Point interactions in acoustics: One-dimensional models (original) (raw)
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Effective equation for a system of mechanical oscillators in an acoustic field
We consider a one dimensional evolution problem modelling the dynamics of an acoustic field coupled with a set of mechanical oscillators. We analyse solutions of the system of ordinary and partial differential equations with time-dependent boundary conditions describing the evolution in the limit of a continuous distribution of oscillators. MSC 2010: 76M50, 35D30, 35M33.
Forced Nonlinear Oscillatory Acoustic Vacuum
2018
In this work, we study the in-plane oscillations of a finite lattice of particles coupled by linear springs under distributed harmonic excitation. Melnikov-type analysis is applied for the persistence of periodic oscillations of a reduced system.
Our work presents an theoretical model and an original experimental method addressed to our eye and brain for understand what is happening to each and every particles that compose a periodical structure when is set into longitudinal oscillations, and to see how the same kind of analysis applies to a system of particles connected by springs along a straight limited line. Physical and mathematical considerations allow us to establish the differential equation and characteristics of compressional traveling waves in helix springs. The particularization of these, for this limited pseudo-continuous medium, gives the time independent wave equation that, by its eigensolutions, can describe not only the standing waves, but also the multiple resonances and the normal modes of vibration in this macroscopic periodical structure, as well as in crystalline materials, along a domain selected direction of the lattice. Our experimental method for visual observation and quantitative study of wave motion became possible trough the use of a long helical spring, stretched in vertical position that is excited at its lower end by an electromagnetic audio-vibrator. Obtained results on long helix springs and on beds rubber strings are agree with the theoretical model, being a convincing experiment for perceiving the intricate widespread phenomenon of sound waves that exist in solids, liquids and gases, but that are directly invisible.
Periodically Forced Nonlinear Oscillatory Acoustic Vacuum
Axioms, 2018
In this work, we study the in-plane oscillations of a finite lattice of particles coupled by linear springs under distributed harmonic excitation. Melnikov-type analysis is applied for the persistence of periodic oscillations of a reduced system.
Wave chaos in acoustics and elasticity
Journal of Physics A: Mathematical and Theoretical, 2007
Interpreting wave phenomena in terms of an underlying ray dynamics adds a new dimension to the analysis of linear wave equations. Forming explicit connections between spectra and wavefunctions on the one hand and the properties of a related ray dynamics on the other hand is a comparatively new research area, especially in elasticity and acoustics. The theory has indeed been developed primarily in a quantum context; it is increasingly becoming clear, however, that important applications lie in the field of mechanical vibrations and acoustics. We provide an overview over basic concepts in this emerging field of wave chaos. This ranges from ray approximations of the Green function to periodic orbit trace formulae and random matrix theory and summarizes the state of the art in applying these ideas in acoustics-both experimentally and from a theoretical/numerical point of view.
Nonlinear acoustic waves in periodic media
A high amplitude acoustic wave travelling in a fluid develops higher harmonics as it propagates through the medium. In a homogeneous fluid, such nonlinearly generated harmonics grow during propagation because of the lack of dispersion. On the other hand, periodic acoustic media such as 1D layered media, or 2D sonic crystals, are known to introduce strong dispersion in wave propagation, even creating forbidden propagation bands or bandgaps where wave propagation at particular frequencies is not allowed. The combined action of nonlinearity (harmonic generation) and periodicity (different propagation velocities and attenuation for the different harmonics) results in novel and unexpected phenomena with respect to the linear counterpart, and opens the door to new mechanisms of acoustic wave control and manipulation. Here we investigate numerically the propagation of a plane acoustic wave in a periodic medium in two situations: a structured fluid, formed by a periodic array of fluid layers with alternating acoustic properties, and a 2D squared array of solid scatterers embedded in a fluid (a sonic crystal). We show how the nonlinear generation and propagation of the second harmonic is strongly affected by the presence of the crystal. PACS no. xx.xx.Nn, xx.xx.Nn
NON-LINER DYNAMICS OF UNDERWATER ACOUSTICS
Journal of Sound and Vibration, 1999
The non-linear dynamic behavior of acoustic wave propagation in an underwater sound channel, described by the Munk's classical sound speed profile perturbed by a single-mode internal wave, is studied using a parabolic ray theory. The amplitude and wavelength of this single-mode wave are used as the branching parameters in bifurcation analysis. The phase plane trajectory of the ray-based system can be periodic, quasi-periodic, and unstable. The regions of instability, located numerically via the bifurcation diagrams, are examined through a sequence of phase diagrams and Poincare´maps. Charts showing the maximum uninterrupted propagation distance reveal instances of anomalous vertical scattering of sound energy. Floquet multipliers were used to investigate instability of periodic orbits.
A Paradigm for Time-periodic Sound Wave Propagation in the Compressible Euler Equations
Methods and Applications of Analysis, 2009
We formally derive the simplest possible periodic wave structure consistent with time-periodic sound wave propagation in the 3 × 3 nonlinear compressible Euler equations. The construction is based on identifying the simplest periodic pattern with the property that compression is counter-balanced by rarefaction along every characteristic. Our derivation leads to an explicit description of shock-free waves that propagate through an oscillating entropy field without breaking or dissipating, indicating a new mechanism for dissipation free transmission of sound waves in a nonlinear problem. The waves propagate at a new speed, (different from a shock or sound speed), and sound waves move through periods at speeds that can be commensurate or incommensurate with the period. The period determines the speed of the wave crests, (a sort of observable group velocity), but the sound waves move at a faster speed, the usual speed of sound, and this is like a phase velocity. It has been unknown since the time of Euler whether or not time-periodic solutions of the compressible Euler equations, which propagate like sound waves, are physically possible, due mainly to the ubiquitous formation of shock waves. A complete mathematical proof that waves with the structure derived here actually solve the Euler equations exactly, would resolve this long standing open problem.
Geometric perturbation theory and acoustic boundary condition dynamics
Physica D: Nonlinear Phenomena
Geometric perturbation theory is universally needed but not recognized as such yet. A typical example is provided by the three-dimensional wave equation, widely used in acoustics. We face vibrating eardrums as binaural auditory input and stemming from an external sound source. In the setup of internally coupled ears (ICE), which are present in more than half of the land-living vertebrates, the two tympana are coupled by an internal air-filled cavity, whose geometry determines the acoustic properties of the ICE system. The eardrums themselves are described by a two-dimensional, damped, wave equation and are part of the spatial boundary conditions of the threedimensional Laplacian belonging to the wave equation in the internal cavity that couples and internally drives the eardrums. In animals with ICE the resulting signal is the superposition of external sound arriving at both eardrums and the internal pressure coupling them. This is also the typical setup for geometric perturbation theory. In the context of ICE it boils down to acoustic boundary-condition dynamics (ABCD) for the coupled dynamical system of eardrums and internal cavity. In acoustics the deviations from equilibrium are extremely small (nm range). Perturbation theory therefore seems natural and is shown to be appropriate. In doing so, we use a time-dependent perturbation theoryà la Dirac in the context of Duhamel's principle. The relaxation dynamics of the tympanic-membrane system, which neuronal information processing stems from, is explicitly obtained in first order. Furthermore, both the initial and the quasi-stationary asymptotic state