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We investigate the propagation of infinitesimal harmonic mechanical waves emitted from a boundary with variable velocity and arriving at a stationary observer. In the classical Doppler effect, Xs(t)=vt is the location of the source with constant velocity v. In the present work, however, we consider a source co-located with a moving boundary x=Xs(t), where Xs(t) can have an arbitrary functional form. For ‘slowly moving’ boundaries (i.e., ones for which the timescale set by the mechanical motion is large in comparison to the inverse of the frequency of the emitted wave), we present a multiple-scale asymptotic analysis of the moving boundary problem for the linear wave equation. We obtain a closed-form leading-order (with respect to the latter small parameter) solution and show that the variable velocity of the boundary results not only in frequency modulation but also in amplitude modulation of the received signal. Consequently, our results extend the applicability of two basic tenets of the theory of a moving source on a stationary domain, specifically that (i) Xs for non-uniform boundary motion can be inserted in place of the constant velocity v in the classical Doppler formula and (ii) that the non-uniform boundary motion introduces variability in the amplitude of the wave. The specific examples of decelerating and oscillatory boundary motion are worked out and illustrated. Copyright © 2017 John Wiley & Sons, Ltd.

We apply the Christov-Galerkin spectral method for the numerical investigation of the interaction of solitons in the Cubic Nonlinear Schrodinger Equation. The issues of convergence are addressed and an algorithm is devised for the application of the method. Results are obtained for the interaction of solitons with different phase velocities and different carrier frequencies. The interactions are shown to be elastic, save for the phase shifts. The latter are extracted from the numerical solution and discussed.

This article presents errors, corrections, and additions to the research outlined in the following citation: Christov, I., Christov, C. I., & Jordan, P. M. (2007). Modeling weakly nonlinear acoustic wave propagation. The Quarterly Journal of Mechanics and Applied Mathematics, 60(4), 473-495.

We show that the linearized equations of the incompressible elastic medium admit a `Maxwell form' in which the shear component of the stress vector plays the role of the electric field, and the vorticity plays the role of the magnetic field. Conversely, the set of dynamic Maxwell equations are strict mathematical corollaries from the governing equations of the incompressible elastic medium. This suggests that the nature of `electromagnetic field' may actually be related to an elastic continuous medium. The analogy is complete if the medium is assumed to behave as fluid in shear motions, while it may still behave as elastic solid under compressional motions. Then the governing equations of the elastic fluid are re-derived in the Eulerian frame by replacing the partial time derivatives by the properly invariant (frame indifferent) time rates. The `Maxwell from' of the frame indifferent formulation gives the frame indifferent system that is to replace the Maxwell system. This new system comprises terms already present in the classical Maxwell equations, alongside terms that are the progenitors of the Biot--Savart, Oersted--Ampere's, and Lorentz--force laws. Thus a frame indifferent (truly covariant) formulation of electromagnetism is achieved from a single postulate that the electromagnetic field is a kind of elastic (partly liquid partly solid) continuum., accepted

Summary Three weakly nonlinear models of lossless, compressible fluid flow—a straightforward weakly nonlinear equation (WNE), the inviscid Kuznetsov equation (IKE) and the Lighthill–Westervelt equation (LWE)—are derived from first principles and their relationship to each other is established. Through a numerical study of the blow-up of acceleration waves, the weakly nonlinear equations are compared to the ‘exact’ Euler equations, and the ranges of applicability of the approximate models are assessed. By reformulating these equations as hyperbolic systems of conservation laws, we are able to employ a Godunov-type finite-difference scheme to obtain numerical solutions of the approximate models for times beyond the instant of blow-up (that is, shock formation), for both density and velocity boundary conditions. Our study reveals that the straightforward WNE gives the best results, followed by the IKE, with the LWE’s performance being the poorest overall.

Boussinesq's original derivation of his celebrated equation for surface waves on a fluid layer opened up new horizons that were to yield the concept of the soliton. The present contribution concerns the set of Boussinesq-like equations under the general title of ‘Boussinesq's paradigm’. These are true bi-directional wave equations occurring in many physical instances and sharing analogous properties. The emphasis is placed: (i) on generalized Boussinesq systems that involve higher-order linear dispersion through either additional space derivatives or additional wave operators (so-called double-dispersion equations); and (ii) on the ‘mechanics’ of the most representative localized nonlinear wave solutions. Dissipative cases and two-dimensional generalizations are also considered. To cite this article: C.I. Christov et al., C. R. Mecanique 335 (2007).

We present a preliminary examination of a new approach to a long-standing problem in non-Newtonian fluid mechanics. First, we summarize how a general implicit functional relation between stress and rate of strain of a continuum with memory is reduced to the well-known linear differential constitutive relations that account for "relaxation" and "retardation." Then, we show that relaxation and retardation are asymptotically equivalent for small Deborah numbers, whence causal pure relaxation models necessarily correspond to ill-posed pure retardation models. We suggest that this dichotomy could be a possible way to reconcile the discrepancy between the theory of and certain experiments on viscoelastic liquids that are conjectured to exhibit only stress retardation., Comment: 5 pages, 1 figure, elsarticle format, v2 corrects some typograpgical errors, published in Mechanics Research Communications

A fictitious time is introduced into the unsteady equation of the stream function rendering it into a higher-order ultra-parabolic equation. The convergence with respect to the fictitious time (we call the latter 'internal iterations') allows one to obtain fully implicit nonlinear scheme in full time steps for the physical-time variable. For particular choice of the artificial time increment, the scheme in full time steps is of second-order of approximation. For the solution of the internal iteration, a fractional-step scheme is proposed based on the splitting of the combination of the Laplace, bi-harmonic and advection operators. A judicious choice for the time staggering of the different parts of the nonlinear advective terms allows us to prove that the internal iterations are unconditionally stable and convergent. We assess the number of operations needed per time step and show computational effectiveness of the proposed scheme. We prove that when the internal iterations converge, the scheme is second-order in physical time and space, nonlinear, implicit and absolutely stable. The performance of the scheme is demonstrated for the flow created by oscillatory motion of the lid of a square cavity. All theoretical findings are demonstrated practically.

A special numerical technique has been developed for identification of solitary wave solutions of Boussinesq and Korteweg–de Vries equations. Stationary localized waves are considered in the frame moving to the right. The original ill-posed problem is transferred into a problem of the unknown coefficient from over-posed boundary data in which the trivial solution is excluded. The Method of Variational Imbedding is used for solving the inverse problem. The generalized sixth-order Boussinesq equation is considered for illustrations.

Generalized Wave Equations containing dispersion, dissipation and energy-production (GDWE) are considered in lieu of dissipative NEE as more suitable models for two-way interaction of localized waves. The quasi-particle behavior and the long-time evolution of localized solutions upon take-over and head-on collisions are investigated numerically by means of an adequate difference scheme which represents faithfully the balance/conservation laws. It is shown that in most cases the balance between energy production/dissipation and nonlinearity plays a similar role to the classical Boussinesq balance between dispersion and nonlinearity, namely it can create and support localized solutions which behave as quasi-particles upon collisions and for a reasonably long time after that.

In this paper, we consider a nonlinear generalized diffusion equation called the Swift-Hohenberg equation (SH) for which a Lyapunov functional is known. We develop a computationally efficient second-order in time implicit difference scheme based on the operator-splitting method. Internal iterations are used to make the scheme both nonlinear and implicit. We prove that the scheme allows strict (independent of the truncation error) implementation of a discrete approximation of the Lyapunov functional. The new scheme is used to investigate the pattern formation from random initial conditions, and spatially chaotic states are found.

The inverse parabolic problem of coefficient identification from over-posed data is embedded into a fourth order in space and second order in time elliptic boundary value problem. The latter is well posed for redundant data at boundaries. The equivalence of the two problems is demonstrated. A difference scheme of splitting-type is employed, and featuring examples are elaborated numerically.

A coordinate-splitting economic difference scheme is proposed for generalized parabolic equations (GPE) containing fourth-order diffusion operators and the algorithm for its implementation is developed. The performance of the scheme is demonstrated for different cases, e.g. for treating bifurcation phenomena. The technique is applied to the numerical solution of Swift-Hohenberg equation describing the Rayleigh-Benard convection and results are obtained for very large system sizes and for very long times on small computational platform.

We consider the long-time evolution of the solution of an energy-consistent system with dispersion and nonlinearity, which is the progenitor of the different Boussinesq equations. Unlike the classical Boussinesq models, the energy-consistent one possesses Galilean invariance. As initial condition we use the superposition of two analytical one-soliton solutions. We use a strongly dynamical implicit difference scheme with internal iterations, which allows us to follow the evolution of the solution at very long times. We focus on the behavior of traveling localized solutions developing from critical initial data. The main solitary waves appear virtually undeformed from the interaction, but additional oscillations are excited at the trailing edge of each one of them. We track their evolution for very long times when they tend to adopt a self-similar shape. We test a hypothesis about the dependence on time of the amplitude and the support of Airy-function shaped coherent structures. The investigation elucidates the mechanism of evolution of interacting solitary waves in the energy-consistent Boussinesq equation.

Pattern formation in a finite layer of fluid induced either by buoyancy or by a surface-tension gradient is considered. The fluid is confined between poor conducting horizontal boundaries, leading to patterns with a characteristic horizontal scale much larger than the fluid depth. The evolution of the system is studied by numerical integration of the (1+2)D equation introduced by Knobloch [1990]: [Formula: see text] Here µ is the scaled bifurcation parameter, κ=1, and a represents the effects of a heat transfer finite Biot number. The coefficients β, δ and γ do not vanish when the boundary conditions at top and bottom are not identical (β≠0, δ≠0) or when non-Boussinesq effects are taken into account (γ≠0). When the conductive state becomes unstable due to surface-tension inhomogeneities, it is shown that the system evolves towards a stationary pattern of hexagons with up or down flow depending on the relative value of the coefficients β and δ. In the case of buoyancy-driven convection (β=δ≠0), the system displays a tesselation of steady squares. Knobloch’s equation also describes time-dependent patterns at high thermal gradients, including spatio-temporal chaos, due to the non-variational character of the equation.

A new class of solutions of three-dimensional equations from the Boussinesq paradigm are considered. The corresponding profiles are not localized functions in the sense of the integrability of the square over an infinite domain. For the new type of solutions, the gradient and the Hessian/Laplacian are square integrable. In the linear limiting case, analytical expressions for the profiles of the pseudolocalized solutions are found. The nonlinear case is treated numerically with a special approximation of the differential operators with spherical symmetry that allows for automatic acknowledgement of the behavioral conditions at the origin of the coordinate system. The asymptotic boundary conditions stem from the $1/r$ behavior at infinity of the pseudolocalized profile. A special approximation is devised that allows us to obtain the proper behavior for much smaller computational box. The pseudolocalized solutions are obtained for both quadratic and cubic nonlinearity., Comment: 8 pages, 3 figures

A higher-order dispersive equation is introduced as a candidate for the governing equation of a field theory. A new class of solutions of the three-dimensional field equation are considered, which are not localized functions in the sense of the integrability of the square of the profile over an infinite domain. For this new class of solutions, the gradient and/or the Hessian/Laplacian are square integrable. In the linear limiting case, an analytical expression for the pseudolocalized solution is found and the method of variational approximation is applied to find the dynamics of the centers of the quasi-particles (QPs) corresponding to these solutions. A discrete Lagrangian can be derived due to the localization of the gradient and the Laplacian of the profile. The equations of motion of the QPs are derived from the discrete Lagrangian. The pseudomass ("wave mass") of a QP is defined as well as the potential of interaction. The most important trait of the new QPs is that at large distances, the force of attraction is proportional to the inverse square of the distance between the QPs. This can be considered analogous to the gravitational force in classical mechanics., Comment: 19 pages, 10 figures, elsarticle format; v2 includes revision in response to referees; accepted for publication in Wave Motion for the special issue on Mathematical Modeling and Physical Dynamics of Solitary Waves: From Continuum Mechanics to Field Theory

ACL-12-43; International audience; Modelling the biomechanics of growing trees is a non-classical problem, as the usual framework of structural mechanics does not take into account the evolution of the domain geometry due to growth processes. Incremental approaches have been used in rod theory to bypass this problem and to model the addition of new material points on an existing deformed structure. However, these approaches are based on the explicit time numerical algorithm of an unknown continuous model, and thus, the accuracy of the numerical results obtained cannot be analysed. A new continuous space-time formulation has been recently proposed to model the biomechanical response of growing rods. The aim of this paper is to discretise the corresponding non-linear system of partial differential equations and the linearised system in order to compare the numerical results with analytical solutions of the linearised problem. The finite element method is implemented to compute the space boundary problem and different time integration schemes are considered to solve the associated initial value problem with a special attention to the forward Euler method which is the analogue of the previously used incremental approach. The numerical results point out that the accuracy of the time integration schemes strongly depends on the value of the parameters. The forward Euler method may present slow convergence property and errors with significant orders of magnitude. Nevertheless, attention must be paid to implicit methods since, for specific values of the parameters and large time steps, they may lead to spurious solutions that may come from numerical instabilities. Hence, the second order Heun's method is an interesting alternative even if it is more time consuming.

Two improved versions of Boussinesq equation (Boussinesq paradigm) have been considered which are well-posed (correct in the sense of Hadamard) as an initial value problem: the Proper Boussinesq Equation (PBE) and the Regularized Long Wave Equation (RLWE). Fully implicit difference schemes have been developed strictly representing, on difference level, the conservation or balance laws for the mass, pseudoenergy or pseudomomentum of the wave system. Thresholds of possible nonlinear blow-up are identified for both PBE and RLWE. The head-on collisions of solitary waves of the sech type (Boussinesq solitons) have been investigated. They are subsonic and negative (surface depressions) for PBE and supersonic and positive (surface elevations) for RLWE. The numerically recovered sign and sizes of the phase shifts are in very good quantitative agreement with analytical results for the two-soliton solution of PBE. The subsonic surface elevations are found to be not permanent but to gradually transform into oscillatory pulses whose support increases and amplitude decreases with time although the total pseudoenergy is conserved within 10−10. The latter allows us to claim that the pulses are solitons despite their “aging” (which is felt on times several times the time-scale of collision). For supersonic phase speeds, the collision of Boussinesq solitons has inelastic character exhibiting not only a significant phase shift but also a residual signal of sizable amplitude but negligible pseudoenergy. The evolution of the residual signal is investigated numerically for very long times.

In the present study we demonstrate the construction of a robust multilevel preconditioner for biquadratic FE elliptic problems. In the general setting of an arbitrary elliptic operator it is well known that the standard hierarchical basis two‐level splittings for higher order FEM elliptic systems deteriorate with increasing the anisotropy ratio. An alternative approach resulting in a robust hierarchical two‐level splitting of the finite element space of continuos piecewise biquadratic functions involves the semi‐coarsening mesh procedure. This evokes us to analyze the behavior of the constant in the strengthened CBS inequality, which is a quality measure for hierarchical two‐level splittings of the FEM stiffness matrices, for the particular case of balanced semi‐coarsening mesh refinement. We present new theoretical estimates which further we support by numerically computed CBS constants over a rich set of parameters (coarsening factor and anisotropy ratio). An optimal order multilevel algorithm is constructed on the basis of the proven uniform estimates and the theory of the Algebraic MultiLevel Iteration (AMLI) methods. Its total computational cost is proportional to the size of the discrete problem with a proportionality constant independent of the anisotropy ratio

In our work, we consider the time dependent Stokes equation on a finite time interval and on a uniform rectangular mesh, written in terms of velocity and pressure. For this problem, a parallel algorithm based on a novel direction splitting approach is developed. Here, the pressure equation is derived from a perturbed form of the continuity equation, in which the incompressibility constraint is penalized in a negative norm induced by the direction splitting. The scheme used in the algorithm is composed of two parts: (i) velocity prediction, and (ii) pressure correction. This is a Crank‐Nicolson‐type two‐stage time integration scheme for two and three dimensional parabolic problems in which the second‐order derivative, with respect to each space variable, is treated implicitly while the other variable is made explicit at each time sub‐step. In order to achieve a good parallel performance the solution of the Poison problem for the pressure correction is replaced by solving a sequence of one‐dimensional secon...

This work consists on the study of an optimal control problem relating to the water pollution. We analyze various questions: existence, uniqueness, control and the regularized formulation of the initial pointwise control problem. We propose also an implementation of an hybrid numerical scheme associated with an algorithm of descent.

This is a further note on the (Guass‐Maxwell) force‐flux construct proposed previously (Goddard, J. D., A note on Eringen's moment balances, Int. J. Eng. Sci., in the press, 2011). Motivated in part by its promise as a homogenization technique for constructing micromorphic continua, the present work is focused rather on some additional representations and on novel applications, such as the derivation of dissipative thermodynamic potentials from force‐flux relations.

Provided here is numerical evidence of localized solutions, solitary waves, in a model equation for Benard convection driven by interfacial stresses (Marangoni effect).

In a 2001 article, Fetecau and Zierep [1] considered Stokes' first problem for a second grade (SG) fluid, unaware that it had already been solved (correctly) by Puri [2] in 1984. These authors used the Fourier sine transform to obtain, what they believed to be, the exact solution of the resulting initial-boundary-value problem (IBVP). Unfortunately, due to their incorrect computation of the distributional derivative of the Heaviside function, an omission of a critical term in the subsidiary equation occurred, an elementary mistake (gracefully explained in [3]) apparently not uncommon in the literature. Therefore, the solution given in [1] is, generally speaking, incorrect., 3 pages, 1 figure (2 images); v2 corrects a few typos. The final publication is available at http://www.springerlink.com

Using solitonic techniques we construct a new exact stationary axisymmetric solution to the 5D Einstein equations in vacuum in asymptotically Kaluza‐Klein spacetime. The solution describes rotating black ring with a single angular momentum surrounded by two Kaluza‐Klein bubbles. It is generated by applying a 2‐soliton Backlund transformation on a static seed. The solution properties are discussed, and its physical characteristics, such as mass, tension, angular velocity and angular momentum, are derived.

We study the class of pulsating strings in AdS_5 x Y^{p,q} and AdS_5 x L^{p,q,r}. Using a generalized ansatz for pulsating string configurations, we find new solutions for this class in terms of Heun functions, and derive the particular case of AdS_5 x T^{1,1}, which was analyzed in arXiv:1006.1539 [hep-th]. Unfortunately, Heun functions are still little studied, and we are not able to quantize the theory quasi-classically and obtain the first corrections to the energy. The latter, due to AdS/CFT correspondence, is supposed to give the anomalous dimensions of operators of the gauge theory dual N=1 superconformal field theory., Comment: 9 pages, talk given at the 2nd Int. Conference AMiTaNS, 21-26 June 2010, Sozopol, Bulgaria, organized by EAC (Euro-American Consortium) for Promoting AMiTaNS, to appear in the Proceedings of 2nd Int. Conference AMiTaNS

Using the dressing Zakharov‐Shabat method we re‐derive the effects of the two‐soliton interactions for the MNLS equations related to the BD.I‐type symmetric spaces. Next we generalize this analysis for the Heisenberg ferromagnet type equations, gauge equivalent to MNLS.

In this article the problem of an interface fracture between two isotropic linear elastic materials is studied within a continuum modeling framework, which incorporates important nanoscale effects. The proposed model of bi‐material crack ascribes curvature‐dependent surface tension to both the fracture surfaces and the solid‐solid interface. Further, it uses as boundary conditions the jump momentum balance equations, which take into account the excess properties ascribed to the material interfaces. Ultimately, the model leads to a 4×4 system of Cauchy‐singular integro‐differential equations which is equivalent to a system of Fredholm integral equations. It is demonstrated herein, using the Method of Integral Transforms and the Theory of Fredholm Integral equations, that this model leads to bounded stresses and strains, in contrast to the square‐root singularities of the stress and strain fields predicted by the classical theory of brittle fracture.

The system of Coupled Nonlinear Schrodinger's Equations (CNLSE) is solved numerically by means of a conservative difference scheme. A new kind of repelling collision is discovered for negative values of the cross-modulation coupling parameter, $\alpha_2$. The results show that as the latter becomes increasingly negative, the behavior of the solitons during interaction change drastically. While for $\alpha_2 >0$, the solitons pass through each other, a negative threshold value $\alpha^*_2 < 0$ is found below which the solitons repell each other. This is a novel result for this kind of models and the conservation of momentum for the system of quasi-particles (QPs) is thoroughly investigated.

We consider a class of multicomponent nonlinear Schrodinger equations (MNLS) related to the symmetric BD.I-type symmetric spaces. As important particular case of these MNLS we obtain the Kulish-Sklyanin model. Some new reductions and their effects on the soliton solutions are obtained by proper modifying the Zakahrov-Shabat dressing method., AIP AMiTaNS'09 Proceedings

We study the dynamics of the sine-Gordon equation's kink soliton solutions under the coarse-grain description via two "collective variables": the position of the "center" of a soliton and its characteristic width ("size"). Integral expressions for the interaction potential and the quasi-particles' cross-masses are derived. However, these cannot be evaluated in closed form when the solitons have varying widths, so we develop a perturbation approach with the velocity of the faster soliton as the small parameter. This enables us to derive a system of four coupled second-order ODEs, one for each collective variable. The resulting initial-value problem is very stiff and numerical instabilities make it difficult to solve accurately, so a semi-empirical iterative approach to its solution is proposed. Then, we demonstrate that, even though it appears the solitons pass through each other, the quasi-particles actually "exchange" their pseudomasses during a collision.

By treating the centers of solitons as point particles and studying their discrete dynamics, we demonstrate a new approach to the quantization of the soliton solutions of the sine-Gordon equation, one of the first model nonlinear field equations. In particular, we show that a linear superposition of the non-interacting shapes of two solitons offers a qualitative (and to a good approximation quantitative) description of the true two-soliton solution, provided that the trajectories of the centers of the superimposed solitons are considered unknown. Via variational calculus, we establish that the dynamics of the quasi-particles obey a pseudo-Newtonian law, which includes cross-mass terms. The successful identification of the governing equations of the (discrete) quasi-particles from the (continuous) field equation shows that the proposed approach provides a basis for the passage from the continuous to a discrete description of the field., 10 pages, 3 figures (6 images); v2: revised and improved the presentation, updated the references, fixed typos; v3: corrected a few minor mistakes and typos, version accepted for publication in Phys. Lett. A

Arguments and numerical evidence are provided for the appearance of solitary waves driven by surface tension tractions (Marangoni effect)

We consider the inverse problem for identification of the coefficient in an elliptic partial differential equation inside the unit square D, with overposed boundary data. This problem is not investigated enough in the literature due to the lack of results about the uniqueness of the inverse problem. Following the main idea of the so-called Method of Variational Imbedding, we imbed the solution of the inverse problem into the elliptic Boundary Value Problem (BVP) stemming from the necessary conditions for minimisation of the quadratic functional of the original equation. The system contains a well-posed fourth-order BVP for the sought function and an explicit equation for the unknown coefficient. We solve imbedding BVP numerically by making use of operator-splitting for the fourth-order BVP. The convergence and stability of the numerical method are established. We introduce an effective way for finding the best approximation for the coefficient. Featuring examples are elaborated numerically.

The random field created by a system of random points which are centers of perfectly disordered equi-sized spheres of a finite radius is introduced and named the PDS-field. The notion of Perfect Disorder is defined statistically correctly taking the cumulants instead of moments to be $\delta $-functions. The Volterra-Wiener functional expansion with the PDS-field as a basis function is considered and a system of orthogonal Wiener functionals is explicitly constructed. The expansion is employed to the problem of specifying the overall conductivity for a random medium containing an array of perfectly disordered spherical inclusions. An infinite hierarchy of coupled equations for the kernels of the Wiener functionals is derived. It is shown that the Wiener expansion with respect to the PDS-field is also a virial one, i.e. the nth order term contributes quantities of order $c^n $, where c is the volume fraction of the inclusions. This allows the full stochastic solution to the problem of heat conduction throu...

We propose an approach to identifying the solutions of the steady incompressible Navier–Stokes equations for large Reynolds numbers. These cannot be obtained as initial-value problems for the unsteady system because of the instability of the latter. Our approach consists of replacing the original steady-state problem for the Navier–Stokes equations by a boundary-value problem for the Euler–Lagrange equations for minimization of the quadratic functional of the original equations. This technique is called Method of Variational Imbedding (MVI) and in this case it leads to a system of higher-order partial differential equations, which is solved by means of an operator-splitting method. As a featuring example we consider the classical flow around a circular cylinder which is known to lose stability as early as for Re=40. We find a stationary solution with recirculation zone for Reynolds numbers as large as Re=200. Thus, new information about the possible hybrid flow regimes is obtained.

We examine the process of the emission of light from an atom that is in a relative translational motion with respect to the medium at rest in which the electromagnetic excitations propagate. The effect of Lorentz contraction of the of electron orbits on the emitted frequency is incorporated in the Rydberg formula, as well as the emitter’s Doppler effect is acknowledged. The result is that the frequency of the emitted light is modified by a factor that is identical with what is called the ‘relativistic Doppler effect’. The new emission formula is applied for reinterpretation of the Ives-Stilwell experiment and shown that within the second order of approximation with respect to the speeds of the atom and the ‘absolute speed’ (Earth’s speed relative to the medium), the absolute motion does not affect the interference. The expression for the modification of the frequency involves both a first and a second-order term with respect to the speed of the atoms in the cathode tube. The latter turns out to be quantitatively the same as if the time would have changed its rate in the frame moving with the atoms. Thus, a new interpretation of the results of this famous experiment is provided without stipulating time dilation.

This work is concerned with the conservation properties of a new vectorial operator splitting scheme for solving the incompressible Navier–Stokes equations. It is proven that the difference approximation of the advection operator conserves square of velocity components and the kinetic energy as the differential operator does, while pressure term conserves only the kinetic energy. Some analytical requirements necessary to be satisfied of difference schemes for incompressible Navier–Stokes equations are formulated and discussed. The properties of the methods are illustrated with results from numerical computations for lid-driven cavity flow.