Elastic wave dispersion equation considering material and geometric nonlinearities

In this letter, we describe the propagation of longitudinal waves in one-dimensional nonlinear elastic thin rods with material nonlinearity and geometric nonlinearity. Mathematical analysis is used to derive the analytical dispersion relationship of longitudinal waves; subsequently, the numerical Fourier spectrum method is used to solve the nonlinear wave equation directly and the results are used to verify the correctness of the derived nonlinear dispersion relation.

Thermodynamical Extension of a Symplectic Numerical Scheme with Half Space and Time Shifts Demonstrated on Rheological Waves in Solids

On the example of the Poynting–Thomson–Zener rheological model for solids, which exhibits both dissipation and wave propagation, with nonlinear dispersion relation, we introduce and investigate a finite difference numerical scheme. Our goal is to demonstrate its properties and to ease the computations in later applications for continuum thermodynamical problems. The key element is the positioning of the discretized quantities with shifts by half space and time steps with respect to each other. The arrangement is chosen according to the spacetime properties of the quantities and of the equations governing them. Numerical stability, dissipative error, and dispersive error are analyzed in detail. With the best settings found, the scheme is capable of making precise and fast predictions. Finally, the proposed scheme is compared to a commercial finite element software, COMSOL, which demonstrates essential differences even on the simplest—elastic—level of modeling.

FEATURES OF NEAR-RESONANT TRIAD INTERACTIONS IN WAVES ON INTERMEDIATE DEPTH

The article is dedicated to studying of nonlinear triad interactions features. Full resonance in such interactions occurs either for triads of frequencies or wavenumbers. For the other parameter mismatch will occur. There is no common opinion whether there is wavenumber or frequency mismatch in waves propagating on intermediate depth. In this work it is shown that mismatch occurs for wavenumbers. Possibility of prediction of spatial periodicity of energy exchange between wave harmonics was also studied. It was found that beat length can be predicted with linear dispersion relation for the relative depth exceeding 0.05. For depths less than 0.05 beat length can be assessed with nonlinear dispersion relation which accounts for wave steepness.

Nonlinear dispersion for ocean surface waves

Two expressions for the nonlinear dispersion relation for gravity waves on water of constant depth are derived, one for wave fields with discrete amplitude spectra, the other for wave fields with continuous wavenumber energy spectra. Numerical examples for wave quartets and for two-dimensional Pierson–Moskowitz spectra are given, and an important possible application is discussed.

Nonlinear Bloch waves and balance between hardening and softening dispersion

The introduction of nonlinearity alters the dispersion of elastic waves in solid media. In this paper, we present an analytical formulation for the treatment of finite-strain Bloch waves in one-dimensional phononic crystals consisting of layers with alternating material properties. Considering longitudinal waves and ignoring lateral effects, the exact nonlinear dispersion relation in each homogeneous layer is first obtained and subsequently used within the transfer matrix method to derive an approximate nonlinear dispersion relation for the overall periodic medium. The result is an amplitude-dependent elastic band structure that upon verification by numerical simulations is accurate for up to an amplitude-to-unit-cell length ratio of one-eighth. The derived dispersion relation allows us to interpret the formation of spatial invariance in the wave profile as a balance between hardening and softening effects in the dispersion that emerge due to the nonlinearity and the periodicity, respectively. For example, for a wave amplitude of the order of one-eighth of the unit-cell size in a demonstrative structure, the two effects are practically in balance for wavelengths as small as roughly three times the unit-cell size.

Holographic Fermions in Anisotropic Einstein-Maxwell-Dilaton-Axion Theory

We investigate the properties of the holographic Fermionic system dual to an anisotropic charged black brane bulk in Einstein-Maxwell-Dilaton-Axion gravity theory. We consider the minimal coupling between the Dirac field and the gauge field in the bulk gravity theory and mainly explore the dispersion relation exponents of the Green functions of the dual Fermionic operators in the dual field theory. We find that along both the anisotropic and the isotropic directions the Fermi momentum will be effected by the anisotropy of the bulk theory. However, the anisotropy has influence on the dispersion relation which is almost linear for massless Fermions with chargeq=2. The universal properties that the mass and the charge of the Fermi possibly correspond to nonlinear dispersion relation are also investigated.

Constructing analytical solutions of linear perturbations of inflation with modified dispersion relations

We develop a technique to construct analytical solutions of the linear perturbations of inflation with a nonlinear dispersion relation, due to quantum effects of the early universe. Error bounds are given and studied in detail. The analytical solutions describe the evolution of the perturbations extremely well even when only the first-order approximations are considered.