An efficient 3-D infinite element for water wave diffraction problems

An efficient and simple infinite element for modelling the far field potential of water wave diffraction problems is presented. The shape functions in the radial direction comprise the first two terms of the asymptotic expansions of Hankel functions. The integrals with infinite limit for calculating the coefficient matrix have been worked out. Numerical tests on the diffraction by a surface-piercing circular cylinder give surprisingly accurate resultant forces even if the infinite elements are placed very near to the cylinder. Other typical three dimensional examples also show that satisfactory results can be obtained by the use of this simple infinite element. A computer program, WALOAD, has been developed for computing the wave forces on fixed two and three dimensional bodies.     

On periodic and solitary pure gravity waves in water of infinite depth

Non-linear two-dimensional waves propagating at a constant velocity at the surface of a fluid of infinite depth are considered. The fluid is assumed to be inviscid and incompressible and the flow irrotational. Gravity is taken into account, but surface tension is neglected. Accurate solutions are computed by boundary integral equation methods. Previous results on irregular waves are confirmed and extended. The computations strongly suggest the existence of non-periodic waves. These waves resemble generalised solitary waves in the sense that they are characterised by a train of oscillations in the far field.     

The optimum optical penetration depth for high diffraction efficiency in guided acousto-optic devices

In guided acousto-optic interaction, diffraction efficiency is a strong function of the overlap integral. Overlap integral with different optical penetration depth is analyzed theoretically. The optimum penetration depth is obtained for high diffraction efficiency.     

Degradation of alachlor in aqueous solution by using hydrodynamic cavitation

Samples of zirconium(IV)iodotungstate have been synthesized under varying mixing order and ratios of aqueous solution of potassium iodate, sodium tungstate and zirconium oxychloride at pH 1. A tentative formula was proposed on the basis of chemical composition, FTIR and thermogravimetric studies. The material shows a capacity of 0.68 meq g⁻¹ (for K⁺) which can be retained up to 200 °C. pH titration data reveal its monofunctional behavior. The distribution coefficient values of metal ions have been determined in various solvent systems. A number of important and analytically difficult quantitative separations of metal ions have been achieved using columns packed with this exchanger. In order to demonstrate practical utility of this material, Hg²⁺ and Pb²⁺ have been selectively separated and determined in the synthetic mixtures. Assay of Al³⁺ and Mg²⁺ in commercial tablets and analysis of lead in the standard reference material have also been attempted.     

Degradation of alachlor in aqueous solution by using hydrodynamic cavitation

The degradation of alachlor aqueous solution by using hydrodynamic cavitation was systematically investigated. It was found that alachlor in aqueous solution can be deomposed with swirling jet-induced cavitation. The degradation can be described by a pseudo-first-order kinetics and the degradation rate was found to be 4.90 × 10⁻² min⁻¹. The effects of operating parameters such as fluid pressure, solution temperature, initial concentration of alachlor and medium pH on the degradation rates of alachlor were also discussed. The results showed that the degradation rates of alachlor increased with increasing pressure and decreased with increasing initial concentration. An optimum temperature of 40 °C existed for the degradation rate of alachlor and the degradation rate was also found to be slightly depend on medium pH. Many degradation products formed during the process, and some of them were qualitatively identified by GC–MS.     

An integral equation method for the diffraction of oblique waves by an infinite cylinder

A hybrid integral equation method is formulated to study the diffraction of oblique waves by an infinite cylinder. The water depth and the geometry of the floating cylinder are assumed to be uniform in the y-direction (one of the horizontal axes). Numerical discretization and integrations are performed in the vertical plane. Analytical solutions are used in far fields such that radiation boundary conditions are satisfied. Numerical results are obtained for the case of wave scattering by a floating rectangular cylinder in a constant water depth. The accuracy and efficiency of present method are compared with those obtained by other numerical techniques.     

An analytical solution for long-wave scattering by a submerged circular truncated shoal

An analytical solution of the long-wave (or shallow-water) equation in closed-form is obtained for simple harmonic waves scattered by a submerged circular truncated shoal. This analytical solution is firstly validated against Longuet-Higgins’s classical analytical solution for a submerged cylinder, and then validated against numerical solutions obtained by using the DRBEM (dual reciprocity boundary-element method) model for a submerged circular truncated cone. Finally, the analytical solution is used to investigate the changing trend of maximum wave amplification, the trace pattern of focal position of wave-energy versus the wave period and the influence of shoal submergence on wave-energy trapping.     

The Dugdale solution for two unequal straight cracks weakening in an infinite plate

A crack arrest model is proposed for an infinite elastic perfectlyplastic plate weakened by two unequal, quasi-static, collinear straight cracks. The Dugdale model solution is obtained for the above problem when the developed plastic zones are subjected to normal cohesive quadratically varying yield point stress. Employing complex variable technique and the superposition principle the desired solution is obtained. A qualitative study of load required to arrest the plastic zones opening with respect to affecting parameters viz. inter-crack distance, plastic zone and crack length is carried out. The results obtained are presented graphically.     

Singular solution of a water wave problem in an ocean of finite depth

The Green's Function of a Water Wave Problem for an Ocean of Finite Depth, bounded internally by a circular cylinder, has been obtained by the use of an appropriate Fourier Series. The technique employed in this investigation may be used when the liquid is internally bounded by cylindrical regions of the form D × I where D is any two dimensional region in the undisturbed free surface and I is the linear interval [0, h].     

Magnetostatic field of a conductor of semicircular cross section in a highly permeable half-space: exact analytic solution for infinite permeability with perturbative correction

Poisson's equation for the magnetic vector potential is solved using complex Fourier (Laplace) transforms in bipolar coordinates, the natural system for the subject two-dimensional geometry. The source is a dc current uniformly distributed over the semicircular cross section of a long conductor that is buried in, and flush with, the otherwise planar boundary of an infinitely permeable material. Exact closed-form potentials are obtained in the conformal mapping of the Neumann boundary value problem that characterizes the case of an infinitely permeable magnetic medium. One term of a perturbative correction that accounts for finite permeability is constructed for both the uniform source distribution and for the associated Green's function.     

Numerical solution for multiple crack problem in an infinite plate under compression

This paper investigates a numerical solution for multiple crack problem in an infinite plate under remote compression. The influence of friction is taken into account. In the first step of the solution, we make a full contact assumption on the crack faces. The full contact assumption means that one component of the dislocation distribution vanishes, and the first mode stress intensity factors (K ₁) at the crack tips become zero. On the above-mentioned assumption, the problem can be solved by using integral equation method, and the second mode stress intensity factors (K ₂) at the crack tips can be evaluated. Meantime, after solving the integral equation the normal contact stress on the crack faces can be evaluated. The next step is to examine the full contact assumption. If the contact stresses on the crack faces are definitely negative, the solution is true. Otherwise, the obtained solution is not true. It is found from present study that in most cases the full contact condition is satisfied, and only in a few cases the full contact condition is violated. Numerical examples are given. It is found that the friction can lower the stress intensity factors at crack tips in general.     

A hybrid element method for diffraction of water waves by three-dimensional bodies

A hybrid element method developed recently for two-dimensional problems of water waves in an infinite fluid is extended to three dimensions. In this method only a limited fluid domain close to irregular bodies is discretized into conventional finite elements, while the remaining infinite domain is treated as one element with analytical representations of high accuracy. Continuity at the junction surface is treated as natural boundary conditions in a variational principle. Computation experience and numerical results for several ocean structures are presented.     

Strip theory for underwater vehicles in water of finite depth

This paper addresses the need to know the unsteady forces and moments on an underwater vehicle in finite-depth water, at small enough submergences for it to be influenced by sea waves. The forces are those due to the waves themselves, as well as the radiation forces due to unsteady vehicle motions. Knowledge of these forces and the mass distribution of the vehicle allow solution of the equations of motion at a single-frequency. Since the theory is linear, any incident wave field can be decomposed into the sum of many individual single-frequency sinusoidal waves. The motions due to each frequency component can then be added together to obtain the total predicted vehicle motions. The wave forces are due to the undisturbed sea wave plus those due to the diffracted wave necessary to satisfy boundary conditions on the vehicle. The long-used strip theory for ships, with the inviscid-flow approximation, is modified for finite depth and inclusion of lift forces on the vehicle fins. The two-dimensional solutions for the forces on each strip are found by a different method than is commonly used for strip theory. This form of the theory is easier to deal with and requires much less computing time than a fully three-dimensional approach. Experiments are conducted and their results are compared with the theory. Excellent agreement is found between the theoretical and experimental wave forces, including the diffracted wave. It is shown that inclusion of the forces on the fins not only improves the theoretical wave forces, but also brings the results of theory for the radiation forces and moments due to vehicle motions much closer to the experimental values that the theory without inclusion of fin lift forces.     

X-ray diffraction of permalloy nanoparticles fabricated by laser ablation in water

Permalloy (NiFeMo) nanoparticles were fabricated by laser ablation of bulk material in water with a UV pulsed laser. Transmission electron microscope images showed that approximately spherical particles about 50 nm in diameter were formed in the ablation process. All diffraction peaks corresponding to the bulk material were present in the nanoparticles. In addition to these peaks several new peaks were observed in the nanoparticles, which were attributed to nickel oxide.     

Green's function solution of a water wave problem in a stratified ocean of finite depth

A stratified ocean consisting of two layers of immissible fluids of finite thickness is considered. The waves are generated by a point source of oscillatory strength lying in the fluid. The Green's function solution is obtained by the use of an appropriately defined Fourier series. The same technique is used to study the waves, when the fluid is bounded internally by a cylinder.     

The analytic solution of near-tip stress fields for perfectly plastic pressure-sensitive material under plane stress condition

Different from dense metals, many engineering materials exhibit pressure-sensitive yielding and plastic volumetric deformation. Adopting a yield criterion that contains a linear combination of the Mises stress and the hydrostatic stress, the analytic solutions of plane-stress mode I perfectly-plastic near-tip stress fields for pressuresensitive materials are derived. Also, the relevant characteristic fields are presented. This perfectly plastic solution, containing a pressure sensitivity parameter , is shown to correspond to the limit of low-hardening solutions, and when =0 it reduces to the perfectly plastic solution of near-tip fields for the Mises material given by Hutchinson [1]. The effects of material pressure sensitivity on the near-tip fields are discussed.     

The solution of a mixed boundary value problem in the theory of diffraction

Summary An exact solution is obtained for the problem of the diffraction of a cylindrical sound wave by an absorbent semi-infinite plane. The two faces of the half-plane have different impedance boundary conditions. The problem which is solved is a mathematical model for a noise barrier whose surface is treated with two different acoustically absorbent materials.The usual Wiener-Hopf method (which is the standard technique for solving half-plane problems) has to be modified to give a solution to the present mixed boundary value problem.     

An analytic solution for groundwater uptake by phreatophytes spanning spatial scales from plant to field to regional

Phreatophytes are important to the overall hydrologic water budget, providing pathways from the uptake of groundwater with its nutrients and chemicals to subsequent discharge to the root zone through hydraulic lift and to the atmosphere through evapotranspiration. An analytic mathematical model is developed to model groundwater uptake by individual plants and fields of plant communities and the regional hydrology of communities of fields. This model incorporates new plant functions developed through aid of Wirtinger calculus. Existing methodology for area-sinks is extended to fields of phreatophytes, and Bell polynomials are employed to extend existing numerical methods to calculate regional coefficients for area-sinks. This model is used to develop capture zones for individual phreatophytes and it is shown that the functional form of groundwater uptake impacts capture zone topology, with groundwater being extracted from greater depths when root water uptake is focused about a taproot. While individual plants siphon groundwater from near the phreatic surface, it is shown that communities of phreatophytes may tap groundwater from greater depths and lateral extent as capture zones pass beneath those of upgradient phreatophytes. Thus, biogeochemical pathways moving chemical inputs from aquifer to ecosystems are influenced by both the distribution of groundwater root uptake and the proximity of neighboring phreatophytes. This provides a computational platform to guide hypothesis testing and field instrumentation and interpretation of their data and to understand the function of phreatophytes in water and nutrient uptake across plant to regional scales.     

Application of the exact solution for scattering by an infinite cylinder to the estimation of scattering by a finite cylinder

A new algorithm for cylindrical Bessel functions that is similar to the one for spherical Bessel functions allows us to compute scattering functions for infinitely long cylinders covering sizes ka = 2πa/λ up to 8000 through the use of only an eight-digit single-precision machine computation. The scattering function and complex extinction coefficient of a finite cylinder that is seen near perpendicular incidence are derived from those of an infinitely long cylinder by the use of Huygens's principle. The result, which contains no arbitrary normalization factor, agrees quite well with analog microwave measurements of both extinction and scattering for such cylinders, even for an aspect ratio p = l/(2a) as low as 2. Rainbows produced by cylinders are similar to those for spherical drops but are brighter and have a lower contrast.     

Finite-difference time-domain solution of light scattering by an infinite dielectric column immersed in an absorbing medium

The two-dimensional (2-D) finite-difference time-domain (FDTD) method is applied to calculate light scattering and absorption by an arbitrarily shaped infinite column embedded in an absorbing dielectric medium. A uniaxial perfectly matched layer (UPML) absorbing boundary condition is used to truncate the computational domain. The single-scattering properties of the infinite column embedded in the absorbing medium, including scattering phase functions and extinction and absorption efficiencies, are derived by use of an area integration of the internal field. An exact solution for light scattering and absorption by a circular cylinder in an absorbing medium is used to examine the accuracy of the 2-D UPML FDTD code. With use of a cell size of 1/120 incident wavelength in the FDTD calculations, the errors in the extinction and absorption efficiencies and asymmetry factors from the 2-D UPML FDTD are generally smaller than approximately 0.1%. The errors in the scattering phase functions are typically smaller than approximately 4%. With the 2-D UPML FDTD technique, light scattering and absorption by long noncircular columns embedded in absorbing media can be accurately solved.