Removal of Crossed Artifacts from Multimodal Dispersion Curves with Modified Frequency–Bessel Method

ABSTRACT Frequency–Bessel (F-J) transform method can obtain higher-mode Rayleigh dispersion curves by multistation ambient noise data superposition (Wang et al., 2019). Because the dispersion curves of the overtones can provide more information compared with the single fundamental mode, the nonuniqueness of surface-wave inversion can be reduced. Because of the limited number of receivers, the integral in the process of transformation cannot be calculated precisely and there exists a kind of crossed artifacts which cuts off the real dispersion curves and contaminates the spectrum. Forbriger (2003) proposed to use the Hankel function instead of the Bessel function to conduct the transformation to remove the crossed artifacts. However, this method can reduce the resolution of the spectrum from ambient noise data. In this article, we give a complete workflow to deal with ambient noises which can eliminate the crossed artifacts without reducing the resolution. The Kramers–Kronig relations are used to obtain complete cross-correlation functions and a modified F-J transform is conducted to finally acquire the spectrum without crossed artifacts.

Method of Singular Integral Equations for Analysis of Strip Structures and Experimental Confirmation

This paper presents a rigorous solution of the Helmholtz equation for regular waveguide structures with the finite sizes of all cross-section elements that may have an arbitrary shape. The solution is based on the theory of Singular Integral Equations (SIE). The SIE method proposed here is used to find a solution to differential equations with a point source. This fundamental solution of the equations is then applied in an integral representation of the general solution for our boundary problem. The integral representation always satisfies the differential equations derived from the Maxwell’s ones and has unknown functions μe and μh that are determined by the implementation of appropriate boundary conditions. The waveguide structures under consideration may contain homogeneous isotropic materials such as dielectrics, semiconductors, metals, and so forth. The proposed algorithm based on the SIE method also allows us to compute waveguide structures containing materials with high losses. The proposed solution allows us to satisfy all boundary conditions on the contour separating materials with different constitutive parameters and the condition at infinity for open structures as well as the wave equation. In our solution, the longitudinal components of the electric and magnetic fields are expressed in the integral form with the kernel consisting of an unknown function μe or μh and the Hankel function of the second kind. It is important to note that the above-mentioned integral representation is transformed into the Cauchy type integrals with the density function μe or μh at certain singular points of the contour of integration. The properties and values of these integrals are known under certain conditions. Contours that limit different materials of waveguide elements are divided into small segments. The number of segments can determine the accuracy of the solution of a problem. We assume for simplicity that the unknown functions μe and μh, which we are looking for, are located in the middle of each segment. After writing down the boundary conditions for the central point of every segment of all contours, we receive a well-conditioned algebraic system of linear equations, by solving which we will define functions μe and μh that correspond to these central points. Knowing the densities μe, μh, it is easy to calculate the dispersion characteristics of the structure as well as the electromagnetic (EM) field distributions inside and outside the structure. The comparison of our calculations by the SIE method with experimental data is also presented in this paper.

Mathematical models of the evolution of classical and solitary cylindrical waves

The evolution of nonlinear elastic cylindrical displacement waves for initial profiles in the form of a Hankel and Macdonald functions is analyzed theoretically and numerically. The difference between the two waves is that the MacDonald function has no hump, decreases monotonically and has a concave downward profile, and the Hankel function is a harmonic attenuating wave. The main novelty is that the evolution of cylindrical waves is studied for two different approaches to the solution of a nonlinear equation. Some significant differences of these waves are shown. First, the features of the Hankel wave, a harmonic wave (symmetrical profile), are briefly described. Then, theoretically and numerically, a single wave with an initial profile in the form of a MacDonald function is analyzed in more detail. Distortion of the initial profile due to the nonlinear interaction of the wave itself and the increase in the maximum amplitude during wave propagation is common to these profiles. Significant features of the McDonald wave are shown - an uncharacteristic initial profile (a profile without a classical hump) evolves in an uncharacteristic way - the profile becomes much steeper and remains convex downwards. Keywords: classical and solitary cylindrical waves; five-constant Murnaghan potential; approximate methods; Hankel and Macdonald initial wave profiles; evolution.

Localization of Small Anomalies via the Orthogonality Sampling Method from Scattering Parameters

We investigate the application of the orthogonality sampling method (OSM) in microwave imaging for a fast localization of small anomalies from measured scattering parameters. For this purpose, we design an indicator function of OSM defined on a Lebesgue space to test the orthogonality relation between the Hankel function and the scattering parameters. This is based on an application of the Born approximation and the integral equation formula for scattering parameters in the presence of a small anomaly. We then prove that the indicator function consists of a combination of an infinite series of Bessel functions of integer order, an antenna configuration, and material properties. Simulation results with synthetic data are presented to show the feasibility and limitations of designed OSM.

Sound Radiation by a Point Source Near a Surface of Aircraft

The problem of sound radiation by a source located in the tail of an aircraft is considered. Three methods of finding acoustic pressure are compared: the boundary element method, the Kirchhoff’s physical theory of diffraction and the ray theory. The simplest model in the form of two-dimensional problem and some thin long shape with acute angle is considered. The diffraction problem for an acoustically solid obstacle lay in the solving Fredholm’s integral equation of the second kind. Due to the boundary element method application, the equation along the entire region is reduced to the equation along the boundary. Discretization by grid nodes, selected on the boundary curve, using the collocation method is applied for numerical solution. A system of linear algebraic equations with real coefficients is formed, then the total acoustic pressure is found. The Kirchhoff’s physical theory of diffraction is based on the fact that on an arbitrary convex body in case of short-wave diffraction in the vicinity of each boundary point in the zone of light the boundary value of pressure is equal to twice pressures in the incident field. By the ray theory the modulus of the acoustic pressure in the scattering field is described by the Hankel function. Argument of this function is equal to the length of full path of the beam when it is reflected once from the border. In conclusion, the pressure in cases, when in the sharp edge there is a split node and when there isn’t, are compared. Also a scattering field calculated by three theories and scattering field in the far receiving point are built.

Equation of state and background geometry for the accelerating universe

By using the observational data, we show that the background geometry for both the early inflation and present accelerating universe can be quasi-de Sitter form containing cosmic fluid with equation of state [Formula: see text], where [Formula: see text]. For the dynamical expanding background of the universe, we obtain the scale factor as a function of conformal time [Formula: see text] and Hankel function index [Formula: see text]. Since the background geometry of the universe is related to the index [Formula: see text], a direct relation between the geometry [Formula: see text] and the equation of state parameter [Formula: see text] will be obtained. By using this fact and the constraints on the scalar spectral index and equation of state from Planck 2015, our calculations show that the cosmic fluid of the present accelerating universe is contained with both phantom [Formula: see text] and non-phantom [Formula: see text] phases, but the cosmic gas during inflation is contained with just non-phantom [Formula: see text] phase. Also, the background geometry for both early inflation and present accelerating universe is inferred to be quasi-de Sitter form with [Formula: see text].