Penetrable Wedge Research Articles

Correction to: Diffraction by a right-angled no-contrast penetrable wedge: recovery of far-field asymptotics

Correction to: Diffraction by a right-angled no-contrast penetrable wedge: recovery of far-field asymptotics

Diffraction by a right-angled no-contrast penetrable wedge: recovery of far-field asymptotics

Abstract We provide a description of the far-field encountered in the diffraction problem resulting from the interaction of a monochromatic plane-wave and a right-angled no-contrast penetrable wedge. To achieve this, we employ a two-complex-variable framework and use the analytical continuation formulae derived in Kunz & Assier (2023, Diffraction by a right-angled No-contrast penetrable wedge: analytical continuation of spectral functions. Q. J. Mech. Appl. Math., 76, 211−241) to recover the wave-field’s geometrical optics components, as well as the cylindrical and lateral diffracted waves. We prove that the corresponding cylindrical and lateral diffraction coefficients can be expressed in terms of certain two-complex-variable spectral functions, evaluated at some given points.

The effects of bathymetry on estimating the depth of a source in range-dependent environments with a single hydrophone

The effects of varying or range-dependent bathymetry, e.g., such as a sloping bottom, on the depth resolution characteristics of single hydrophone matched field processing (MFP), e.g., width of depth main-lobe as a function of source and receiver nominal positions, is not well understood. A major challenge is that in the case of range-dependent environments there are generally no analytical representations for the pressure field, thus requiring numerical methods for their evaluation. To provide some theoretical insights into how the shape of the bathymetry and locations of the source and receiver influences the MFP depth ambiguity function, we study the case of an ideal wedge with perfectly reflective pressure-release boundaries using the analytical solutions of Buckingham [1]. We derive a simple approximation for the depth ambiguity function main lobe width and compare it against the theoretical result for an ideal range-independent environment [2] and then numerically for a penetrable wedge and more realistic shelf-like environments. M.K. Buckingham, “Acoustic propagation in a wedge-shaped ocean with perfectly reflecting boundaries,” NRL report 8793, March 1984. M. Cheney and I.P. Kirsteins, “Resolution of matched field processing for a single hydrophone in a rigid waveguide,” J. Acoust Soc. Am. 151, pp. 3186–3197, 2022.

Diffraction by a Right-Angled No-Contrast Penetrable Wedge: Analytical Continuation of Spectral Functions

Summary We study the problem of diffraction by a right-angled no-contrast penetrable wedge by means of a two-complex-variable Wiener–Hopf approach. Specifically, the analyticity properties of the unknown (spectral) functions of the two-complex-variable Wiener–Hopf equation are studied. We show that these spectral functions can be analytically continued onto a two-complex dimensional manifold, and unveil their singularities in C2. To do so, integral representation formulae for the spectral functions are given and thoroughly used. It is shown that the novel concept of additive crossing holds for the penetrable wedge diffraction problem, and that we can reformulate the physical diffraction problem as a functional problem using this concept.

Exploration of Newtonian Heating, Viscous Dissipation Effects on MHD Mixed Convection Flow of Williamson Fluid Against Radially Stretched Penetrable Wedge: A Numerical Study

An investigation on framing the features of Newtonian heating, suction/blowing, viscous dissipation on magneto hydrodynamic mixed convective Williamson fluid flow over a porous stretching wedge that is influenced by thermal radiation. The con-temporary model is employed to simulate the non-Newtonian behavior of fluid over porous wedge which stretches faster or slower than free stream velocity. Practical applications of wedge flow are drug delivery, cancer treatment, biotechnology and biomedicine. Initially, the major partial differential equations are transformed into ordinary differential equations via a suitable set of similarity transformation and then solved by the developed bvp4c algorithm in MATLAB. A comprehensive analysis of definite parameters on the profiles of velocity, temperature, [Formula: see text], and skin friction coefficient is explained and scrutinized. The results indicate that elaboration in [Formula: see text] and We shows a decline in velocity profile at [Formula: see text] but an opposite trend for [Formula: see text]. However, temperature increases for both [Formula: see text]. Furthermore, (at [Formula: see text]), an augmentation in Pr results in uphill local Nusselt number. The consequences of the present exploration are compared with the existing works in specific situations and more agreement has been perceived.

MHD radiative ohmic heating nanofluid flow of a stretching penetrable wedge: A numerical analysis

AbstractThe present numerical investigation has focused on the magnetohydrodynamics flow of a viscous nanofluid over a stretching wedge with the boundary convective conditions, thermal radiation, and ohmic heating. Buongiorno's two‐component nonhomogeneous nanoscale model was used and a dilute nanofluid with spherical type particles is considered. Similarity transformations are used to render the system of governing partial differential equations into a system of coupled similarity equations. The transformed equations are solved numerically with the BVP4C method. Validation of solutions with previous studies based on special cases of the general model is included. The salient features of fluid velocity profile, temperature as well as concentration profiles are discussed in a graphical manner for various values of selected governing factors. The skin friction coefficient, mass, and heat transfer rates are calculated and summarized. It is worthwhile noticing that the validation results exhibit an excellent agreement with already existing reports. The modeling of the present problem is useful in the thermal processing of sheet‐like substances that is a necessary operation in paper procurement, wire drawing, drawing of plastic films, polymeric sheets, and metal spinning.

Wide-angle mode parabolic equations for the modelling of horizontal refraction in underwater acoustics and their numerical solution on unbounded domains

The modelling of sound propagation in the ocean by the solution of mode parabolic equations is discussed. Mode parabolic equations can be obtained as the one-way approximation to horizontal refraction equations for modal amplitudes. Their wide-angle capabilities depend on the order of the Padé approximation of the involved pseudo-differential operators. Various aspects of numerical solution methods for wide-angle mode parabolic equations are considered in detail, including artificial domain truncation and Cauchy initial data for the point source field approximation. The capabilities of the discussed numerical approaches are demonstrated in several important test cases, including the problems of sound propagation in a penetrable wedge and in a sea with an underwater canyon.

High-contrast approximation for penetrable wedge diffraction

Abstract The important open canonical problem of wave diffraction by a penetrable wedge is considered in the high-contrast limit. Mathematically, this means that the contrast parameter, the ratio of a specific material property of the host and the wedge scatterer, is assumed small. The relevant material property depends on the physical context and is different for acoustic and electromagnetic waves for example. Based on this assumption, a new asymptotic iterative scheme is constructed. The solution to the penetrable wedge is written in terms of infinitely many solutions to (possibly inhomogeneous) impenetrable wedge problems. Each impenetrable problem is solved using a combination of the Sommerfeld–Malyuzhinets and Wiener–Hopf techniques. The resulting approximated solution to the penetrable wedge involves a large number of nested complex integrals and is hence difficult to evaluate numerically. In order to address this issue, a subtle method (combining asymptotics, interpolation and complex analysis) is developed and implemented, leading to a fast and efficient numerical evaluation. This asymptotic scheme is shown to have excellent convergent properties and leads to a clear improvement on extant approaches.

Benchmarking ocean acoustic models

In this special session honoring the work by Buckingham, we review some of his early research from the 1980s where he developed analytic solutions to various benchmark problems. In particular, he developed elegant solutions for (1) the perfect wedge; (2) a penetrable wedge; and (3) a conical seamount. These were very valuable at the time in resolving errors in early parabolic equation models; they have continued to be important as computational models for fully three-dimensional environments have become more widely used.In this special session honoring the work by Buckingham, we review some of his early research from the 1980s where he developed analytic solutions to various benchmark problems. In particular, he developed elegant solutions for (1) the perfect wedge; (2) a penetrable wedge; and (3) a conical seamount. These were very valuable at the time in resolving errors in early parabolic equation models; they have continued to be important as computational models for fully three-dimensional environments have become more widely used.

On the Method of Source Images for the Wedge Problem Solution in Ocean Acoustics: Some Corrections and Appendices

In this study the method of source images for the problem of sound propagation in a penetrable wedge [G. Deane and M. Buckingham, J. Acoust. Soc. Am. 93 (1993) 1319–1328] is revisited. This solution is very important three-dimensional (3D) benchmark in computational underwater acoustics, since a wedge bounded from above by the sea surface and overlying a sloping penetrable bottom is the simplest model of a shallow-sea waveguide near the coastline. The corrected formulae for the positions of the source images and bottom images are presented together with the explanation of their derivation. The problem of branch choice in the reflection coefficient is thoroughly discussed, and the corresponding explicit formulae are given. In addition, numerical validation of the proposed branch choice schemes and the resulting wedge problem solutions are presented. Finally, source images solution is computed for a series of examples with different ratios of shear and bulk moduli in the bottom. The interplay between the acoustic-elastic waves coupling and the horizontal refraction in the wedge is demonstrated.

Generalized ray method for three-dimensional propagation in a penetrable wedge

A generalized ray solution is presented for the three-dimensional (3-D) acoustic field produced by a transient point source in a rigid-bottom, fast-speed fluid-bottom, and fast-speed elastic-bottom wedge modeling a continental shelf environment. The effect of diffraction (scattering) at the wedge apex is ignored in the solution, but nevertheless the solution is exact and complete for the image component of the field, expressed by a sum of partial waves including a pulse emitted from the source plus a finite number of pulses reflected off the wedge boundaries. Time records of the acoustic pressure illustrating 3-D acoustical propagation effects are evaluated at five receivers located at equal radial range in the horizontal from the source, but different orientation to the source, measured by the azimuthal angle assuming values: 0^{circ } (down-slope), 45^{circ } (obliquely down-slope), 90^{circ } (cross-slope), 135^{circ } (obliquely up-slope), and 180^{circ } (up-slope). The arrival times of the spherical and head waves propagating along various paths are evaluated at each receiver for each bottom. It was found that the time interval between the first arrival and the ultimate arrival diminished, and the pulses became more peaked, as the azimuthal angle of the receiver increased. For the rigid-bottom wedge, we have found that all backscattered pulses are significant at the up-slope receiver. For the fluid-bottom wedge, we have found that: at each receiver, the source-pulse is preceded by the ground wave which is much weaker than the water wave; and, at the up-slope receiver, the backscattered pulses are insignificant. For the elastic-bottom wedge, we have found that: at each receiver, the ground-wave-response begins earlier than that in the fluid-bottom wedge, and thus the record separates out into distinct ground-wave and water-wave phases; and, at the up-slope receiver, the backscattered pulses are significant.

An explicit analytical solution for sound propagation in a three-dimensional penetrable wedge with small apex angle.

A problem of sound propagation in a shallow-water waveguide with a weakly sloping penetrable bottom is considered. The adiabatic mode parabolic equations are used to approximate the solution of the three-dimensional (3D) Helmholtz equation by modal decomposition of the acoustic pressure field. The mode amplitudes satisfy parabolic equations that admit analytical solutions in the special case of the 3D wedge. Using the analytical formula for modal amplitudes, an explicit and remarkably simple expression for the acoustic pressure in the wedge is obtained. The proposed solution is validated by the comparison with a solution of the 3D penetrable wedge problem obtained using a fully 3D parabolic equation that includes a leading-order cross term correction.

Leading-order cross term correction of three-dimensional parabolic equation models.

The issue of handling a leading-order cross-multiplied term in three-dimensional (3D) parabolic equation (PE) based models is addressed. In particular, numerical results obtained incorporating a leading-order cross-term correction in an existing 3D PE model, written in cylindrical coordinates, based on higher-order Padé approximations in both depth and azimuth, and a splitting operator technique are reported. Note that the numerical algorithm proposed in this paper could be used in the future to update any 3D PE codes that neglect cross terms and use a splitting numerical technique. The 3D penetrable wedge benchmark problem is chosen to illustrate the accuracy of the now-fully wide-angle enhanced 3D PE model. The comparisons with a 3D reference solution based on the image source clearly show that handling the leading-order cross term in the 3D PE computation is sufficient to remove the phase errors inherent to any 3D PE models that neglect cross terms in their formulations.

Numerically exact three-dimensional propagation

Solving propagation in a three-dimensional environment is one of the most challenging problems in computational physics. However, for cases where the environment possesses some kind of symmetry be it rotational or translational, the propagation problem can be simplified considerably. It can be shown that in problems where the environment is translationally invariant, the 3D wave equation can be Fourier transformed along the direction of translational symmetry to reduce it to a 2D equation for each spectral component. The 3D solution can be obtained by solving the 2D wave equation for each spectral component and performing the inverse Fourier transform. In this paper, we use the above technique to compute propagation in an ideal and a penetrable wedge. For the ideal wedge, the pressure-release boundary condition is applied to both boundaries and for the penetrable wedge, the pressure-release boundary condition is applied to the horizontal surface and the continuity of pressure and normal velocity is imposed on the sloped interface. To obtain a numerically exact solution, we use the virtual source technique to solve the 2D problem for each spectral component.

High-Frequency Evaluation of the Field Inside and Outside an Acute-Angled Dielectric Wedge

Explicit closed form solutions are presented for the high-frequency evaluation of the electromagnetic field produced inside and outside an acute-angled lossless penetrable wedge. Both cases of E- and H-polarized incident plane waves are addressed in the study. The problem is tackled and solved in the framework of the uniform theory of diffraction, so that the total field at the observation point is determined by adding the geometrical optics contributions and the diffraction one. This last is obtained by performing a uniform asymptotic evaluation of the radiation integrals arising from a physical optics approximation for the equivalent electric and magnetic surface currents lying on the infinite wedge boundaries. No limitation exists on the refractive index of the structure. The accuracy of the solution is assessed by comparisons with data produced by numerical tools. The physical optics approximation gives rise to inaccuracies when using the solution at grazing incidence and in correspondence of the dielectric interfaces.

A coupled-mode method for sound propagation in a range-dependent penetrable waveguide

The coupled-mode method based on the direct global matrix (DGMCM) approach for sound propagation in range-dependent waveguides [Luo et al., "A numerically stable coupled-mode formulation for acoustic propagation in range-dependent waveguides," Sci. China-Phys. Mech. Astron. 55, 572 (2012)] is further developed. The normal mode model KRAKEN is adopted to provide local modal solutions and their associated coupling matrices. As a result, the model DGMCM is capable of providing full two-way solutions for the two-dimensional realistic problems characterized by a penetrable bottom and a depth-varying sound speed profile. In addition, the closed-form expressions of coupling matrices for sound propagation in a range-dependent, two-layer waveguide are proposed. The numerical solutions of the coupling matrices by DGMCM agree well with the analytical solutions. Sound propagation in a penetrable wedge is solved by the updated DGMCM model. The numerical results indicate that the updated DGMCM model is numerically stable and accurate, and can provide benchmark solutions for realistic range-dependent problems.

The Wiener-Hopf Formulation of the Penetrable Wedge Problem. Part III: The Skew Incidence Case

Earlier works obtained the semi-analytical solution of the generalized Wiener-Hopf equations that formulate the diffraction of a plane wave by a dielectric wedge at normal incidence. This solution permits the separating, recognizing, and understanding of the different components of the dielectric wedge field: reflected and refracted plane waves, surface waves, lateral waves, diffraction coefficients. This article considers the general case of a plane wave impinging on an arbitrary penetrable wedge at skew incidence. Comparing previous findings with the skew incident case, no new conceptual and numerical difficulties are present. However, the involved algebraic manipulations become very cumbersome, and to handle them, the MATHEMATICA (Wolfram Research, Inc., Champaign, Illinois, USA) computer program was used.

The Wiener-Hopf Solution of the Isotropic Penetrable Wedge Problem: Diffraction and Total Field

The diffraction of an incident plane wave by an isotropic penetrable wedge is studied using generalized Wiener-Hopf equations, and the solution is obtained using analytical and numerical-analytical approaches that reduce the Wiener-Hopf factorization to Fredholm integral equations of second kind. Mathematical aspects are described in a unified and consistent theory for angular region problems. The formulation is presented in the general case of skew incidence and several numerical tests at normal incidence are reported to validate the new technique. The solutions consider engineering applications in terms of GTD/UTD diffraction coefficients and total fields.

Plane-wave diffraction by an obtuse-angled dielectric wedge

Uniform high-frequency solutions in closed form are derived for the diffraction of a plane wave normally impacting on a penetrable wedge having an obtuse apex angle and arbitrary dielectric permittivity. The approach used here takes advantage of a physical optics approximation for the electric and magnetic equivalent surface currents in the scattering integrals related to the inner region of the wedge and the surrounding space. Numerical tests and comparisons with finite-difference time-domain results demonstrate the accuracy and effectiveness of the proposed solutions.

A Symmetric Riemann-Hilbert Problem for Order-4 Vectors in Diffraction Theory

Summary A class of n × n matrices of the form G = R1 + a R2 (a is a scalar Hfunction defined on a contour L, R1 and R2 are rational matrices), which admit a closed-form Wiener-Hopf factorization is analyzed. It is shown that the diffraction problem ( E-polarization) for a penetrable right-angled wedge with an electrically resistive and a perfectly magnetically conductive sides can be formulated as a Riemann-Hilbert problem for two order-4 vectors with a 4 × 4 matrix coefficient G = R1 + a R2. This problem reduces to two scalar Riemann-Hilbert problems and one vector problem for two order-2 vectors with a certain constraint. The order-2 vector problem is solved by employing the theory of the scalar Riemann-Hilbert problem on a genus-1 Riemann surface. The component Ez of the electric field and the diffraction coefficient are determined.