We derive a new completely integrable dispersive shallow water equation that is bi-Hamiltonian and thus possesses an infinite number of conservation laws in involution. The equation is obtained by using an asymptotic expansion directly in the Hamiltonian for Euler's equations in the shallow water regime. The soliton solution for this equation has a limiting form that has a discontinuity in the first derivative at its peak.

An integrable shallow water equation with peaked solitons

The Classical Field Theories

Matter is commonly found in the form of materials. Analytical mechanics turned its back upon this fact, creating the centrally useful but abstract concepts of the mass point and the rigid body, in which matter manifests itself only through its inertia, independent of its constitution; “modern” physics likewise turns its back, since it concerns solely the small particles of matter, declining to face the problem of how a specimen made up of such particles will behave in the typical circumstances in which we meet it. Materials, however, continue to furnish the masses of matter we see and use from day to day: air, water, earth, flesh, wood, stone, steel, concrete, glass, rubber, ... All are deformable. A theory aiming to describe their mechanical behavior must take heed of their deformability and represent the definite principles it obeys.

The non-linear field theories of mechanics

A concise, self-contained introduction to solid polymers, the mechanics of their behavior and molecular and structural interpretations. This updated edition provides extended coverage of recent developments in rubber elasticity, relaxation transitions, non-linear viscoelastic behavior, anisotropic mechanical behavior, yield behavior of polymers, breaking phenomena, and other fields.

Mechanical properties of solid polymers

The basic physical concepts of classical continuum mechanics are body, configuration of a body, and force system acting on a body. In a formal rational development of the subject, one first tries to state precisely what mathematical entities represent these physical concepts: a body is regarded to be a smooth manifold whose elements are the material points; a configuration is defined as a mapping of the body into a three-dimensional Euclidean space, and a force system is defined to be a vector-valued function defined for pairs of bodies1. Once these concepts are made precise one can proceed to the statement of general principles, such as the principle of objectivity or the law of balance of linear momentum, and to the statement of specific constitutive assumptions, such as the assertion that a force system can be resolved into body forces with a mass density and contact forces with a surface density, or the assertion that the contact forces at a material point depend on certain local properties of the configuration at the point. While the general principles are the same for all work in classical continuum mechanics, the constitutive assumptions vary with the application in mind and serve to define the material under consideration.

The Thermodynamics of Elastic Materials with Heat Conduction and Viscosity

In a previous chapter, the foundations of the classical theory of elasticity have been presented. It is concerned with the description and explanation in terms of a unified theory of the relations which are observed between load and deformation for elastic solids of various shapes, sizes, and compostions. The elastic character of the materials to which the theory is applicable may be loosely described as follows: If a body of elastic material is subjected to a load, it will be deformed and on the removal of the load will regain its initial dimensions and shape. In the concept of the elastic solid is also contained the assumption that the mechanical system constituted by an elastic solid and a system of forces applied to it is a system in which energy is conserved. The work done by the applied forces during the iso thermal deformation of the body is balanced by potential energy stored in the elastically deformed body and the kinetic energy of the various parts of the body and the members through which the deforming forces are applied.

Large elastic deformations

The connexion between the statistical representation of turbulence and dissipation of energy has been discussed in relation to the decay of the isotropic turbulence which is produced in a wind tunnel by means of regular grids. It was shown that a length λ can be defined which may be taken as a measure of the scale of the small eddies which are responsible for dissipation. This λ can be found by measuring the correlation R y between the indications of two hot wire anemometers set at a distance y apart on a line perpendicular to the axis of the tunnel. Then 1/ λ 2 = Lt y→0 1 - R y / y 2, and the mean rate of dissipation of energy per unit volume is W¯ = 15 μ u 2¯/ λ 2, (1) where u2¯ is the mean of the square of one component of velocity. When turbulence is generated in a wind stream by a grid of regularly spaced bars it may be expected to possess a definite scale proportional to the linear dimensions of the grid. In any complete statistical description of turbulence this scale must be implicitly or explicitly involved. One way in which the scale can be defined is to measure the distance y apart by which the two hot wires must be separated before the correlation between the indications disappears. Another way is to define the scale as l 2 = ∫ y R y d y . (2)

Mechanism of the production of small eddies from large ones

Within the scope of the three-dimensional theory of homogeneous incompressible inviscid fluids, this paper contains a derivation of a system of equations for propagation of waves in water of variable depth. The derivation is effected by means of the incompressibility condition, the energy equation, the invariance requirements under superposed rigid-body motions, together with a single approximation for the (three-dimensional) velocity field.

/pdf/a-derivation-of-equations-for-wave-propagation-in-water-of-1p4hfd4v2f.pdf

A. E. Green

Papers

Large elastic deformations

Mechanism of the production of small eddies from large ones

A derivation of equations for wave propagation in water of variable depth

Large elastic deformations and non-linear continuum mechanics

A derivation of equations for wave propagation in water of variable depth