Harmonisation of Classical Wave Equation

In this short note we present a technique using which one attributes frequency and wavevector to (almost) arbitrary scalar fields. Our proposed definition is then applied to the classical wave equation to yield a novel nonlinear PDE.

A simple way of unifying the formulas for the Coulomb’s law and Newton’s law of the universal gravitation: An approach based on membranes

In this work, a new approach is presented with the aim of showing a simple way of unifying the classical formulas for the forces of the Coulomb’s law of electrostatic interaction <mml:math display="inline"> <mml:mrow> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>C</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> and the Newton’s law of universal gravitation <mml:math display="inline"> <mml:mrow> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>G</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . In this approach, these two forces are of the same nature and are ascribed to the interaction between two membranes that oscillate according to different curvature functions with spatial period <mml:math display="inline"> <mml:mrow> <mml:mfrac> <mml:mrow> <mml:mi>ξ</mml:mi> <mml:mi>π</mml:mi> </mml:mrow> <mml:mi>k</mml:mi> </mml:mfrac> </mml:mrow> </mml:math> , where <mml:math display="inline"> <mml:mi>ξ</mml:mi> </mml:math> is a dimensionless parameter and <mml:math display="inline"> <mml:mi>k</mml:mi> </mml:math> is a wave number. Both curvature functions are solutions of the classical wave equation with wavelength given by the de Broglie relation. This new formula still keeps itself as the inverse square law, and it is like <mml:math display="inline"> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>C</mml:mi> </mml:msub> </mml:mrow> </mml:math> when the dimensionless parameter <mml:math display="inline"> <mml:mrow> <mml:mi>ξ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>274</mml:mn> </mml:mrow> </mml:math> and like <mml:math display="inline"> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>G</mml:mi> </mml:msub> </mml:mrow> </mml:math> when <mml:math display="inline"> <mml:mrow> <mml:mi>ξ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1.14198</mml:mn> <mml:mo>×</mml:mo> <mml:msup> <mml:mrow> <mml:mn>10</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>45</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . It was found that the values of the parameter <mml:math display="inline"> <mml:mi>ξ</mml:mi> </mml:math> quantize the formula from which <mml:math display="inline"> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>C</mml:mi> </mml:msub> </mml:mrow> </mml:math> and <mml:math display="inline"> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>G</mml:mi> </mml:msub> </mml:mrow> </mml:math> are obtained as particular cases.

Hereditariness and non-locality in wave propagation modeling

The classical wave equation is generalized within the framework of fractional calculus in order to account for the memory and non-local effects that might be material features. Both effects are included in the constitutive equation, while the equation of motion of the deformable body and strain are left unchanged. Memory effects in viscoelastic materials are modeled through the distributed-order fractional constitutive equation that generalizes all linear models having differentiation orders up to order one. The microlocal approach in analyzing singularity propagation is utilized in the case of viscoelastic material described by the fractional Zener model, as well as in the case of two non-local models: non-local Hookean and fractional Eringen.

Slow-roll versus stochastic slow-roll inflation

We consider the classical wave equation with a thermal and Starobinsky–Vilenkin noise which in the slow-roll and long wave approximation describes the quantum fluctuations of the gravity-inflaton system in an expanding metric. We investigate the resulting consistent stochastic Einstein-Klein-Gordon system in the slow-roll regime. We show in some models that the slow-roll requirements (of the negligence of $$\partial _{t}^{2}\phi $$∂t2ϕ) can be satisfied in the probabilistic sense for the stochastic system with quantum and thermal noise for arbitrarily large time and an infinite range of fields. We calculate expectation values of some inflationary variables taking into account quantum and thermal noise. We show that the mean acceleration $$\langle \partial _{t}^{2}a\rangle $$⟨∂t2a⟩ can be negative or positive (depending on the model) when the random fields take values beyond the classical range of inflation.

Wave-equation traveltime inversion with multifrequency bands: Synthetic and land data examples

We have developed a wave-equation traveltime inversion method with multifrequency bands to invert for the shallow or intermediate subsurface velocity distribution. Similar to the classical wave-equation traveltime inversion, this method searches for the velocity model that minimizes the squared sum of the traveltime residuals using source wavelets with progressively higher peak frequencies. Wave-equation traveltime inversion can partially avoid the cycle-skipping problem by recovering the low-wavenumber parts of the velocity model. However, we also use the frequency information hidden in the traveltimes to obtain a more highly resolved tomogram. Therefore, we use different frequency bands when calculating the Fréchet derivatives so that tomograms with better resolution can be reconstructed. Results are validated by the zero-offset gathers from the raw data associated with moderate geometric irregularities. The improved wave-equation traveltime method is robust and merely needs a rough estimate of the starting model. Numerical tests on the synthetic and field data sets validate the above claims.

Complex fractional Zener model of wave propagation in ℝ

The classical wave equation is generalized within fractional framework, by using fractional derivatives of real and complex order in the constitutive equation, so that it describes wave propagation in one dimensional infinite viscoelastic rod. We analyze existence, uniqueness and properties of solutions to the corresponding initial-boundary value problem for generalized wave equation. Also, we provide a comparative analysis with the case of the same equation but considered on a bounded or half-bounded spatial domain. We conclude our investigation with a numerical example that illustrates obtained results.

The Classical Connection

This chapter discusses the relationship between the elegance of the classical Lagrangian and Hamiltonian formulations of mechanics and optics. In physics, action is a mathematical functional which takes the trajectory, or path, of the system as its argument and has a real number as its result. Classical mechanics postulates that the path actually followed by a physical system is that for which the action is minimized, or, more generally, is stationary. The action is defined by an integral, and the classical equations of motion of a system can be derived by minimizing the value of that integral. The chapter first provides an overview of Lagrangians, action, and Hamiltonians in order to draw out an alternative approach to finding equations of motion. It then considers the classical wave equation and classical scattering and concludes with an analysis of the classical inverse scattering problem.

An Outline of Cosmology Based on Interpretation of the Johannine Prologue

As we know there are two main paradigms concerning the origin of the Universe: the first is Big-Bang Theory, and the other is Creation paradigm. But those two main paradigms each have their problems, for instance Big Bang Theory assumes that the first explosion was triggered by chance alone, therefore it says that everything emerged out of vacuum fluctuation caused by pure statistical chance. By doing so, its proponents want to avoid the role of the Prime Mover (God). Of course there are also other propositions such as the Steady State theory or Cyclical universe, but they do not form the majority of people in the world. On the other side, Creation Theory says that the Universe was created by God in 6x24 hours according to Genesis chapter 1, although a variation of this theory says that it is possible that God created the Universe in longer period of thousands of years or even billions of years. But such a proposition seems not supported by Biblical texts. To overcome the weaknesses of those main paradigms, I will outline here another choice, namely that the Universe was created by Logos (Christ in His pre-existence). This is in accordance with the Prolegomena of the Gospel of John, which says that the Logos was there in the beginning (John 1:1). I describe 3 applications of the classical wave equation according to Shpenkov, i.e. hydrogen energy states, periodic table of elements, and planetary orbit distances. For sure, Shpenkov derived many more results beside these 3 phenomena as discussed in his 3 volume books and many papers, but these 3 phenomena are selected to give clear examples of what can be done with the classical wave equation. And then I extend further the classical wave equation to fractal vibrating string. While of course this outline is not complete, this article is written to stimulate further investigations in this direction.