Right-angled Wedge Research Articles

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.

Impedance Variation of an Induction Coil Near the Edge of a Right-Angled Steel Wedge

Impedance Variation of an Induction Coil Near the Edge of a Right-Angled Steel Wedge

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.

Exact Scattering by Metal Structures With Strips and Right-Angle Wedges

Exact Scattering by Metal Structures With Strips and Right-Angle Wedges

Wave diffraction by an impedance wedge

A new diffraction field expression is proposed for a wedge with different face impedances. The modified theory of physical optics based diffraction coefficient is taken into account. The reflection coefficients are expressed in terms of the split functions that occur in the double integral solution of the resistive half-plane problem. The effects of wedge’s each faces are represented separately. The uniform field expression is obtained and compared with the literature for a right-angled impedance wedge numerically.

Asymptotic Stress Fields for Complete Contact Between Dissimilar Materials

We investigate the influence of material dissimilarity on the traction fields at the corners of a contact between an elastic right-angle wedge and an elastic half-plane. The local asymptotic fields are characterized in terms of the properties of the leading eigenvalue for cases of slip and stick as a function of the Dundurs bimaterial parameters α and β, and the coefficient of friction f. Permissible values of α and β are partitioned into two possible ranges, one where behavior is qualitatively similar to the case where the indenting wedge is rigid [α = 1] and the other where behavior is similar to the case where the materials are the same [α = β = 0]. The results give insight into the high local stresses at the edge of a contact between elastically dissimilar bodies and can also be used to evaluate the effectiveness of mesh refinement in corresponding finite element models.

Instability of a vortex sheet leaving a right-angled wedge

We examine the dynamics of a semi-infinite vortex sheet attached not to a semi-infinite plate but instead to a rigid right-angled wedge, with the sheet aligned along one of its edges. Our approach to this problem, which was suggested by David Crighton, accords well with the fundamental ethos of Crighton’s work, which was characterized by ‘the application of rigorous mathematical approximations to fluid mechanical idealizations of practically relevant problems’ (Ffowcs Williams, Annu. Rev. Fluid Mech., vol. 34, 2002, pp. 37–49). The resulting linearised unsteady potential flow is forced by an oscillatory dipole in the uniform stream passing along the top of the wedge, while there is stagnant fluid in the remaining quadrant. Spatial instability is considered according to well-established methods: causality is enforced by allowing the frequency to become temporarily complex. The essentially quadrant-type geometry replaces the usual Wiener–Hopf technique by the Mellin transform. The core difficulty is that a first-order difference equation of period 4 requires a solution of period unity. As a result, the complex fourth roots $(\pm 1\pm \text{i})$ of $-4$ appear in the complementary function. The Helmholtz instability wave is excited and requires careful handling to obtain explicit results for the amplitude of the instability wave.

Influence of intermolecular forces at critical-point wedge filling.

We use microscopic density functional theory to study filling transitions in systems with long-ranged wall-fluid and short-ranged fluid-fluid forces occurring in a right-angle wedge. By changing the strength of the wall-fluid interaction we can induce both wetting and filling transitions over a wide range of temperatures and study the order of these transitions. At low temperatures we find that both wetting and filling transitions are first order in keeping with predictions of simple local effective Hamiltonian models. However close to the bulk critical point the filling transition is observed to be continuous even though the wetting transition remains first order and the wetting binding potential still exhibits a small activation barrier. The critical singularities for adsorption for the continuous filling transitions depend on whether retarded or nonretarded wall-fluid forces are present and are in excellent agreement with predictions of effective Hamiltonian theory even though the change in the order of the transition was not anticipated.

Diffraction by a right-angled impedance wedge: an edge source formulation.

This paper concerns the frequency domain problem of diffraction of a plane wave incident on an infinite right-angled wedge on which impedance (absorbing) boundary conditions are imposed. It is demonstrated that the exact Sommerfeld-Malyuzhinets contour integral solution for the diffracted field can be transformed to a line integral over a physical variable along the diffracting edge. This integral can be interpreted as a superposition of secondary point sources (with directivity) positioned along the edge, in the spirit of the edge source formulations for rigid (sound-hard) wedges derived by Svensson et al. [Acta Acust. Acust. 95, 568-572 (2009)]. However, when surface waves are present the physical interpretation of the edge source integral must be altered: it no longer represents solely the diffracted field, but rather it includes surface wave contributions.

A point vortex model for the formation of ocean eddies by flow separation

A simple model for the formation of ocean eddies by flow separation from sharply curved horizontal boundary topography is developed. This is based on the Brown–Michael model for two-dimensional vortex shedding, which is adapted to more realistically model mesoscale oceanic flow by including a deforming free surface. With a free surface, the streamfunction for the flow is not harmonic so the conformal mapping methods used in the standard Brown–Michael approach cannot be used and the problem must be solved numerically. A numerical scheme is developed based on a Chebyshev spectral method for the streamfunction partial differential equation and a second order implicit timestepping scheme for the vortex position ordinary differntial equations. This method is used to compute shed vortex trajectories for three background flows: (A) a steady flow around a semi-infinite plate, (B) a free vortex moving around a semi-infinite plate, and (C) a free vortex moving around a right-angled wedge. In (A), the inclusion of surface deformation dramatically slows the vortex and changes its trajectory from a straight path to a curved one. In (B) and (C), without the inclusion of flow separation, free vortices traverse fully around the tip along symmetrical trajectories. With the effects of flow separation included, very different trajectories are found: for all values of the model parameter—the Rossby radius—the free and shed vortices pair up and move off to infinity without passing around the tip. Their final propagation angle depends strongly and monotonically on the Rossby radius.

High-frequency diffraction of an electromagnetic plane wave by an imperfectly conducting rectangular cylinder at grazing incidence

We shall derive new results for the electromagnetic scattered far wave field produced when a high frequency plane E-polarized wave is at grazing incident on an imperfectly conducting rectangular cylinder. The solution to the problem is obtained by using the geometrical theory of diffraction, multiple diffraction methods, the canonical solution for the problem of the diffraction of a plane wave by a right-angled impedance wedge, in conjunction with a novel analytic approach.

Diffraction of an obliquely incident electromagnetic wave by an impedance right-angled concave wedge

Scattering of a plane electromagnetic wave by an anisotropic impedance right-angled concave wedge at skew incidence is analysed. A closed-form solution is derived by reducing the problem to a symmetric order-2 vector Riemann–Hilbert problem (RHP) on the real axis. The problem of matrix factorization leads to a scalar RHP on a genus-3 Riemann surface. Its solution is derived by the Weierstrass integrals. Due to a special symmetry of the problem the associated Jacobi inversion problem is solved in terms of elliptic integrals, not a genus-3 Riemann θ-function. The electric and magnetic field components are expressed through the Sommerfeld integrals, and the incident and reflected waves are recovered.

Critical Point Wedge Filling

We present results of a microscopic density functional theory study of wedge filling transitions, at a right-angle wedge, in the presence of dispersionlike wall-fluid forces. Far from the corner the walls of the wedge show a first-order wetting transition at a temperature T(w) which is progressively closer to the bulk critical temperature T(c) as the strength of the wall forces is reduced. In addition, the meniscus formed near the corner undergoes a filling transition at a temperature T(f)<T(w), the value of which is found to be in excellent agreement with macroscopic predictions. We show that the filling transition is first order if it occurs far from the critical point but is continuous if T(f) is close to T(c) even though the walls still show first-order wetting behavior. For this continuous transition the distance of the meniscus from the apex grows as ℓ(w)≈(T(f)-T)(-β(w)) with the critical exponent β(w)≈0.46±0.05 in good agreement with the phenomenological effective Hamiltonian prediction. Our results suggest that critical filling transitions, with accompanying large scale universal interfacial fluctuation effects, are more generic than thought previously, and are experimentally accessible.

High-frequency diffraction of a electromagnetic plane wave by an imperfectly conducting rectangular cylinder

We shall consider the problem of determining the scattered far-wave field produced when a plane E-polarized wave is incident on an imperfectly conducting rectangular cylinder. On the basis of the uniform asymptotic solution for the problem of the diffraction of a plane wave by a right-angled impedance wedge, in conjunction with Keller’s method and multiple diffraction, a high-frequency far-field solution to the problem is given for two edge diffractions.

Time Domain Diffraction by a Right-Angled Penetrable Wedge

The electromagnetic diffraction by a right-angled dielectric wedge normally illuminated by a plane wave is considered in the time domain. The proposed approach exploits the knowledge of uniform asymptotic diffraction coefficients in the frequency domain, and uses the inverse Laplace transform for obtaining the response to an impulse plane wave. Then, the response to an arbitrary function plane wave is determined in the dielectric material and the surrounding space via a convolution integral. Comparisons with FDTD data prove the validity of the approach. At best authors' knowledge, time domain solutions are not available to date for the considered diffraction problem.

Existence, uniqueness and asymptotic behaviour of the solution of axi-symmetric flow equations past a right-angled wedge

This paper deals with the existence, uniqueness and asymptotic behaviours, as the similarity variable tends to infinity, of the solutions of equations governing a two-dimensional axi-symmetric flow past a right-angled wedge. The results pertaining to the existence and uniqueness of the solutions are based on topological arguments, whereas the results on the asymptotic behaviour are based on the asymptotic integration of second order linear differential equations.

Mode properties of flat-top silver nanoridge surface plasmon waveguides

Propagating two-dimensional plasmon modes can be guided along nanoscale flat-top metal ridges. Mode field distributions, dispersion curves, travel ranges, and figures-of-merit of various flat-top silver nanoridge plasmon waveguide modes have been calculated and are presented in this paper. It has been found that flat-top metal nanoridge plasmon modes are quasi-transverse electromagnetic (TEM) modes with longitudinal electromagnetic field components 3 orders of magnitude smaller than the transverse field components. The quasi-TEM nature of the plasmon modes reveals that the propagation of the free electron oscillation on the top of the nanoridge contributes mainly to the tightly confined ridge mode. It is also found that as the width of the nanoridge decreases, the ridge plasmon mode becomes more tightly confined on the ridge top. As the nanoridge width increases, the plasmon mode approaches two decoupled right-angle wedge plasmon modes.

A Uniform Asymptotic Solution for the Diffraction by a Right-Angled Dielectric Wedge

We propose a uniform asymptotic solution for the field diffracted by a lossless right-angled dielectric wedge illuminated by a plane wave at normal incidence. The diffraction problem is solved by splitting the observation domain in the inner region of the wedge and the surrounding free-space. The scattered electric field in each region is assumed to be originated by a set of equivalent electric and magnetic surface currents involved in the well-known radiation integral. Such currents are localized on the interior and exterior faces of the wedge, and expressed in terms of the corresponding geometrical optics field. Useful analytical manipulations and asymptotic evaluations of the resulting integrals allow one to obtain the diffraction coefficients in terms of the Fresnel's reflection and transmission coefficients of the structure and the transition function of the uniform geometrical theory of diffraction. The related diffracted field compensates the discontinuities of the geometrical optics field and gives total field levels in good agreement with finite difference time domain results.

FDTD Calculations of the Diffraction Coefficient of Vibrating Wedges

The full-vector Maxwell's equations Unite-difference time-domain (FDTD) method is employed to calculate diffraction coefficients of vibrating conducting and dielectric wedges. Two-dimensional FDTD models of right-angle wedges are constructed to include total-field scattered-field incident plane-wave source conditions as well as convolutional perfectly matched-layer boundary conditions. These models are first validated by calculating the diffraction coefficient of a stationary perfect electrical conducting (PEC) right-angle wedge for comparison to uniform geometrical theory of diffraction (UTD) analytical solutions. Next, a brute-force FDTD technique for modeling wedge vibrations is utilized to calculate the Doppler diffraction coefficients of vibrating lossless and lossy wedges.

Flexural edge waves in semi-infinite elastic plates

This paper presents a detailed analysis of the dispersion for flexural edge waves in semi-infinite isotropic elastic plates. A solution to the dynamic equations of motion is constructed by the superposition of two partial solutions, each providing zero shear stresses at the plate faces. A dispersion equation is expressed via the determinant of an infinite system of linear algebraic equations. The system is reduced to a finite one by taking into account the asymptotic behaviour of unknown coefficients. The accuracy of the solution is confirmed by a good agreement with the available experimental data and by a proper satisfaction of the prescribed boundary conditions.A detailed analysis of dispersion properties for the edge wave and corresponding displacements at various frequencies is carried out. In addition to the well-known results it is shown that the plate height does not influence the existence of the edge wave at high frequencies and, as the frequency increases, the phase velocity of the edge wave in a semi-infinite plate asymptotically approaches the velocity of an edge wave in a right-angled wedge. The performed analysis allows evaluating the plate theories such as the Kirchhoff theory or other refined plate theories developed for modeling edge waves in semi-infinite elastic plates at low frequencies.