The wave equation

The special and general relativity theories are used to demonstrate that the velocity of an unradiative particle in a Schwarzschild metric background, and in an electrostatic field, is the group velocity of a wave that we call a “particle wave,” which is a monochromatic solution of a standard equation of wave motion and possesses the following properties. It generalizes the de Broglie wave. The rays of a particle wave are the possible particle trajectories, and the motion equation of a particle can be obtained from the ray equation. The standing particle wave equation generalizes the Schrodinger equation of wave amplitudes. The particle wave motion equation generalizes the Klein–Gordon equation; this result enables us to analyze the essence of the particle wave frequency. The equation of the eikonal of a particle wave generalizes the Hamilton–Jacobi equation; this result enables us to deduce the general expression for the linear momentum. The Heisenberg uncertainty relation expresses the diffraction of the particle wave, and the uncertainty relation connecting the particle instant of presence and energy results from the fact that the group velocity of the particle wave is the particle velocity. A single classical particle may be considered as constituted of geometrical particle wave; reciprocally, a geometrical particle wave may be considered as constituted of classical particles. The expression for a particle wave and the motion equation of the particle wave remain valid when the particle mass is zero. In that case, the particle is a photon, the particle wave is a component a classical electromagnetic wave that is embedded in a Schwarzschild metric background, and the motion equation of the wave particle is the motion equation of an electromagnetic wave in a Schwarzschild metric background. It follows that a particle wave possesses the same physical reality as a classical electromagnetic wave. This last result and the fact that the particle velocity is the group velocity of its wave are in accordance with the opinions of de Broglie and of Schrodinger. We extend these results to the particle subjected to any static field of forces in any gravitational metric background. Therefore we have achieved a synthesis of undulatory mechanics, classical electromagnetism, and gravitation for the case where the field of forces and the gravitational metric background are static, and this synthesis is based only on special and general relativity.

The multiple-scale wave equation

A generally covariant wave equation is derived geometrically for grand unified field theory. The equation states most generally that the covariant d'Alembertian acting on the vielbein vanishes for the four fields which are thought to exist in nature: gravitation, electromagnetism, weak field and strong field. The various known field equations are derived from the wave equation when the vielbein is the eigenfunction. When the wave equation is applied to gravitation the wave equation is the eigenequation of wave mechanics corresponding to Einstein's field equation in classical mechanics, the vielbein eigenfunction playing the role of the quantized gravitational field. The three Newton laws, Newton's law of universal gravitation, and the Poisson equation are recovered in the classical and nonrelativistic, weak-field limits of the quantized gravitational field. The single particle wave-equation and Klein-Gordon equations are recovered in the relativistic, weak-field limit of the wave equation when scalar components are considered of the vielbein eigenfunction of the quantized gravitational field. The Schrodinger equation is recovered in the non-relativistec, weak-field limit of the Klein-Gordon equation). The Dirac equation is recovered in this weak-field limit of the quantized gravitational field (the nonrelativistic limit of the relativistic, quantezed gravitational field when the vielbein plays the role of the spinor. The wave and field equations of O(3) electrodynamics are recovered when the vielbein becomes the relativistic dreibein (triad) eigenfunction whose three orthonormal space indices become identified with the three complex circular indices (1), (2), (3), and whose four spacetime indices are the indices of non-Euclidean spacetime (the base manifold). This dreibein is the potential dreibein of the O(3) electromagnetic field (an electromagnetic potential four-vector for each index (1), (2), (3)). The wave equation of the parity violating weak field is recovered when the orthonormal space indices of the relativistic dreibein eigenfunction are identified with the indices of the three massive weak field bosons. The wave equation of the strong field is recovered when the orthonormal space indices of the relativistic vielbein eigenfunction become the eight indices defined by the group generators of the SU (3) group.

A new numerical approach for the time-dependent wave and heat equations as well as for the time-independent Poisson equation developed in Part 1 is applied to the simulation of 1-D and 2-D test problems on regular and irregular domains. Trivial conforming and non-conforming Cartesian meshes with 3-point stencils in the 1-D case and 9-point stencils in the 2-D case are used in calculations. The numerical solutions of the 1-D wave equation as well as the 2-D wave and heat equations for a simple rectangular plate show that the accuracy of the new approach on non-conforming meshes is practically the same as that on conforming meshes for both the Dirichlet and Neumann boundary conditions. Moreover, very small distances ( $$0.1 h - 10^{-9}h$$ where h is the grid size) between the grid points of a Cartesian mesh and the boundary do not decrease the accuracy of the new technique. The application of the new approach to the 2-D problems on an irregular domain shows that the order of accuracy is close to four for the wave and heat equations and is close to five for the Poisson equation. This is in agreement with the theoretical results of Part 1 of the paper. The comparison of the numerical results obtained by the new approach and by FEM shows that at similar 9-point stencils, the accuracy of the new approach on irregular domains is two orders higher for the wave and heat equations and three orders higher for the Poisson equation than that for the linear finite elements. Moreover, the new approach yields even much more accurate results than the quadratic and cubic finite elements with much wider stencils. An example of a problem with a complex irregular domain that requires a prohibitively large computation time with the finite elements but can be easily solved with the new approach is presented.

This paper describes the use of the One-Dimensional Wave Equation for the confirmation of pile driving equipment and for field evaluation of pile axial load capacity for North Sea Forties Field Platforms FA and FC. The wave equation proved to be a powerful tool in the design and construction of the platforms. During the early design stages, the wave equation was used along with foundation exploration and evaluation techniques to select the pile size and thickness which could be driven to design penetration without recourse to drilling or jetting. Various hammers were evaluated by use of the wave equation before selecting the Menck 7000 for final pile driving with smaller hammers used for initial driving of each pile. The wave equation was also used as an aid to the design and selection of pile chasers, pile make-up, and driving cushions to maintain effective pile driving. During construction of Platform FA, strain gages were installed at the head of of a pile and recorded during driving, and the recordings were used to evaluate the driving efficiency of the hammers. Using the observed hammer efficiency, the pile driving record of blows per foot, and appropriate curves (determined in advance by wave equation analysis) relating soil resistance to blows per foot, it was possible for the soil resistance during driving to be estimated. By driving piles to almost final penetration, then delaying final driving several days, an evaluation of the soil "set- up " was obtained. A technique is presented which relates soil "set-up" to the ultimate capacity predicted by the wave equation. Soil "set-up" was included in the determination of the ultimate axial capacity of the pile. INTRODUCTION In 1950, Smith (1) developed a tractable solution to the wave equation which could be used to analyze complex pile driving problems. In 1969, Lowery, et al (2) developed a computer program which used the wave equation to predict soil resistance acting on a pilft during driving. More recently Coyle, et al (3, 4 and 5) has developed more precise techniques in using soil properties in the wave equation to predict pile axial load capacity considering soil "set-up". The Texas Highway Department has used this technique to evaluate pile load capacity on several coastal bridge projects during the last several years. This paper describes some of the techniques employed in using the wave equation in selecting pile driving hammers and other accessories for effective pile driving. In addition it describes how the wave equation was used to verify or evaluate the pile axial load capacity on Platforms FA and FC. SELECTION OF PILE' DRIVING HAMMERS AND OTHER ACCESSORIES Early in 1972 during the design of the Forties Field platforms it became apparent that to drive 5500 kip ultimate capacity piles, a large hammer and a stiff pile would be required. Since the project was to be conducted under a tight time schedule, it was desired to develop a pile make-up which could be quickly assembled and expeditiously driven to the required depth and capacity without the need to drill or jet.

The modified intermediate long wave (MILW) equation is a (1+1)-dimensional nonlinear singular integro-differential equation that possesses soliton solutions. In an appropriate limit the MILW equation reduces to the well-known modified Korteweg-de Vries equation. In this paper we solve the initial value problem for the MILW equation through a suitable implementation of the inverse scattering transform and use of the Miura-type transformation that maps solutions of the MILW equation into solutions of a complexified version of the standard intermediate long wave (ILW) equation. The initial value used for the MILW equation is assumed to be real valued, sufficiently smooth, and decaying to zero as the absolute value of the spatial variable approaches large values. An interesting feature of the procedure we develop is that soliton solutions for the ILW and MILW equations can be derived by appropriate specializations of a master set of equations.

The initial-boundary value problems for the wave equations with local and non-local linear boundaryconditions at the ends of a general segment are considered. To solve them, a generalize functions method has beendeveloped, which translates the original boundary value problems to solving the wave equation with a singular right-handside containing a singular simple and double layers, the densities of which are determined by the boundary and initial valuesof the desired function and its derivatives. Received integral representation of the solution in terms of boundary functions,which are a generalization of Green’s formula for solutions of the wave equation. To determine the unknown boundaryfunctions, it is built in space Fourier transforms in time, a two-leaf resolving system linear algebraic equations, whichconnects 4 boundary values solution and its derivatives. Together with two boundary conditions of local and non-localtype, a resolving system of equations is built for solving the stated initial-boundary value problems. On its basis, givenanalytical solutions for classical three boundary value problems with conditions Dirichlet, Neumann and mixed at the endsof the segment. The developed method allows solving boundary value problems with different local and nonlocal boundaryconditions and must find an application change in solving wave and other equations on graphs of different structures.

Exact solutions and invariants of motion for general types of regularized long wave equations

A fourth order cubic B-spline collocation method for the numerical study of the RLW and MRLW equations

Following the tradition of the nano-and picosecond optics, the basic theoretical studies continue to investigate the processes of propagation of femtosecond and attosecond laser pulses through the corresponding envelope equation for narrow-band laser pulses, working in paraxial approximation. We should point out here that this approximation is not valid for broad-band band pulses. In air due to the small dispersion the wave equation as well as the 3D + 1 amplitude equation describe more accurate the pulse dynamics. New exact localized solutions of the linear wave and amplitude equations are presented. The solutions discover non-paraxial semi-spherical diffraction of single-cycle and half-cycle laser pulses and a new class of spherically symmetric solutions of the wave equation.

The ordinary 3D wave equation for nondissipative, homogeneous, isotropic media admits solutions where the point sources are permitted to move, but as shown in this paper, it does not admit solutions where the receiver is allowed to move. To overcome this limitation, a new wave equation that permits both the receiver and the source to move is derived in this paper. This new wave equation is a generalization of the standard wave equation, and it reduces to the standard wave equation when the receiver is at rest. To derive this new wave equation, we first mathematically define a diverging spherical wave caused by a stationary point source. From this purely mathematical definition, the wave equation for a stationary source and a moving receiver is derived, together with a corresponding free-space Green function. Utilizing the derived Green function, it is shown that unlike the standard wave equation this new wave equation also permits solutions where both the receiver and the source are permitted to move. In conclusion, this paper demonstrates that, instead of an ordinary wave equation, the wave equation for a moving source and a moving receiver governs the waves emitted by moving point sources and received by moving receivers. This new wave equation has possible applications in acoustics, electrodynamics, and other physical sciences.

We discuss dissipative stochastic wave and diffusion equations resulting from an interaction of the inflaton with an environment in an external expanding homogeneous metric. We show that a diffusion equation well approximates the wave equation in a strong friction limit. We calculate the long wave power spectrum of the wave equation under the assumption that the perturbations are slowly varying in time and the expansion is almost exponential. Under the assumption that the noise has a form invariant under the coordinate transformations we obtain the power spectrum close to the scale invariant one. In the diffusion approximation we go beyond the slow variation assumption. We calculate the power spectrum exactly in models with exponential inflation and polynomial potentials and with power-law inflation and exponential potentials.

In these notes we analyze some problems related to the controllabilityand observability of partial differential equations and its space semidiscretizations.First we present the problems under consideration in the classicalexamples of the wave and heat equations and recall some well knownresults. Then we analyze the $1-d$ wave equation with rapidly oscillating coefficients,a classical problem in the theory of homogenization. Then we discussin detail the null and approximate controllability of the constant coefficientheat equation using Carleman inequalities. We also show how a fixed pointtechnique may be employed to obtain approximate controllability results forheat equations with globally Lipschitz nonlinearities. Finally we analyze thecontrollability of the space semi-discretizations of some classical PDE models:the Navier-Stokes equations and the $1-d$ wave and heat equations. We alsopresent some open problems.

The sech2 solitary wave solution of the regularized long wave equation is reobtained via the inverse scattering transform. The wave function of the eigenvalue problem of the relevant Schrödinger equation is proved reflectionless with sech2 potential of arbitrary amplitude. Moreover, the nonexistence of N-solitary wave solution (sech2 form and N≥2) is confirmed for the regularized long wave (RLW) equation and the method provides the possibility of solving some incompletely integrable equations via the inverse scattering transform.

Attenuation of compressional and shear waves in sediments often follows power laws with near linear variation with frequency. This cannot be modeled with viscous or relaxation wave equations, but more general temporal memory operators in the wave equation can describe such behavior. These operators can be justified in four ways: 1) Power laws for attenuation with exponents other than two correspond to the use of convolution operators with a kernel which is a power law in time. 2) The corresponding constitutive equation is also a convolution, often with a temporal power law function. 3) It is also equivalent to an infinite set of relaxation processes which can be formulated via the complex compressibility. 4) The constitutive equation can also be expressed as an infinite sum of higher order derivatives. We also analyze a grain-shearing model for propagation of waves in saturated, unconsolidated granular materials. It is expressed via a spring damper model with time-varying damping. It turns out that it results in a fractional Kelvin-Voigt wave equation and a fractional diffusion equation for the compressional and shear waves respectively, giving a new perspective for understanding and interpreting this model.