of diffraction, break down. Thus our results remain valid in regions containing geometrical shadow boundaries or caustics of diffracted rays. To illustrate their general applicability, we apply them to two diffraction problems involving a circular aperture: For a diverging spherical incident wave we compute its diffracted field, and for a converging incident wave we compute both its angular spectrum and its diffracted field far from focus.

Separation and Analysis of the Acoustic Field Scattered by a Rigid Sphere

The scattered field calculated when a plane acoustic wave is incident upon an ideally rigid sphere is considered as a composite and resolved into two parts. One part, which includes the specularly reflected contribution to the field, arises out of scattering by the bright hemisphere—the scattering sphere being divided into two regions by the boundary of the geometrical shadow. The second part, owing to scattering by the shadow side of the sphere, includes the major part of the radiation due to creeping waves. This separation is consimilar with the separation of the scattered field that is made in the creeping-wave formulation of scattering theory. The separation is made here, however, by treating the scattering sphere as a spherical radiator and solving a pair of boundary-value problems by classical means. Each of the boundary-value problems is related to one part of the scattered field. Both parts are found to yield scattering components that are related to “image pulses” such as are predicted when scattering problems involving transient signals are analyzed using theory based on the Kirchhoff approximation. It is found, however, that the image-pulse-type returns, which arise out of the exact classical theory, cancel when the two separate parts of the scattered field are added together, so that, in the complete scattered field, no image-pulse-type return is detectable. It is also found that a secondary creeping-wave component originates in the scattering from the bright side of the sphere. This result indicates that the generation of creeping waves may not be a phenomenon taking place solely at the geometrical shadow boundary. Moreover, it is found that the part of the scattered field which yields the specularly reflected return can be very closely approximated by the scattered field that would be predicted by calculations incorporating the Kirchhoff approximation. Interference between the various components, identified as a result of the separation, explain a number of features of the scattering behavior observed when long pulses are incident on the sphere.

THE FRESNEL FIELD IN KOTTLER'S DIFFRACTION THEORY

The Kottler theory of electromagnetic diffraction has been shown to be in good agreement near the boundary of geometrical shadow with the Kirchhoff scalar approach.

Shift of the Shadow Boundary and Scattering Cross Section of an Opaque Object

When a wave of wavelength λ is incident upon an opaque object of typical dimension a, a shadow is formed in the geometric optics limit λ/a=0. If λ/a is small and not zero, the shadow boundary is shifted slightly from the geometrical shadow boundary as was first shown by Artmann. He found the shift to be asymptotic to α(λ2a)⅓ for a circular cylinder, where α is positive or negative according as the field or its normal derivative vanishes on the cylinder. The same result was obtained by Rice for a parabolic cylinder, but for the hard cylinder his α differed from Artmann's. We have redetermined α for the circular cylinder and found it to agree with the result for the parabolic cylinder in both cases. We have also determined the shift for a circular cylinder on which the field satisfies an impedance boundary condition. The former result is implicit in the work of Goriainov and both results are implicit in the work of Wait and Conda. We have also determined the scattering cross section of a circular cylinder with an impedance boundary condition. These results lead us to propose two formulas, one for the shift of the shadow boundary and one for the scattering cross section, of any smooth two- or three-dimensional object. The latter expresses the deviation from the geometrical optics cross section as an integral, around a normal section of the shadow, of a multiple of the shift. This formula is verified for a sphere and for oblique incidence on a circular cylinder. Both electromagnetic and scalar waves are considered.