Oblique wave scattering by a semi-infinite rigid dock in the presence of bottom undulations

The problem of oblique wave scattering by a semi-infinite rigid dock in the presence of varying bottom topography is investigated here using linear water wave theory. Employing a simplified perturbation analysis together with appropriate use of Green’s integral theorem, the reflection coefficient up to first order is obtained in terms of an integral involving the shape function representing the bottom topography. The zero-order reflection coefficient is obtained by using the residue calculus method of complex variable. The bottom undulations are described by sinusoidal and an exponentially decaying profile. The first order correction to the reflection coefficient is depicted graphically in a number of figures for the two shape functions characterizing the bottom undulations and appropriate conclusions are drawn.     

The coarse shape groups

The (pointed) coarse shape category Sh^* (), having (pointed) topological spaces as objects and having the (pointed) shape category as a subcategory, was recently constructed. Its isomorphisms classify (pointed) topological spaces strictly coarser than the (pointed) shape type classification. In this paper we introduce a new algebraic coarse shape invariant which is an invariant of shape and homotopy, as well. For every pointed space (X,?) and for every k∈N₀, the coarse shape group , having the standard shape group for its subgroup, is defined. Furthermore, a functor is constructed. The coarse shape and shape groups already differ on the class of polyhedra. An explicit formula for computing coarse shape groups of polyhedra is given. The coarse shape groups give us more information than the shape groups. Generally, does not imply (e.g. for solenoids), but from pro-πk(X,?)=0 follows . Moreover, for pointed metric compacta (X,?), the n-shape connectedness is characterized by , for every k?n.     

Scattering of oblique waves by bottom undulations in a two-layer fluid

Scattering of waves obliquely incident on small cylindrical undulations at the bottom of a two-layer fluid wherein the upper layer has a free surface and the lower layer has an undulating bottom, is investigated here assuming linear theory. There exists two modes of time-harmonic waves propagating at each of the free surface and the interface. Due to an obliquely incident wave of a particular mode, reflected and transmitted waves of both the modes are created in general by the bottom undulations. For small undulations, a simplified perturbation analysis is used to obtain first-order reflection and transmission coefficients of both the modes due to oblique incidence of waves of again both modes, in terms of integrals involving the shape function describing the bottom. For sinusoidal undulations, these coefficients are plotted graphically to illustrate the energy transfer between the waves of different modes induced by the bottom undulations.     

A finite difference method for the wide‐angle “parabolic” equation in a waveguide with downsloping bottom

We consider the third‐order wide‐angle “parabolic” equation of underwater acoustics in a cylindrically symmetric fluid medium over a bottom of range‐dependent bathymetry. It is known that the initial‐boundary‐value problem for this equation may not be well posed in the case of (smooth) bottom profiles of arbitrary shape, if it is just posed e.g. with a homogeneous Dirichlet bottom boundary condition. In this article, we concentrate on downsloping bottom profiles and propose an additional boundary condition that yields a well‐posed problem, in fact making it L² ‐conservative in the case of appropriate real parameters. We solve the problem numerically by a Crank–Nicolson‐type finite difference scheme, which is proved to be unconditionally stable and second‐order accurate and simulates accurately realistic underwater acoustic problems. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Oblique wave scattering by an impermeable ocean-bed of variable depth in a two-layer fluid with ice-cover

Within the framework of linearized theory, obliquely incident water wave scattering by an uneven ocean-bed in the form of a small bottom undulation in a two-layer fluid, where the upper layer has a thin ice-cover while the lower one has the undulation, is investigated here. In such a two-layer fluid, there exist two modes of time-harmonic waves—the one with lower wave number propagating just below the ice-cover and the one with higher wave number along the interface. An incident wave of a particular mode gets reflected and transmitted by the bottom undulations into waves of both the modes. Assuming irrotational motion, a perturbation technique is employed to solve the first-order corrections to the velocity potentials in the two-layer fluid by using Fourier transform appropriately and also to calculate the reflection and transmission coefficients in terms of integrals involving the shape function representing the bottom undulation. For a sinusoidal bottom topography, these coefficients are depicted graphically against the wave number. It is observed that when the oblique wave is incident on the ice-cover surface, we always find energy transfer to the interface, but for interfacial oblique incident waves, there are parameter ranges for which no energy transfer to the ice-cover surface is possible.     

Geometric lower bounds for the spectrum of elliptic PDEs with Dirichlet conditions in part

An extension of the lower-bound lemma of Boggio is given for the weak forms of certain elliptic operators, which are in general nonlinear and have partially Dirichlet and partially Neumann boundary conditions. Its consequences and those of an adapted Hardy inequality for the location of the bottom of the spectrum are explored in corollaries wherein a variety of assumptions are placed on the shape of the Dirichlet and Neumann boundaries.     

Nonlinear Stationary Surface Waves Above an Underwater Ridge

Given an ideal incompressible heavy irrotational fluid, we consider the exact statement of the problem on gravitational-capillary surface waves of small amplitude travelling along an underwater ridge. We show that, under some requirements on the shape of the bottom and the surface tension, the equations of an ideal incompressible fluid have smooth solutions periodic in the variable directed along the underwater ridge and decreasing exponentially with a small positive exponent in the perpendicular direction.     

Numerical response of an arbitrarily shaped harbour

The numerical development of resonance of a harbour of arbitrary shape and depth is studied. The harbour is subdivided into subregions according to the variations of bottom topography such that each subregion is of uniform depth. The Helmholtz wave equation is formulated in each subregion as an integral equation of the Green's theorem. The solution to the entire harbour basin is obtained by a matching procedure at the subregion boundaries. Here, we consider a harbour with basins of constant depths connected in series successively to accommodate a more complicated harbour geometry. An application of this study is made to Kincardine harbour with five basins connected in series successively.     

The steady two-dimensional flow over a rectangular obstacle lying on the bottom

We study a plane problem with mixed boundary conditions for a harmonic function in an unbounded Lipschitz domain contained in a strip. The problem is obtained by linearizing the hydrodynamic equations which describe the steady flow of a heavy ideal fluid over an obstacle lying on the flat bottom of a channel. In the case of obstacles of rectangular shape we prove unique solvability for all velocities of the (unperturbed) flow above a critical value depending on the obstacle depth. We also discuss regularity and asymptotic properties of the solutions.     

Investigation of the Stress State of Pressure Vessels of Inhomogeneous Structure Made of Composite Materials

The problem on the stress-strain state of layered cylindrical shells with bottoms of intricate shape under the action of internal pressure is considered. The elastic system examined is formed by spiral-circular winding. Two variants of the shell bottom structure are investigated. In the first variant, one spiral layer is installed, which leads to great variations in the bottom thickness along the meridian. In the second one, the bottoms are formed according to the zone-winding scheme. The stress state of the shell constructions of the classes considered is determined by solving boundary-value problems for systems of ordinary differential equations. The solution results for cylindrical shells with elliptic bottoms for the two types of winding are given. It is shown that the zone winding leads to smaller deflections and stresses than the conventional ways of reinforcing shell bottoms.     

Comparison theorems for dam foundation design from back pressure diagram

Comparison theorems are established for the determination of the subsurface contour of the dam foundation from the seepage back pressure p(x). The corresponding problems are considered in a general setting, which allows curvilinear boundary sections, nonhomogeneous soil, and multiply connected regions. The theorems analyze the sensitivity of the contour to changes in initial data, such as the back pressure diagram, aquifer geometry, race bottom shape, and distribution of seepage coefficients. Similarly to Polozhii's comparison theorems [8] for problems of seepage under head, the proposed theorems can be applied to obtain majorizing bounds for inverse boundary-value problems.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 65, pp. 76–82, 1988.     

Analysis of the Nonlinear Shallow Water Equations Over Nonplanar Topography

The role played by the beach bottom profile on coastal inundation phenomena is analyzed here by means of approximate analytical solutions of the nonlinear shallow water equations (NSWEs) over uneven bottoms. These are obtained by only using the assumptions of small waves at the seaward boundary and small topographic forcing. Our work, built on the Carrier and Greenspan [ 1 ] hodographic transformation and on the solution of the boundary value problem (BVP) for the NSWEs proposed by Antuono and Brocchini [ 2 ], focuses on the propagation of nonlinear non-breaking waves over quasi-planar beaches. Since the terms associated with the perturbed bottom only appear in the second-order perturbed solutions, the breaking conditions for the planar-beach bathymetry also predict well the breaking occurring on the nonplanar beaches analyzed here. The most important results, concerning the shoreline position and the near-shoreline velocity, are given for both pulse-like and periodic input waves propagating over two types of nonplanar bathymetries. The solution proposed here is a fundamental benchmark for any numerical and theoretical analyzes concerned with estimates of wave run-up on beaches of complex shape.     

Modeling Sediment Deposition Patterns Arising From Suddenly Released Fixed-Volume Turbulent Suspensions

Models presented in several recent papers [1–3] dealing with particle transport by, and deposition from, bottom gravity currents produced by the sudden release of dilute, well‐mixed fixed‐volume suspensions have been relatively successful in duplicating the experimentally observed long‐time, distal, areal density of the deposit on a rigid horizontal bottom. These models, however, fail in their ability to capture the experimentally observed proximal pattern of the areal density with its pronounced dip in the region initially occupied by the well‐mixed suspension and its equally pronounced local maximum at roughly the one‐third point of the total reach of the deposit. The central feature of the models employed in [1–3] is that the particles are always assumed to be vertically well‐mixed by fluid turbulence and to settle out through the bottom viscous sublayer with the Stokes settling velocity for a fluid at rest with no re‐entrainment of particles from the floor of the tank. Because this process is assumed from the outset in the models of [1–3], the numerical simulations for a fixed‐volume release will not take into account the actual experimental conditions that prevail at the time of release of a well‐mixed fixed‐volume suspension. That is, owing to the vigorous stirring that produces the well‐mixed suspension, the release volume will initially possess greater turbulent energy than does an unstirred release volume, which may only acquire turbulent energy as a result of its motion after release through various instability mechanisms. The eddy motion in the imposed fluid turbulence reduces the particle settling rates from the values that would be observed in an unstirred release volume possessing zero initial turbulent energy. We here develop a model for particle bearing gravity flows initiated by the sudden release of a fixed‐volume suspension that takes into account the initial turbulent energy of mixing in the release volume by means of a modified settling velocity that, over a time scale characteristic of turbulent energy decay, approaches the full Stokes settling velocity. Thereafter, in the flow regime, we assume that the turbulence persists and, in accord with current understanding concerning the mechanics of dense underflows, that this turbulence is most intense in the wall region at the bottom of the flow and relatively coarse and on the verge of collapse (see [22]) at the top of the flow where the density contrast is compositionally maintained. We capture this behavior by specifying a “shape function” that is based upon experimental observations and provides for vertical structure in the volume fraction of particles present in the flow. The assumption of vertically well‐mixed particle suspensions employed in [1–5] corresponds to a constant shape function equal to unity. Combining these two refinements concerning the settling velocity and vertical structure of the volume fraction of particles into the conservation law for particles and coupling this with the fluid equations for a two‐layer system, we find that our results for areal density of deposits from sudden releases of fixed‐volume suspensions are in excellent qualitative agreement with the experimentally determined areal densities of deposit as reported in [1, 3, 6]. In particular, our model does what none of the other models do in that it captures and explains the proximal depression in the areal density of deposit.     

An instability criterion for activator-inhibitor systems in a two-dimensional ball II

Let B be a two-dimensional ball with radius R. We continue to study the shape of the stable steady states to

     

Nonparametric inference for extrinsic means on size-and-(reflection)-shape manifolds with applications in medical imaging

For all p>2,k>p, a size-and-reflection-shape space of k-ads in general position in R^p, invariant under translation, rotation and reflection, is shown to be a smooth manifold and is equivariantly embedded in a space of symmetric matrices, allowing a nonparametric statistical analysis based on extrinsic means. Equivariant embeddings are also given for the reflection-shape-manifold , a space of orbits of scaled k-ads in general position under the group of isometries of R^p, providing a methodology for statistical analysis of three-dimensional images and a resolution of the mathematical problems inherent in the use of the Kendall shape spaces in p-dimensions, p>2. The Veronese embedding of the planar Kendall shape manifold is extended to an equivariant embedding of the size-and-shape manifold , which is useful in the analysis of size-and-shape. Four medical imaging applications are provided to illustrate the theory.     

On vortex perturbations in a free-interacting boundary layer

Large-scale structures with an inviscid, non-linear subdomain (deck) on the bottom of a boundary layer in the case of subsonic and transonic free stream velocities are considered. A class of locally inviscid perturbations with an internal line of discontinuity of the tangential velocity, which leads to the appearance of a free term on the right-hand side of the Benjamin-Ono equations, is investigated. The shape of the above-mentioned line is sought and it is determined from the solution of a system of one-dimensional non-stationary equations in which, apart from the Benjamin-Ono equation, a kinematic condition and an equation for the inviscid deck close to the wall also occur. An example of a periodic, non-linear solution is constructed and amplitude constraints which ensure its realization are formulated.     

The shapes of bodies with maximum lift-to-drag ratio in supersonic flow

Assuming that the pressure coefficient on the body surface is defined by the angle between the local normal to it and the velocity vector of the undisturbed flow, the problem of the shape of a body which possesses the maximum lift-to-drag ratio is solved. When the bottom section area and the constant coefficient of friction are given, the optimal body has a plane windward surface positioned at the angle of attack to the undisturbed flow. The leeward surface of the optimal body is parallel to the velocity vector of the undisturbed flow. The absolutely optimal body is a two-dimensional wedge. When additional constraints on the external dimensions of the body are specified, solutions of variational problems are obtained on the basis of which bodies which have the maximum lift-to-drag ratio in supersonic flow are designed.     

On cylindrical symmetric flows through pipe-like domains

We prove existence of cylindrical symmetric solutions to the steady Navier-Stokes equations in bounded pipe-like domains in with the slip boundary conditions. The result is shown for any large flows across the boundary assuming only a geometrical constraint on the shape of the domain which is independent of data. The simply connectedness of the domain is not required. The technique is based on a reformulation of the original problem and delivers us a new type of estimates in the Hölder spaces for this class of the solutions.