The nonlinear forcing terms for the wave equation in general curvilinear coordinates are derived based on an isotropic homogeneous weakly nonlinear elastic material. The expressions for the nonlinear part of the first Piola-Kirchhoff stress are specialized for axisymmetric torsional and longitudinal fundamental waves in a circular cylinder. The matrix characteristics of the nonlinear forcing terms and secondary mode wave structures are manipulated to analyze the higher harmonic generation due to the guided wave mode self-interactions and mutual interactions. It is proved that both torsional and longitudinal secondary wave fields can be cumulative by a specific type of guided wave mode interactions. A method for the selection of preferred fundamental excitations that generate strong cumulative higher harmonics is formulated, and described in detail for second harmonic generation. Nonlinear finite element simulations demonstrate second harmonic generation by T(0,3) and L(0,4) modes at the internal resonance points. A linear increase of the normalized modal amplitude ratio A2/A1(2) over the propagation distance is observed for both cases, which indicates that mode L(0,5) is effectively generated as a cumulative second harmonic. Counter numerical examples demonstrate that synchronism and sufficient power flux from the fundamental mode to the secondary mode must occur for the secondary wave field to be strongly cumulative.
A plate ray perspective for elastic wave propagation in hollow circular cylinders is presented in order to excite a predominant flexural mode, which in turn generates higher order harmonics due to nonlinear material behavior. The scattering angles are determined for the internally resonant higher order harmonics due to the interactions of two collimated waves. Primary waves that can generate strongly cumulative higher order harmonics are identified for mode self interactions and mutual interactions. A helical inter-digital transducer has been designed for the excitation of a single dominant flexural mode. Numerical evaluations that demonstrate cumulative second harmonic generation are undertaken for both torsional and longitudinal flexural waves. Quadratic sum and difference harmonic generation is observed for the mutual interaction between two primary torsional flexural wave modes.
In recent years, the effects of Lorentz symmetry breaking in cosmology has attracted considerable amount of attention. In cosmological context several topics can be affected by Lorentz violation,e.g., inflationary scenario, CMB, dark energy problem and barryogenesis. In this paper we consider the cosmological particle creation due to Lorentz violation (LV). We consider an exactly solvable model for finding the spectral properties of particle creation in an expanding space-time exhibiting Lorentz violation. In this model we calculate the spectrum and its variations with respect to the rate and the amount of space-time expansion.
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