Explore the vast range of titles available.

The existence of local terrain has a great influence on the scattering and diffraction of seismic waves. The wave function expansion method is a commonly used method for studying terrain effects, because it can reveal the physical process of wave scattering and verify the accuracy of numerical methods. An exact, analytical solution of two-dimensional scattering of plane SH (shear-horizontal) waves by an elliptical-arc canyon on the surface of the elastic half-space is proposed by using the wave function expansion method. The problem of transforming wave functions in multi-ellipse coordinate systems was solved by using the extra-domain Mathieu function addition theorem, and the steady-state solution of the SH wave scattering problem of elliptical-arc depression terrain was reduced to the solution of simple infinite algebra equations. The numerical results of the solution are obtained by truncating the infinite equation. The accuracy of the proposed solution is verified by comparing the results obtained when the elliptical arc-shaped depression is degraded into a semi-ellipsoidal depression or even a semi-circular depression with previous results. Complicated effects of the canyon depth-to-span ratio, elliptical axis ratio, and incident angle on ground motion are shown by the numerical results for typical cases.

We consider Laplacian eigenfunctions in circular, spherical, and elliptical domains in order to discuss three kinds of high-frequency localization: whispering gallery modes, bouncing ball modes, and focusing modes. Although the existence of these modes has been known for a class of convex domains, the separation of variables for circular, spherical, and elliptical domains helps us to better understand the \"mechanism\" of localization, i.e., how an eigenfunction is getting distributed in a small region of the domain and decays rapidly outside this region. Using the properties of Bessel and Mathieu functions, we derive inequalities which imply and clearly illustrate localization. Moreover, we provide an example of a nonconvex domain (an elliptical annulus) for which the high-frequency localized modes are still present. At the same time, we show that there is no localization in most rectangle-like domains. This observation leads us to formulate an open problem of localization in polygonal domains and, more generally, in piecewise smooth convex domains.

Two new windows for signal processing are introduced in this work. The windows are based on radial Mathieu functions and their performance is compared with well-known ones such as Kaiser, cosh, exponential, Hamming, prolate, and other windows. Most of the results are focused on 1D systems (or windows) and extensions to 2D and multidimensional cases are straightforward. These new Mathieu windows have three adjusting parameters and, by a proper choice of them, it is possible to have a wide span of window shapes that cover most of the existing ones and they might be more appropriate for some applications. An example dealing with the identification of three tones plus normal noise is studied, and the performance of these Mathieu windows is among the best ones. So, they are highly recommended for spectral or harmonic analysis applications. Moreover, due to the high flexibility in adjusting Mathieu windows shapes, these new proposed windows may be applicable for smoothing finite response filters, and related applications as well.

We report a new family of time-dependent single-qubit radiation fields for which the correspondent evolution operator can be disentangled in an exact way via the Wei–Norman formalism. Such fields are characterized in terms of the Mathieu functions. We show that the regions of stability of the Mathieu functions determine the nature of the driving fields: For parameters in the stable region, the fields are oscillating, being able to be periodic under certain conditions. Whereas, for parameters in the instability region, the fields are pulse-like. In addition, in the stability region, this family admits solutions for evolution loops in quantum control. We obtain some prescriptions to reach such a control effect. Geometric phases in the evolution are also analyzed and discussed.

An analytic solution is presented to the scattering of a left circularly polarized (LCP) plane wave from a chiral elliptic cylinder placed in another infinite chiral medium, using the method of separation of variables. The incident, scattered, as well as the transmitted electromagnetic fields are expressed using appropriate angular and radial Mathieu functions and expansion coefficients. The unknown scattered and transmitted field expansion coefficients are subsequently determined by imposing proper boundary conditions at the surface of the elliptic cylinder. Numerical results are presented graphically as normalized scattering widths for elliptic cylinders of different sizes and chiral materials, to show the effects of these on the scattering widths.

This paper presents an analytic solution to the scattering properties of chiral elliptic cylinder embedded in infinite chiral medium due to incident plane wave. The external electromagnetic fields as well as the internal electromagnetic fields are written in terms Mathieu functions and expansion coefficients. In order to obtain both the internal and external unknown field expansion coefficients, the boundary conditions are applied rigorously at the surface of different chiral/chiral material. Results are plotted graphically for the normalized scattering widths for elliptic cylinders of different sizes and chiral materials to show the effects of these parameters on scattering cross widths. It is shown numerically by adding the external chiral material to elliptic cylinder provides more parameters to control the RCS.

The active Brownian particle (ABP) model exemplifies a wide class of active matter particles. In this work, we demonstrate how this model can be cast into a field theory in both two and three dimensions. Our aim is manifold: we wish both to extract useful features of the system, as well as to build a framework which can be used to study more complex systems involving ABPs, such as those involving interaction. Using the two-dimensional model as a template, we calculate the mean squared displacement exactly, and the one-point density in an external potential perturbatively. We show how the effective diffusion constant appears in the barometric density formula to leading order, and determine the corrections to it. We repeat the calculation in three dimensions, clearly a more challenging setup. Comparing different ways to capture the self-propulsion, we find that its perturbative treatment results in more tractable derivations without loss of exactness, where this is accessible.

Radiation characteristics of an axially slotted circular or elliptical antenna coated by biaxial anisotropic material are investigated using series solution. The fields inside and outside the coating regions are expressed in terms of appropriate Mathieu functions with unknown field coefficients. The boundary conditions at the conducting and dielectric coating surfaces are invoked to obtain the unknown field expansion coefficients. Numerical results are presented graphically for the radiation pattern with various geometries and electrical parameters.

This paper considers the effects of smoothly varying chordwise porosity of a finite perforated plate on turbulence–aerofoil interaction noise. The aeroacoustic model is made possible through the use of a novel Mathieu function collocation method, rather than a traditional Wiener–Hopf approach which would be unable to deal with chordwise-varying quantities. The main focus is on two bio-inspired porosity distributions, modelled from air flow resistance data obtained from the wings of barn owls (tyto alba) and common buzzards (buteo buteo). Trailing-edge noise is much reduced for the owl-like distribution, but, perhaps surprisingly, so too is leading-edge noise, despite both wings having similar porosity values at the leading edge. A general monotonic variation is then considered indicating that there may indeed be a significant acoustic impact of how the porosity is distributed along the whole chord of the plate, not just its values at the scattering edges. Through this investigation, it is found that a plate whose porosity continuously decreases from the trailing edge to a zero-porosity leading edge can, in fact, generate lower levels of trailing-edge noise than a plate whose porosity remains constant at the trailing-edge value.

An analytical method for calculating the matrix elements of the Hamiltonian of the torsion Schrödinger equation in a basis of Mathieu functions is developed. The matrix elements are represented by integrals of the product of three Mathieu functions, and also the derivatives of these functions. Analytical expressions for the matrix elements are obtained by approximating the Mathieu functions by Fourier series and are products of the corresponding Fourier expansion coefficients. It is shown that replacing high-order Mathieu functions by one harmonic leads to insignificant errors in the calculation.