Showing papers on "Boundary value problem published in 1971"

Optimal Control of Systems Governed by Partial Differential Equations

Abstract: Principal Notations.- I Minimization of Functions and Unilateral Boundary Value Problems.- 1. Minimization of Coercive Forms.- 1.1. Notation.- 1.2. The Case when ?: is Coercive.- 1.3. Characterization of the Minimizing Element. Variational Inequalities.- 1.4. Alternative Form of Variational Inequalities.- 1.5. Function J being the Sum of a Differentiable and Non-Differentiable Function.- 1.6. The Convexity Hypothesis on $$ {U_{ad}} $$.- 1.7. Orientation.- 2. A Direct Solution of Certain Variational Inequalities.- 2.1. Problem Statement.- 2.2. An Existence and Uniqueness Theorem.- 3. Examples.- 3.1. Function Spaces on ?.- 3.2. Function Spaces on ?.- 3.3. Subspaces of Hm(?).- 3.4. Examples of Boundary Value Problems.- 3.5. Unilateral Boundary Value Problems (I).- 3.6. Unilateral Boundary Value Problems (II).- 3.7. Unilateral Boundary Value Problems (III).- 3.8. Unilateral Boundary Value Problems Case of Systems.- 3.9. Elliptic Operators of Order Greater than Two.- 3.10. Non-differentiable Functionals.- 4. A Comparison Theorem.- 4.1. General Results.- 4.2. An Application.- 5. Non Coercive Forms.- 5.1. Convexity of the Set of Solutions.- 5.2. Approximation Theorem.- Notes.- II Control of Systems Governed by Elliptic Partial Differential Equations.- 1. Control of Elliptic Variational Problems.- 1.1. Problem Statement.- 1.2. First Remarks on the Control Problem.- 1.3. The Set of Inequalities Defining the Optimal Control.- 2. First Applications.- 2.1. System Governed by the Dirichlet Problem Distributed Control.- 2.2. The Case with No Constraints.- 2.3. System Governed by a Neumann Problem Distributed Control.- 2.4. System Governed by a Neumann Problem Boundary Control.- 2.5. Local and Global Constraints.- 2.6. System Governed by a Differential System.- 2.7. System Governed by a 4th Order Differential Operator.- 2.8. Orientation.- 3. A Family of Examples with N = 0 and $$ {U_{ad}} $$ Arbitrary.- 3.1. General Case.- 3.2. Application (I).- 3.3. Application (II).- 4. Observation on the Boundary.- 4.1. System Governed by a Dirichlet Problem (I).- 4.2. Some Results on Non-homogeneous Dirichlet Problems.- 4.3. System Governed by a Dirichlet Problem (II).- 4.4. System Governed by a Neumann Problem.- 5. Control and Observation on the Boundary. Case of the Dirichlet Problem.- 5.1. Orientation.- 5.2. Boundary Control in L2(?).- 5.3. A "Controllability-Like" Problem.- 5.4. Pointwise Control and Observation.- 6. Constraints on the State.- 6.1. Orientation.- 6.2. Control and Constraints on the Boundary.- 7. Existence Results for Optimal Controls.- 7.1. Orientation.- 7.2. Distributed Control.- 7.3. Singular Perturbation of the System.- 7.4. Boundary Control.- 7.5. Control of Systems Governed by Unilateral Problems.- 8. First Order Necessary Conditions.- 8.1. Statement of the Theorem.- 8.2. Proof of the Theorem.- 8.2.1. "Algebraic" Transformation.- 8.2.2. General Remarks on the Utilization of (8.13.).- 8.2.3. Proof that dj,?0.- Notes.- III Control of Systems Governed by Parabolic Partial Differential Equations.- 1. Equations of Evolution.- 1.1. Data.- 1.2. Evolution Problems.- 1.3. Proof of Uniqueness.- 1.4. Proof of Existence.- 1.5. Some Examples.- 1.6. Semi-groups.- 2. Problems of Control.- 2.1. Notation. Immediate Properties.- 2.2. Set of Inequalities Characterizing the Optimal Control.- 2.3. Case (i). Set of Inequalities.- 2.4. Case (ii). Set of Inequalities.- 2.5. Orientation.- 3. Examples.- 3.1. Mixed Dirichlet Problem for a Second Order Parabolic Equation.- 3.1.1. C = Injection Map of L2(0, T V)?L2(Q).- 3.1.2. C = Identity Map of L2(0, T V) into itself.- 3.1.3. Observation of the Final State.- 3.2. Mixed Neumann Problem for a Parabolic Equation of Second Order.- 3.2.1. Case (i).- 3.2.2. Case (ii).- 3.3. System of Equations and Equations of Higher Order.- 3.3.1. System of Equations.- 3.3.2. Higher Order Equations.- 3.4. Additional Results.- 3.5. Orientation.- 4. Decoupling and Integro-Differential Equation of Riccati Type (I).- 4.1. Notation and Assumptions.- 4.2. Operator P(t), Function r(t).- 4.3. Formal Calculations.- 4.4. The Finite Dimensional Case Approximation.- 4.5. Passage to the Limit.- 4.6. Integro-Differential Equation of Riccati Type.- 4.7. Connections with the Hamilton-Jacobi Theory.- 4.8. The Case where Constraints are Present.- 4.9. Various Remarks.- 4.9.1. Direct Study of the "Riccati Equation".- 4.9.2. Another Approach to the Direct Study of the "Riccati Equation".- 4.9.3. Yet Another Approach to the Direct Study of the "Riccati Equation".- 5. Decoupling and Integro-Differential Equation of Riccati Type (II).- 5.1. Application of the Schwartz-Kernel Theorem.- 5.2. Example of a Mixed Neumann Problem with Boundary Control.- 5.3. Example of a Mixed Neumann Problem with Observation of the Final State.- 5.4. Mixed Neumann Problem, Observation of the Final State and Constraints in a Vector Space.- 5.5. Remarks on Decoupling in the Presence of Constraints.- 6. Behaviour as T ? + ?.- 6.1. Orientation and Hypotheses.- 6.2. The Case T = ?.- 6.3. Passage to the Limit as T ? + ?.- 7. Problems which are not Necessarily Coercive.- 7.1. Distributed Observation.- 7.2. Observation of the Final State.- 7.3. Examples where N = 0 and $$ {U_{ad}} $$ is not Bounded.- 8. Other Observations of the State and other Types of Control.- 8.1. Pointwise Observation of the State.- 8.2. Pointwise Control.- 8.3. Control and Observation on the Boundary.- 9. Boundary Control and Observation on the Boundary or of the Final State for a System Governed by a Mixed Dirichlet Problem.- 9.1. Orientation and Problem Statement.- 9.2. Non Homogeneous Mixed Dirichlet Problem.- 9.3. Definition of $$ \frac{{\partial y}}{{\partial {v_A}}} $$ Observation.- 9.4. Cost Function Equations of Optimal Control.- 9.5. Regular Control.- 9.6. Observation of the Final State.- 9.7. Observation of the Final State, Second Order Parabolic Operator.- 10. Controllability.- 10.1. Problem Statement.- 10.2. Controllability and Uniqueness.- 10.3. Super-Controllability and Super-Uniqueness.- 11. Control via Initial Conditions Estimation.- 11.1. Problem Statement. General Results.- 11.2. Examples.- 11.3. Controllability.- 11.4. An Estimation Problem.- 12. Duality.- 12.1. General Remarks.- 12.2. Example.- 13. Constraints on the Control and the State.- 13.1. A General Result.- 13.2. Applications (I).- 13.3. Applications (II).- 14. Non Quadratic Cost Functions.- 14.1. Orientation.- 14.2. An Example.- 14.3. Remarks on Decoupling.- 15. Existence Results for Optimal Controls.- 15.1. Orientation.- 15.2. Non-linear Problem with Distributed Control (I).- 15.3. Non-linear Problem with Distributed Control. Singular Perturbation.- 15.4. Non-linear Problem. Boundary Control.- 15.5. Utilization of Convexity and the Maximum Principle for Second Order Parabolic Equations.- 15.6. Control of Systems Governed by Evolution Inequalities.- 16. First Order Necessary Conditions.- 16.1. Statement of the Theorem.- 16.2. Proof of Theorem 16.1.- 16.2.1. "Algebraic" Transformation.- 16.2.2. Utilization of (16.11.).- 16.2.3. Proof of (16.12.).- 16.3. Remarks.- 17. Time Optimal Control.- 17.1. Problem Statement.- 17.2. Existence Theorem.- 17.3. Bang-Bang Theorem.- 18. Miscellaneous.- 18.1. Equations with Delay.- 18.1.1. Definition of the State.- 18.1.2. Control Problem.- 18.2. Spaces which are not Normable.- Notes.- IV Control of Systems Governed by Hyperbolic Equations or by Equations which are well Posed in the Petrowsky Sense.- 1. Second Order Evolution Equations.- 1.1. Notation and Hypotheses.- 1.2. Problem Statement. An Existence and Uniqueness Result.- 1.3. Proof of Uniqueness.- 1.4. Proof of Existence.- 1.5. Examples (I).- 1.6. Examples (II).- 1.7. Orientation.- 2. Control Problems.- 2.1. Notation. Immediate Properties.- 2.2. Case (2.5.).- 2.3. Case (2.6.).- 2.4. Case (2.7.).- 2.5. Case (2.8.).- 3. Transposition and Applications to Control.- 3.1. Transposition of Theorem 1.1.- 3.2. Application (I).- 3.3. Application (II).- 3.4. Application (III).- 4. Examples.- 4.1. Examples of Hyperbolic Problems. Distributed Control, Distributed Observation.- 4.2. Examples of Hyperbolic Systems. Distributed Control, Observation of the Final State.- 4.3. Petrowsky Type Equation. Distributed Control. Distributed Observation.- 4.4. Petrowsky Type Equation. Distributed Control. Observation of the Final State.- 4.5. Orientation.- 5. Decoupling.- 5.1. Problem Statement. Rewriting as a System of First Order Equations.- 5.2. Rewriting of the Set of Equations Determining the Optimal Control.- 5.3. Decoupling.- 5.4. Riccati Integro-differential Equation.- 5.5. Another Optimal Control Problem. Decoupling.- 6. Control via Initial Conditions. Estimation.- 6.1. Problem Statement.- 6.2. Coercivity of J(?).- 6.3. System of Equations Determining the Optimal Control.- 7. Boundary Control (I).- 7.1. Problem Statement.- 7.2. Definition of the State of the System.- 7.3. Distributed Observation.- 7.4. Boundary Observation.- 8. Boundary Control (II).- 8.1. Problem Statement.- 8.2. Control ? Regular.- 8.3. Examples.- 9. Parabolic-Hyperbolic Systems.- 9.1. Recapitulation of Some General Results.- 9.2. Complement.- 9.3. Control Problems.- 9.4. Example (I).- 9.5. Example (II).- 9.6. Decoupling.- 10. Existence Theorems.- 10.1. Orientation.- 10.2. Example. Introduction of a "Viscosity" Term.- 10.3. Time Optimal Control.- Notes.- V Regularization, Approximation and Penalization.- 1. Regularization.- 1.1. Parabolic Regularization.- 1.2. Application to Optimal Control.- 1.3. Application to Decoupling.- 1.4. Various Remarks.- 1.5. Regularization of the Control.- 2. Approximation in Terms of Systems of Cauchy-Kowaleska Type.- 2.1. Evolution Equation on a Variety.- 2.2. Approximation by a System of Cauchy-Kowaleska Type.- 2.3. Linearized Navier-Stokes Equation.- 3. Penalization.- Notes.

On the Boundary Condition at the Surface of a Porous Medium

Abstract: A theoretical justification is given for an empirical boundary condition proposed by Beavers and Joseph [1]. The method consists of first using a statistical approach to extend Darcy's law to non homogeneous porous medium. The limiting case of a step function distribution of permeability and porosity is then examined by boundary layer techniques, and shown to give the required boundary condition. In an Appendix, the statistical approach is checked by using it to derive Einstein's law for the viscosity of dilute suspensions.

Surface Thermal Boundary Condition for Ocean Circulation Models

Abstract: By employing a heat budget analysis appropriate to zonally and time averaged conditions within the atmosphere, it is shown that the net downward heat flux Q at the ocean's surface can be expressed as Q = Q2 (TA*–Ts), where TA* is an apparent atmospheric equilibrium temperature, Ts the sea surface temperature, and Q2 a coefficient determined from the zonally and time averaged data. The latter coefficient, which is of the order of 70 ly day−1 (°C)−1, varies with latitude by as much as 20%. It is suggested that the use of the above relation as a flux-type thermal boundary condition would allow for large-scale thermal coupling of ocean and atmosphere. The more common use of specified Ts as a boundary condition clearly does not allow for such coupling.

Calculation of plane steady transonic flows

Abstract: Transonic small disturbance theory is used to solve for the flow past thin airfoils including cases with imbedded shock waves. The small disturbance equations and similarity rules are presented, and a boundary value problem is formulated for the case of a subsonic freestream Mach number. The governing transonic potential equation is a mixed (elliptic-hyperbolic) differential equation which is solved numerically using a newly developed mixed finite difference system. Separate difference formulas are used in the elliptic and hyperbolic regions to account properly for the local domain of dependence of the differential equation. An analytical solution derived for the far field is used as a boundary condition for the numerical solution. The difference equations are solved with a line relaxation algorithm. Shock waves, if any, and supersonic zones appear naturally during the iterative process. Results are presented for nonlifting circular arc airfoils and a shock free Nieuwland airfoil. Agreement with experiment for the circular arc airfoils, and exact theory for the Nieuwland airfoil is excellent.

Boundary problems for pseudo-differential operators

Abstract: which will contain at least the operator describing a classical boundary problem, and also i ts parametr ix in the elliptic case. In fact what we construct there is one of the smallest possible \"algebras\" tha t will work. In tha t respect, our result is less general than tha t of Vi~ik and Eskin [10]. The difference lies in the fact tha t in our problem, the pseudo-differential appearing in (0.1) (coefficient A) has to satisfy a supplementary condition along the boundary: the transmission property. (1) The operators tha t arise in (0.1) have already been described in [6] (where we also require analyticity). In this work, we only require tha t the operators preserve locally Coo functions. The symbolic calculus is developed further than in [6], and we derive an index formula for elliptic problems, extending tha t of [3]. Roughly speaking, the coefficient A in (0.1)is a sum A = P + G , whereP is a pseudodifferential operator satisfying the transmission condition (w 2), and G (which we call a singular Green operator -w 3) is an operator which takes any distribution into a function which is C ~ inside ~ (but may be irregular at the boundary): such operators arise for

Numerical Simulation of Incompressible Flows Within Simple Boundaries. I. Galerkin (Spectral) Representations

Abstract: Galerkin (spectral) methods are explored for the numerical simulation of incompressible flows within simple boundaries. A major part of the paper is devoted to the development of transform methods for efficient simulation of flows in box geometries with periodic and free-slip boundary conditions. Techniques for incorporating known symmetries and invariances into transform methods are illustrated for the Taylor-Green vortex. Galerkin methods for accurate and efficient representation of rigid no-slip boundary conditions are also explained. A class of pseudospectral approximations is introduced in order to handle more complicated dynamical interactions in more complicated geometries. Later papers in this series will demonstrate the important advantages of spectral methods over finite-difference methods for simulation of many of the flows of current interest and will present specific numerical results for various transition and turbulent flows.

Stress singularities in a two-material wedge

Abstract: A method for the determination of stresses in a two-material wedge-shaped region is presented. The method is applicable for plane strain or plane stress problems and treats the general case where each region is a wedge of arbitrary angle. The results are obtained by the use of the Mellin transform and the theory of residues. The characteristic equation is investigated to determine the stress singularity resulting from certain combination of geometry and material properties. A formal solution is then presented for the case where the loading is in the form of a point dislocation along the interface. This solution is the Green's function for the more general mismatch problems and therefore has applications in solving other problems with compatible boundary conditions. The results obtained show that for small values of r the dominant effect is due to geometry and the secondary effect is caused by the choice of elastic constants of the materials.

Vibration Analysis of Structures by Component Mode Substitution

Abstract: A method is presented to determine the vibration modes of a complex structural system by using component vibration modes. The structural system is considered to be an assemblage of subsystems or components. The vibration modes for each component are determined separately and then used to synthesize the system modes. The number of component modes used may be truncated to reduce the number of generalized coordinates required for a vibration analysis. Only component vibration modes are retained as generalized coordinates when the system modes are obtained; hence, the method is particularly suitable for structures with a large number of component interface coordinates, such as finiteelement shell models. The boundary conditions used for determining component vibration modes can be either free-free or constrained. An optional technique to modify the component modes is included in order to obtain more accurate system modes. Numerical results from two examples are included.

Boundary Energy of a Bose Gas in One Dimension

Abstract: By the superposition of Bethe's wave functions, using the Lieb's solution for the system of identical bosons interacting in one dimension via a $\ensuremath{\delta}$-function potential, we construct the wave function of the corresponding system enclosed in a box by imposing the boundary condition that the wave function must vanish at the two ends of an interval. Coupled equations for the energy levels are derived, and approximately solved in the thermodynamic limit in order to calculate the boundary energy of this Bose gas in its ground state. The method of superposition is also applied to the analogous problem of the Heisenberg-Ising chain (not the ring).

Multiple scattering calculations for technology.

Abstract: A many-flux (discrete ordinate) radiative transfer calculation procedure is described with the goal of making the mathematics easy to learn and use. The major approximation is the neglect of polarization. Emission within the scattering medium is not included, and the formulas are restricted to a scattering medium bounded by parallel planes. The boundary conditions allow for a variety of kinds of illumination, and the surface reflection coefficients at the boundaries of the scattering medium are accurately determined. A comparison is made with the two-flux (Kubelka-Munk) and four-flux calculation methods, and this leads to empirical expressions for the scattering and absorption coefficients in these simple theories, which make them give nearly the same results as exact theories. These empirical expressions provide a very simple method for estimating the absolute reflectance and transmittance of turbid media and greatly increase the utility of the two-flux and four-flux calculation methods. The two-flux equations give excellent results provided the absorption is small compared to scattering and the optical thickness is greater than 5. A comparison with experimental data taken with collimated illumination shows that the four-flux equations give good results at any optical thickness even if the absorption is strong.

Effect of Width and Length on Stress Intensity Factors of Internally Cracked Plates Under Various Boundary Conditions

Abstract: This paper is concerned with the analysis of stress intensity factors of a strip with a longitudinal crack subject to tension and bending along its edges, and the tension of rectangular plates with a central crack. For both problems three types of boundary conditions, that is, stress conditions, displacement conditions and their combinations are considered. Analysis is based on Laurent expansions of the complex potentials satisfying the stress free relations along the crack. The expansion coefficients are determined from boundary conditions along outer edges, by using a perturbation technique in the first problem and a boundary collocation procedure based on resultant forces and mean displacements in the second problem. Numerical calculations are performed for various plate configurations, and the results are summarized in forms ready for practical use. The accuracy of numerical results are also examined, and they are regarded as correct up to four figures.

Uniqueness Theorems in Linear Elasticity

Abstract: 1 Introduction.- 2 Basic Equations.- 2.1 Formulation of Initial-Boundary Value Problems.- 2.2 The Classical and Weak Solutions.- 2.3 The Homogeneous Isotropic Body. Plane Elasticity.- 2.4 Definiteness Properties of the Elasticities.- 3 Early Work.- 4 Modern Uniqueness Theorems in Three-Dimensional Elastostatics.- 4.1 The Displacement Boundary Value Problem for Bounded Regions.- 4.1.1 General Anisotropy.- 4.1.2 A Homogeneous Anisotropic Material.- 4.1.3 A Homogeneous Isotropic Material.- 4.1.4 The Implication of Strong Ellipticity for Uniqueness.- 4.1.5 The Non-Homogeneous Isotropic Material with no Definiteness Assumptions on the Elasticities.- 4.1.6 The Displacement Boundary Value Problem for a Homogeneous Isotropic Sphere.- 4.1.7 Fichera's Maximum Principle.- 4.2 Exterior Domains.- 4.3 The Traction Boundary Value Problem.- 4.3.1 General Anisotropy.- 4.3.2 A Homogeneous Isotropic Material.- 4.3.3 The Traction Boundary Value Problem for a Homogeneous Isotropic Elastic Sphere.- 4.3.4 Necessary Conditions for Uniqueness in the Traction Boundary Value Problem for Three-Dimensional Homogeneous Isotropic Elastic Bodies.- 4.4 Mixed Boundary Value Problems.- 4.4.1 General Anisotropy.- 4.4.2 A Homogeneous Isotropic Material.- 5 Uniqueness Theorems in Homogeneous Isotropic Two-Dimensional Elastostatics.- 5.1 Kirchhoff's Theorem in Two-Dimensions. The Displacement and Traction Boundary Value Problems.- 5.2 Uniqueness in Plane Problems with Special Geometries.- Appendix: Uniqueness of Three-Dimensional Axisymmetric Solutions.- 6 Problems in the Whole- and Half-Space.- 6.1 Specification of the Various Boundary Value Problems. Continuity onto the Boundary and in the Neighbourhood of Infinity.- 6.2 Uniqueness of Problems (a)-(d). Corollaries for the Space EN.- 6.3 Uniqueness for the Mixed-Mixed Problem of Type (e).- 6.3.1 A Complete Representation of the Biharmonic Displacement in a Homogeneous Isotropic Body Occupying the Half-Space.- 6.3.2 Uniqueness in the Mixed-Mixed Problem (e).- 7 Miscellaneous Boundary Value Problems.- 7.1 Problems for a Sphere.- 7.2 The Cauchy Problem for Isotropic Elastostatics.- 7.3 The Signorini Problem. Other Problems with Ambiguous Conditions.- 8 Uniqueness Theorems in Elastodynamics. Relations with Existence, Stability, and Boundedness of Solutions.- 8.1 The Initial Displacement and Mixed-Boundary Value Problems. Energy Arguments.- 8.2 The Initial-Displacement Boundary Value Problem. Analyticity Arguments.- 8.3 The Initial-Mixed Boundary Value Problem for Bounded Regions. Further Arguments.- 8.4 Summary of Existing Results in the Uniqueness of Elastodynamic Solutions.- 8.5 Non-Standard Problems, including those with Ambiguous Conditions.- 8.6 Stability, Boundedness, Existence and Uniqueness.- References.

A model for the boundary condition of a porous material. Part 2

Abstract: The present paper contains an analysis of the model of a porous material proposed in part 1, and carries out calculations which allow comparison between theory and the experiments described therein. The relevant boundary conditions to be applied at an interface between a fluid and such a material are considered.

A numerical study of laminar separation bubbles using the Navier-Stokes equations

Abstract: The flow in a two-dimensional laminar separation bubble is analyzed by means of finite-difference solutions to the Navier-Stokes equations for incompressible flow. The study was motivated by the need to analyze high-Reynolds-number flow fields having viscous regions in which the boundary-layer assumptions are questionable. The approach adopted in the present study is to analyze the flow in the immediate vicinity of the separation bubble using the Navier-Stokes equations. It is assumed that the resulting solutions can then be patched to the remainder of the flow field, which is analyzed using boundary-layer theory and inviscid-flow analysis. Some of the difficulties associated with patching the numerical solutions to the remainder of the flow field are discussed, and a suggestion for treating boundary conditions is made which would permit a separation bubble to be computed from the Navier-Stokes equations using boundary conditions from inviscid and boundary-layer solutions without accounting for interaction between individual flow regions. Numerical solutions are presented for separation bubbles having Reynolds numbers (based on momentum thickness) of the order of 50. In these numerical solutions, separation was found to occur without any evidence of the singular behaviour at separation found in solutions to the boundary-layer equations. The numerical solutions indicate that predictions of separation by boundary-layer theory are not reliable for this range of Reynolds number. The accuracy and validity of the numerical solutions are briefly examined. Included in this examination are comparisons between the Howarth solution of the boundary-layer equations for a linearly retarded freestream velocity and the corresponding numerical solutions of the Navier-Stokes equations for various Reynolds numbers.

The motion of floating bodies

Abstract: We shall restrict ourselves here to floating bodies without any means of propelling themselves. The body may, of course, be a ship lying dead in the water, but there is no real limitation to practical shapes of any particular sort except that we shall suppose the body to be hydrostatically stable. This will restrict the extent of this survey in an important way: we are able to slough off all effects associated with an average velocity of the body. Since mathematical solution of problems almost inevitably proceeds by way of linearization of the boundary conditions, this means that we may avoid introducing a linearization parameter whose smallness expresses the fact that the body doesn't disturb the water much during its forward motion, for example, slenderness or thinness. If we do introduce such a geometrical assumption, it will be an additional approximation, not one forced upon us by the physical situation. Fortunately, Newman's (1970) article treats, among other things, the recent advances in the theory of motion of slender ships under way. More can be found in a paper by Ogilvie (1964) . We shall assume from the beginning that motions are small and take this into account in formulating equations and boundary conditions. Further­ more, we shaH assume the fluid inviscid, and without surface tension. It is not difficult to write down equations and boundary conditions for a less restricted problem. However, since most results are for the case of small motions and since the perturbation expansions associated with the deriva­ tion of the linearized problem from the more exact one do not present any special points of interest, it seems more efficient to start with the simpler problem. Even so, some account will be given of recent attempts to consider nonlinear problems.

Theory of HF and VHF Propagation Across the Rough Sea, 1, The Effective Surface Impedance for a Slightly Rough Highly Conducting Medium at Grazing Incidence

Abstract: One method of analyzing radiation and propagation above a surface is to employ an effective surface impedance to describe the effect of the boundary In this paper an expression is derived for the effective impedance at grazing incidence of a slightly rough interface between air and a finitely conducting medium for which the Leontovich boundary condition is applicable The perturbation technique of Rayleigh and Rice is employed, and attention is restricted to vertical polarization The resulting effective surface impedance consists of two terms, the impedance of the lower medium when the surface is perfectly smooth and a term accounting for roughness The latter term can be complex in general and depends on the strengths of the roughness spectral components present The result is applicable to either deterministic periodic surfaces or random rough surfaces Various alternate definitions of the effective surface impedance are examined and are seen to be equivalent The analysis of power flow at the surface permits the interpretation of the interaction process in terms of scattered fields

Two problems in the gravity flow of granular materials

Abstract: Two problems representative of the gravity flow of granular materials are considered in the context of a theory presented by Goodman (1970). The problems consist of steady fully-developed flow of a granular material down an inclined plane and between vertical parallel plates. It is shown that the dynamical behaviour of these materials is quite different from that of a viscous fluid. For the inclined flow problem, the normal stresses are not only unequal but vary non-linearly with depth. Also the maximum value of the mass flux distribution does not necessarily occur at the upper surface. For the vertical channel-flow problem, the material behaves somewhat like a Bingham fluid in that a plug region exists in the central part of the channel. The interesting feature of this problem is that the concentration of material volume in the shearing region outside the plug may either increase or decrease from the plug to the channel wall, depending on the boundary conditions. Experimental evidence for these phenomena in real granular materials is cited.The results of this investigation suggest that the gravity flow of granular materials is essentially governed by two factors-a material characteristic length, which is possibly related to the grain size, and the externally imposed constraints such as the gravity field or the pressure exerted upon the granular material from the confining plates.

Gravitational Radiation Damping of Slowly Moving Systems Calculated Using Matched Asymptotic Expansions

Abstract: This paper treats the slow‐motion approximation for radiating systems as a problem in singular perturbations. By using the method of matched asymptotic expansions, we can construct approximations valid both in the near zone and the wave zone. The outgoing‐wave boundary condition applied to the wave‐zone expansion leads, by matching, to a unique and easily calculable radiation resistance in the near zone. The method is developed and illustrated with model problems from mechanics and electromagnetism; these should form a useful and accessible introduction to the method of matched asymptotic expansions. The method is then applied to the general relativistic problem of gravitational radiation from gravitationally bound systems, where a significant part of the radiation can be attributed to nonlinear terms in the expansion of the metric. This analysis shows that the formulas derived from the standard linear approximation remain valid for gravitationally bound systems. In particular, it shows that, according to general relativity, bodies in free‐fall motion do indeed radiate. These results do not depend upon any definition of gravitational field energy.

Overland Flow on an Infiltrating Surface

Abstract: The partial differential equation for vertical, one-phase, unsaturated moisture flow in soils is employed as a mathematical model for infiltration rate. Solution of this equation for the rainfall-ponding upper boundary condition is proposed as a sensitive means to describe infiltration rate as a dependent upper boundary condition. A nonlinear Crank-Nicholson implicit finite difference scheme is used to develop a solution to this equation that predicts infiltration under realistic upper boundary and soil matrix conditions. The kinematic wave approximation to the equations of unsteady overland flow on cascaded planes is solved by a second order explicit difference scheme. The difference equations of infiltration and overland flow are then combined into a model for a simple watershed that employs computational logic so that boundary conditions match at the soil surface. The mathematical model is tested by comparison with data from a 40-foot laboratory soil flume fitted with a rainfall simulator and with data from the USDA Agricultural Research Service experimental watershed at Hastings, Nebraska. Good agreement is obtained between measured and predicted hydrographs, although there are some differences in recession lengths. The results indicate that a theoretically based model can be used to describe simple watershed response when appropriate physical parameters can be obtained.