An Asymptotic Method in Ship Hydrodynamics

Ship hydrodynamics in a constant waterway have been well studied. With a coordinate system fixed on the moving ship, the boundary value problem (BVP) is usually treated as a steady one. However, the hydrodynamics of a ship moving in confined waterways with abrupt changes in width or depth are complex due to their unsteady nature. The studies on such unsteady problems are insufficient. In the engineering practice, the hydrodynamic unsteadiness can be fully reflected by a scenario when a ship enters a lock. Prior studies have predominantly focused on predicting the ships' hydrodynamic forces without considering the unsteady terms on the free-surface boundary conditions. Obviously, such steady or quasi-steady methods overlooked the crucial unsteady phenomena on free water surface. To address this gap, the present study introduces a novel three-level difference scheme to discretize the free surface condition, preserving unsteady terms while maintaining temporal continuity of cells on the free surface. With the implementation of such fully unsteady BVP, we observed some interesting unsteady free surface motions, which were not well documented in the existing literature. To verify our new observations, as well as to validate the numerical method proposed in this study, two physical model tests were designed and conducted in a towing tank: a submerged ellipsoid enters a deep lock at relatively high speeds, and a box enters a shallow and narrow lock at very low speeds. The discussion is highlighted on the unsteady waves in front of the moving bodies, as well as the unsteady resistance induced by such waves.

It is shown that there exists a category of two-dimensional diffraction problems, which can be put into a ‘standard form‘ of dual integral equations. These diffraction problems include: diffraction of electromagnetic waves by a finite strip, a finite slit, the diffraction of scalar or vector elastic waves by a rigid strip or crack, etc. A general method for solving such dual integral equations is given by the artifice of constructing a set of functions of compact support biorthogonal to another given set of functions. The sufficient conditions for a given dual integral equations to be solvable in this manner are also determined. Hence, the method forms a complement to the Weiner-Hopf method. To illustrate the method solutions are obtained for a bench-mark problem : the diffraction of light by a finite perfectly conducting strip (or equivalently the diffraction of SH waves by a crack). Comparison with results obtained by others for low, intermediate and high frequencies show the utility and accuracy of the method for the entire range of frequencies. Both the near field and the far field are obtained, the latter is shown to correspond to the Fraunhoffer diffraction pattern for high frequency. It is also shown that for the equivalent crack problem the stress intensity factor (SIF) fluctuates rapidly with changes in the angle of incidence for high frequencies, thus making the SIF especially sensitive to angle of incidence at high frequencies.

An exact analytic solution of the problem of diffraction of a plane acoustic wave on a cone of arbitrary aperture angle was obtained and studied in [1]. For the case of spherical wave diffraction on a cone a formula is known [2] which relates the solutions of the spherical and plane wave diffraction problems. This study will employ the results of [1, 2] to derive and investigate an exact analytical solution of the problem of diffraction of a spherical acoustic wave on a cone of arbitrary aperture angle. Results of numerical calculations will be presented and compared with analogous results for a plane wave.

For a long time, numerous problems in various applied fields [1-3] have been investigated with the use of singular integral equations. For example, some problems of aerodynamics [1], diffraction of electromagnetic and acoustic waves [2], elasticity [3], etc. can be reduced to the solution of such equations. The theory of singular integral equations was developed long ago [3], but related applied problems required the solution of these equations in the class of absolutely integrable functions. It was mentioned in [4] that, under specific conditions imposed on the coefficients of a singular integral equation, one can find solutions in the class of fimctions with nonintegrable singularity of the form 1/x as x -* 0. But the absence of applied problems leading to such solutions, numerical methods for their construction, and a theory of distributions developed for singular integral equations hindered the investigation of such singular solutions. However, in the framework of aerodynamics [5, 6], the mechanics of wings has recently been analyzed with the use of ejection of an external flow. It was shown in [6] that, in the case of a thin profile, this problem can be reduced to solving a singular integral equation of the first kind on a closed interval in the class of fimctions that have a nonintegrable singularity of the above-mentioned type at the ejection point. By [7], the investigation of the diffraction of an electromagnetic wave on an ideal conductor with a current thread on the surface leads to a similar mathematical problem. Such problems were numerically solved in [1, 7, 8] in sorne special cases with the use of the discrete vortex and discrete current methods. However, the integral operators constructed in these papers are continuously invertible in related pairs of spaces but have the property that their squares map the entire subspace of singular solutions to zero. In the present paper, in view of [9, 10], we give another mathematical treatment of the same applied problems, which permits one to consider them in the framework of the theory of pseudodifferential operators on a closed curve. Then we describe the construction of Sobolev type weighted spaces and develop the theory of linear operators (similar to pseudodifferential ones) in these spaces for open curves. We present examples illustrating the difference between the notions of integral in the sense of the Cauchy principal value and pseudodifferential operators on distributions that are singular solutions. Similar linear operators are constructed for singular integral equations of the second kind with Hilbert kernel and constant coefficients on a closed interval. We indicate the possibility of the generalization of the suggested construction of linear operators to some spatial problems.

The radiation and diffraction of steady-state elastic waves in a periodically perforated orthotropic plane

Diffraction of elastic waves in the plane multiply-connected region and dynamic stress concentration

Diffraction of a spherical acoustic wave due to the coupling of pressure release and absorbing half planes in a moving fluid

Diffraction of Acoustic Surface Waves in Anisotropic Media

The surface signatures of meandering fronts and eddies have been regularly observed and documented in synthetic aperture radar (SAR) images. Wave‐current interactions, the suppression of short wind waves by natural film, and the varying wind field resulting from atmospheric boundary layer changes across an oceanic temperature front all contribute to the radar image manifestation of such mesoscale features. The corresponding imaging mechanisms are quantitatively explored using a new radar imaging model (Kudryavtsev et al., 2005) that solves the energy balance equation where wind forcing, viscous and wave breaking dissipation, wave‐wave interactions, and generation of short waves by breaking waves are taken into account. High‐quality and synoptic in situ observations of the surface conditions should ideally be used in this model. However, such data are rarely available. Instead, the fields of temperature and ocean current are herein derived from two distinct numerical ocean models. SAR image expressions of current fronts and eddies are then simulated based on these fields. The comparison of simulated images with European Remote Sensing (ERS) SAR and Envisat advanced SAR (ASAR) images is favorable. We consequently believe that the new radar imaging model provides promising capabilities for advancing the quantitative interpretation of current features manifested in SAR images.

Acoustic transients generated by high voltage pulses in longitudinal KDP modulator crystals were investigated. Photographic method has been developed to visualize the acoustic waves. The experiments directly demonstrate the piezoelectric acoustic wave generation, the acoustic wave propagation, the reflexion, diffusion and diffraction of acoustic waves and the interaction of acoustic waves with crystal imperfections. From the experimental results the velocity of shear waves and the related elastic constant can be evaluated.

The field of a vertical electric dipole over a flat impedance surface is considered. Formulas for calculating the attenuation function of the ground wave field over real impedance underlying media are presented. The reduced surface impedance is determined at the boundary of the “ground-air” media and is expressed by the ratio of the tangential components of the electric and magnetic fields. For layered underlying media, the surface impedance is calculated using a recurrent formula that takes into account the thickness and electrical properties of each layer (dielectric constant and resistivity). The effects associated with the presence of a surface electromagnetic wave (SEW), when the electromagnetic field exhibits a clearly expressed surface character, are established. It is shown that the conditions for the propagation of long, medium and short radio waves over the “ice-sea” structure with a highly inductive impedance are more favorable than over an ideally conducting medium due to the occurrence of SEW.

In this work, we study numerically and theoretically phase-shifted Bragg gratings (PSBG) for Bloch surface waves (BSW) propagating along the interfaces between a 1D photonic crystal and a homogeneous medium. The studied on-chip structure consists of a set of dielectric ridges located on the photonic crystal surface constituting two symmetrical onchip Bragg gratings separated by a defect layer. Rigorous simulation results demonstrate that the surface wave diffraction on the proposed on-chip PSBG is close to the diffraction of plane electromagnetic waves on conventional PSBG. For the considered examples, the correlation coefficient between the spectra of conventional PSBG and on-chip PSBG exceeds 0.99 near the resonance corresponding to the excitation of the eigenmodes localized in the defect layer. Conventional PSBG are widely used for spectral filtering as well as for temporal and spatial transformations of optical pulses and beams including differentiation and integration of pulse envelope or beam profile. In the present work, we discuss the capability of on-chip PSBG to implement the operations of temporal and spatial differentiation of BSW pulses and beams. The presented examples demonstrate the possibility of using the proposed structure for high-quality differentiation. The obtained results can be applied for the design of the prospective integrated systems for on-chip alloptical analog computing.

The problem on the diffraction of the electromagnetic plane wave on the impedance interface between two media is investigated. The impedance is assumed to be different from a constant on a segment of the interface, where the impedance is described by piecewise-linear, quadratic, or step functions. Bibliography: 4 titles.

The diffraction of a spherical acoustic wave by a half plane in an infinite plane, porous barrier is considered. The barrier is modeled as a rigid material filled with narrow pores, more or less normal to the plane, that provide sound damping. However, the compressibility of the gas in the pores is also considered so that some sound transmission takes place. An approximate boundary condition is derived that models both these effects. The principal purpose of the barrier is to reduce the transmitted sound so that the flow resistance of the pores is large. This consideration is important in the sense that point sources are also regarded as reasonable substitutes for real sources. The far field is calculated using integral transforms, the Wiener-Hopf technique and asymptotic methods.

: LONG-TERM GOAL. The first goal of this effort is to achieve a better understanding of the role of capillary waves in ocean surface phenomena. A second goal is to investigate the propagation of ocean waves into shallow water, as on beaches and reefs, and the generation of capillary waves from these. OBJECTIVES. Objectives of this research are to demonstrate a physical model and a practical computational model for investigating capillary wave interactions with longer waves and to apply this to applications, such as generation and spectra in both wavenumber and frequency. Further objectives are to model nonlinear waves in shallow water, including the generation of capillary waves by these.