Propagation of Low-Frequency Elastic Disturbances in a Composite Material

1974 ◽  
Vol 41 (1) ◽  
pp. 97-100 ◽  
Author(s):  
W. Kohn
Keyword(s):  
Composite Material ◽  
Wave Equation ◽  
Dispersion Relation ◽  
Displacement Field ◽  
Low Frequency ◽  
Local Strain ◽  
Airy Functions ◽  
Low Frequencies ◽  
One Dimensional ◽  
Long Wavelength

In the limit of low frequencies the displacement u(x, t) in a one-dimensional composite can be written in the form of an operator acting on a slowly varying envelope function, U(x, t): u(x, t) = [1 + v1(x)∂/∂x + …] U(x, t). U(x, t) itself describes the overall long wavelength displacement field. It satisfies a wave equation with constant, i.e., x-independent, coefficients, obtainable from the dispersion relation ω = ω(k) of the lowest band of eigenmodes: (∂2/∂t2 − c¯2∂2/∂x2 − β∂4/∂x4 + …) U(x, t) = 0. Information about the local strain, on the microscale of the composite laminae, is contained in the function v1(x), explicitly expressible in terms of the periodic stiffness function, η(x), of the composite. Appropriate Green’s functions are constructed in terms of Airy functions. Among applications of this method is the structure of the so-called head of a propagating pulse.

Analysis of the characteristics‐integration method for one‐dimensional wave inversion

Geophysics ◽  
1988 ◽  
Vol 53 (8) ◽  
pp. 1034-1044 ◽  
Author(s):  
Nei‐Mao Chen ◽  
Yu‐Hua Chu ◽  
John T. Kuo
Keyword(s):  
Wave Equation ◽  
Inverse Method ◽  
Integration Method ◽  
Low Frequency ◽  
Real Data ◽  
Frequency Component ◽  
One Dimensional ◽  
Reflection Data ◽  
Wave Inversion ◽  
Impedance Profile

Basing our work on the one‐dimensional (1-D) wave equation, we present an inverse method which we call the characteristics‐integration method. The method is derived from integration along characteristic families of straight lines of the wave equation in the time domain. With the source function known and reflection data recorded on the surface, the characteristics‐integration method can efficiently and economically recover the subsurface impedance profile, provided that the structure is inhomogeneous only in the depth direction. In general, when seismic data are contaminated by noise, the characteristics‐integration method, like any other 1-D inverse method, suffers from instability. We find that, for a smoothly varying impedance profile, the instability of inversions using characteristic methods depends heavily on the bandwidth of the source wavelet. We devised a resampling technique to stabilize the inverse scheme and to suppress the growth of errors. Numerical examples, including data contaminated by noise, data missing the low‐frequency component, and real data cases, show the feasibility of recovering impedance profiles using the characteristics‐integration method.

scholarly journals Electromagnetic responses of relativistic electrons

2018 ◽  
Vol 84 (1) ◽  
Author(s):  
C. A. A. de Carvalho ◽  
D. M. Reis
Keyword(s):  
Dispersion Relation ◽  
Quantum Electrodynamics ◽  
Relativistic Electrons ◽  
Zero Temperature ◽  
Low Frequencies ◽  
One Dimensional ◽  
Electromagnetic Responses ◽  
Analytic Expressions ◽  
Weak Fields ◽  
Recent Claim

We compute the real and imaginary parts of the electric permittivities and magnetic permeabilities of relativistic electrons from quantum electrodynamics at finite temperatures and densities, for weak fields, neglecting electron–electron interactions. For non-zero temperatures, electromagnetic responses are reduced to one-dimensional integrals computed numerically. For zero temperature, we find analytic expressions for both their real/dispersive and imaginary/absorptive parts. As an application of our results, we obtain the dispersion relation for longitudinal electric plasmons. Present calculations support our recent claim that, at low frequencies and long wavelengths, the system will exhibit simultaneously negative electric and magnetic responses.

Propagation of Low Frequency Elastic Disturbances in a Three-Dimensional Composite Material

1975 ◽  
Vol 42 (1) ◽  
pp. 159-164 ◽  
Author(s):  
W. Kohn
Keyword(s):  
Secular Equation ◽  
Displacement Vector ◽  
Three Dimensional ◽  
Low Frequency ◽  
Three Dimensions ◽  
Low Frequencies ◽  
One Dimensional ◽  
Coupled Partial Differential Equations ◽  
Constant Coefficients ◽  
The Mean

This paper is a generalization to three dimensions of an earlier study for one-dimensional composites. We show here that in the limit of low frequencies the displacement vector ui(r,t) can be written in the form ui (r,t) = (∂ij + vijl (r) ∂/∂xl + …) Uj (r,t). Here Uj (r,t) is a slowly varying vector function of r and t which describes the mean displacement of each cell of the composite. Its components satisfy a set of three coupled partial differential equations with constant coefficients. These coefficients are obtainable from the three-by-three secular equation which yields the low-lying normal mode frequencies, ω(k). Information about local strains is contained in the function vijl(r), which is characteristic of static deformations, and is discussed in detail. Among applications of this method is the structure of the head of a pulse propagating in an arbitrary direction.

scholarly journals Effect of Tunnel Diameter on Ground Vibrations Generated by Underground

2000 ◽  
Vol 19 (1) ◽  
pp. 17-25 ◽  
Author(s):  
Trains Qinhua Lin ◽  
Victor V. Krylov
Keyword(s):  
Theoretical Model ◽  
Green’S Function ◽  
Green's Function ◽  
Displacement Field ◽  
Low Frequency ◽  
Zero Approximation ◽  
Ground Vibrations ◽  
Reciprocity Principle ◽  
Low Frequencies ◽  
Tunnel Diameter

A theoretical model has been developed of generating ground vibrations by underground trains travelling in idealised circular tunnels of finite diameter. By means of the reciprocity principle, the displacement field radiated by a point force applied to the bottom of the tunnel, i.e., the Green's function of the problem, has been derived in zero and first approximations versus tunnel diameter. This more precise Green's function has been applied to carry out calculations of railway-generated ground vibrations using earlier developed methods. The results show that the velocities of generated low-frequency ground vibrations increase with the increase in tunnel diameter. It is also shown that zero approximation is accurate only at very low frequencies.

scholarly journals THE DELOCALIZED EFFECTIVE DEGREES OF FREEDOM OF A BLACK HOLE AT LOW FREQUENCIES

2008 ◽  
Vol 17 (13n14) ◽  
pp. 2617-2624 ◽  
Author(s):  
BARAK KOL
Keyword(s):  
Field Theory ◽  
Black Hole ◽  
Effective Field Theory ◽  
Degrees Of Freedom ◽  
World Line ◽  
Low Frequency ◽  
Effective Field ◽  
Theory Approach ◽  
Low Frequencies ◽  
Long Wavelength

Identifying the fundamental degrees of freedom of a black hole poses a long-standing puzzle. Recently Goldberger and Rothstein forwarded a theory of the low frequency degrees of freedom within the effective field theory approach, where they are relevancy-ordered but of unclear physical origin. Here these degrees of freedom are identified with near-horizon but non-compact gravitational perturbations which are decomposed into delocalized multipoles. Their world-line (kinetic) action is determined within the classical effective field theory (CLEFT) approach and their interactions are discussed. The case of the long-wavelength scattering of a scalar wave off a Schwarzschild black hole is treated in some detail, interpreting within the CLEFT approach the equality of the leading absorption cross section with the horizon area.

scholarly journals Wave dispersion in pulsar plasma. Part 1. Plasma rest frame

2019 ◽  
Vol 85 (3) ◽  
Author(s):  
M. Z. Rafat ◽  
D. B. Melrose ◽  
A. Mastrano
Keyword(s):  
Dispersion Relation ◽  
Rest Frame ◽  
Resonance Condition ◽  
Low Frequency ◽  
Phase Speed ◽  
Wave Dispersion ◽  
Relativistic Plasma ◽  
Dispersion Function ◽  
One Dimensional ◽  
Mode Dispersion

Wave dispersion in a pulsar plasma (a one-dimensional, strongly magnetized, pair plasma streaming highly relativistically with a large spread in Lorentz factors in its rest frame) is discussed, motivated by interest in beam-driven wave turbulence and the pulsar radio emission mechanism. In the rest frame of the pulsar plasma there are three wave modes in the low-frequency, non-gyrotropic approximation. For parallel propagation (wave angle$\unicode[STIX]{x1D703}=0$) these are referred to as the X, A and L modes, with the X and A modes having dispersion relation$|z|=z_{\text{A}}\approx 1-1/2\unicode[STIX]{x1D6FD}_{\text{A}}^{2}$, where$z=\unicode[STIX]{x1D714}/k_{\Vert }c$is the phase speed and$\unicode[STIX]{x1D6FD}_{\text{A}}c$is the Alfvén speed. The L mode dispersion relation is determined by a relativistic plasma dispersion function,$z^{2}W(z)$, which is negative for$|z|<z_{0}$and has a sharp maximum at$|z|=z_{\text{m}}$, with$1-z_{\text{m}}<1-z_{0}\ll 1$. We give numerical estimates for the maximum of$z^{2}W(z)$and for$z_{\text{m}}$and$z_{0}$for a one-dimensional Jüttner distribution. The L and A modes reconnect, for$z_{\text{A}}>z_{0}$, to form the O and Alfvén modes for oblique propagation ($\unicode[STIX]{x1D703}\neq 0$). For$z_{\text{A}}<z_{0}$the Alfvén and O mode curves reconnect forming a new mode that exists only for$\tan ^{2}\unicode[STIX]{x1D703}\gtrsim z_{0}^{2}-z_{\text{A}}^{2}$. The L mode is the nearest counterpart to Langmuir waves in a non-relativistic plasma, but we argue that there are no ‘Langmuir-like’ waves in a pulsar plasma, identifying three features of the L mode (dispersion relation, ratio of electric to total energy and group speed) that are not Langmuir like. A beam-driven instability requires a beam speed equal to the phase speed of the wave. This resonance condition can be satisfied for the O mode, but only for an implausibly energetic beam and only for a tiny range of angles for the O mode around$\unicode[STIX]{x1D703}\approx 0$. The resonance is also possible for the Alfvén mode but only near a turnover frequency that has no counterpart for Alfvén waves in a non-relativistic plasma.

Multicell Homogenization of One-Dimensional Periodic Structures

2010 ◽  
Vol 132 (1) ◽  
Author(s):  
Stefano Gonella ◽  
Massimo Ruzzene
Keyword(s):  
Dynamic Behavior ◽  
Periodic Structures ◽  
Low Frequency ◽  
Structural Response ◽  
Fourier Series Expansion ◽  
Low Frequencies ◽  
One Dimensional ◽  
Physical Constraints ◽  
Unit Cells ◽  
Taylor Series Expansions

Much attention has been recently devoted to the application of homogenization methods for the prediction of the dynamic behavior of periodic domains. One of the most popular techniques employs the Fourier transform in space in conjunction with Taylor series expansions to approximate the behavior of structures in the low frequency/long wavelength regime. The technique is quite effective, but suffers from two major drawbacks. First, the order of the Taylor expansion, and the corresponding frequency range of approximation, is limited by the resulting order of the continuum equations and by the number of boundary conditions, which may be imposed in accordance with the physical constraints on the system. Second, the approximation at low frequencies does not allow capturing bandgap characteristics of the periodic domain. An attempt at overcoming the latter can be made by applying the Fourier series expansion to a macrocell spanning two (or more) irreducible unit cells of the periodic medium. This multicell approach allows the simultaneous approximation of low frequency and high frequency dynamic behavior and provides the capability of analyzing the structural response in the vicinity of the lowest bandgap. The method is illustrated through examples on simple one-dimensional structures to demonstrate its effectiveness and its potentials for application to complex one-dimensional and two-dimensional configurations.

Dynamic Response of a Buried Pipeline at Low Frequencies

1985 ◽  
Vol 107 (1) ◽  
pp. 44-50 ◽  
Author(s):  
P. M. O’Leary ◽  
S. K. Datta
Keyword(s):  
Plane Waves ◽  
Low Frequency ◽  
Buried Pipeline ◽  
Low Frequencies ◽  
Long Wavelength ◽  
Induced Stresses ◽  
Dynamic Amplification ◽  
Hoop Stresses ◽  
Incident Waves ◽  
Pipe Materials

A long wavelength and low-frequency analysis is presented here for the dynamic behavior of a long continuous pipeline embedded in an elastic medium. Using a shell model for the pipe, it is shown that the dynamic amplification of axial and hoop stresses induced in the pipe due to traveling plane waves (longitudinal and shear) depends crucially on the ratio of the rigidities of the surrounding soil and the pipe. Results are presented showing the dependence of the induced stresses on the direction of propagation of the incident waves, the Poisson’s ratios and rigidities of the ground and pipe materials.

Thermally corrected solutions of the one-dimensional wave equation for the laser-induced ultrasound

2021 ◽  
Vol 130 (2) ◽  
pp. 025104
Author(s):  
Misael Ruiz-Veloz ◽  
Geminiano Martínez-Ponce ◽  
Rafael I. Fernández-Ayala ◽  
Rigoberto Castro-Beltrán ◽  
Luis Polo-Parada ◽  
...  
Keyword(s):  
Wave Equation ◽  
One Dimensional ◽  
The One ◽  
Dimensional Wave
Sign in / Sign up

Export Citation Format

Share Document