Harmonic Wave Propagation in a Periodically Layered, Infinite Elastic Body: Plane Strain, Analytical Results

1979 ◽  
Vol 46 (1) ◽  
pp. 113-119 ◽  
Author(s):  
T. J. Delph ◽  
G. Herrmann ◽  
R. K. Kaul
Keyword(s):  
Wave Propagation ◽  
Asymptotic Behavior ◽  
Plane Strain ◽  
Elastic Body ◽  
Harmonic Wave ◽  
Wave Number ◽  
Periodic Medium ◽  
Frequency Wave ◽  
Wave Numbers ◽  
Band Characteristic

The problem of harmonic wave propagation in an unbounded, periodically layered elastic body in a state of plane strain is examined. The dispersion spectrum is shown to be governed by the roots of an 8 × 8 determinant, and represents a surface in frequency-wave number space. The spectrum exhibits the typical stopping band characteristic of wave propagation in a periodic medium. The dispersion equation is shown to uncouple along the ends of the Brillouin zones, and also in the case of wave propagation normal to the layering. The significance of this uncoupling is examined. Also, the asymptotic behavior of the spectrum for large values of the wave numbers is investigated.

Harmonic Wave Propagation in a Periodically Layered, Infinite Elastic Body: Antiplane Strain

1978 ◽  
Vol 45 (2) ◽  
pp. 343-349 ◽  
Author(s):  
T. J. Delph ◽  
G. Herrmann ◽  
R. K. Kaul
Keyword(s):  
Wave Propagation ◽  
Elastic Medium ◽  
Elastic Body ◽  
Dispersion Equation ◽  
Harmonic Wave ◽  
Wave Number ◽  
Harmonic Waves ◽  
Antiplane Strain ◽  
Frequency Wave ◽  
Asymptotic Values

The propagation of horizontally polarized shear waves through a periodically layered elastic medium is analyzed. The dispersion equation is obtained by using Floquet’s theory and is shown to define a surface in frequency-wave number space. The important features of the surface are the passing and stopping bands, where harmonic waves are propagated or attenuated, respectively. Other features of the spectrum, such as uncoupling at the ends of the Brillouin zones, conical points, and asymptotic values at short wavelengths, are also examined.

Harmonic Wave Propagation in a Periodically Layered, Infinite Elastic Body: Plane Strain, Numerical Results

1980 ◽  
Vol 47 (3) ◽  
pp. 531-537 ◽  
Author(s):  
T. J. Delph ◽  
G. Herrmann ◽  
R. K. Kaul
Keyword(s):  
Wave Propagation ◽  
Plane Strain ◽  
Elastic Body ◽  
Harmonic Wave ◽  
Numerical Results ◽  
Analytical Properties

Numerical results are presented for the dispersion spectrum for harmonic wave propagation in an unbounded, periodically layered elastic body in a state of plane strain. Both real and complex branches are considered. The spectrum is shown to be multiple-valued and quite intricate in detail. Some analytical properties of the Floquet surface are also discussed.

A Spectral Finite Element Model for Wave Propagation Analysis in Laminated Composite Plate

2006 ◽  
Vol 128 (4) ◽  
pp. 477-488 ◽  
Author(s):  
A. Chakraborty ◽  
S. Gopalakrishnan
Keyword(s):  
Finite Element ◽  
Wave Propagation ◽  
Shear Deformation ◽  
Mechanical Response ◽  
Laminated Composite ◽  
Wave Number ◽  
Single Element ◽  
Element Formulation ◽  
Frequency Wave ◽  
Spectral Finite Element

A new spectral plate element (SPE) is developed to analyze wave propagation in anisotropic laminated composite media. The element is based on the first-order laminated plate theory, which takes shear deformation into consideration. The element is formulated using the recently developed methodology of spectral finite element formulation based on the solution of a polynomial eigenvalue problem. By virtue of its frequency-wave number domain formulation, single element is sufficient to model large structures, where conventional finite element method will incur heavy cost of computation. The variation of the wave numbers with frequency is shown, which illustrates the inhomogeneous nature of the wave. The element is used to demonstrate the nature of the wave propagating in laminated composite due to mechanical impact and the effect of shear deformation on the mechanical response is demonstrated. The element is also upgraded to an active spectral plate clement for modeling open and closed loop vibration control of plate structures. Further, delamination is introduced in the SPE and scattered wave is captured for both broadband and modulated pulse loading.

Wave Propagation Analysis in Anisotropic Plate Using Wavelet Spectral Element Approach

2008 ◽  
Vol 75 (1) ◽  
Author(s):  
Mira Mitra ◽  
S. Gopalakrishnan
Keyword(s):  
Finite Element ◽  
Wave Propagation ◽  
Differential Equations ◽  
Laminated Composite ◽  
Composite Plates ◽  
Spectral Element ◽  
Wave Number ◽  
Frequency Wave ◽  
Propagation Analysis ◽  
Spectral Finite Element

In this paper, a 2D wavelet-based spectral finite element (WSFE) is developed for a anisotropic laminated composite plate to study wave propagation. Spectral element model captures the exact inertial distribution as the governing partial differential equations (PDEs) are solved exactly in the transformed frequency-wave-number domain. Thus, the method results in large computational savings compared to conventional finite element (FE) modeling, particularly for wave propagation analysis. In this approach, first, Daubechies scaling function approximation is used in both time and one spatial dimensions to reduce the coupled PDEs to a set of ordinary differential equations (ODEs). Similar to the conventional fast Fourier transform (FFT) based spectral finite element (FSFE), the frequency-dependent wave characteristics can also be extracted directly from the present formulation. However, most importantly, the use of localized basis functions in the present 2D WSFE method circumvents several limitations of the corresponding 2D FSFE technique. Here, the formulated element is used to study wave propagation in laminated composite plates with different ply orientations, both in time and frequency domains.

scholarly journals On the climatological probability of the vertical propagation of stationary planetary waves

2016 ◽  
Vol 16 (13) ◽  
pp. 8447-8460 ◽  
Author(s):  
Khalil Karami ◽  
Peter Braesicke ◽  
Miriam Sinnhuber ◽  
Stefan Versick
Keyword(s):  
Wave Propagation ◽  
Climate Model ◽  
Reanalysis Data ◽  
Planetary Waves ◽  
Superior Performance ◽  
Wave Number ◽  
Wave Numbers ◽  
Membership Value ◽  
Environmental Prediction ◽  
Strong Vortex

Abstract. We introduce a diagnostic tool to assess a climatological framework of the optimal propagation conditions for stationary planetary waves. Analyzing 50 winters using NCEP/NCAR (National Center for Environmental Prediction/National Center for Atmospheric Research) reanalysis data we derive probability density functions (PDFs) of positive vertical wave number as a function of zonal and meridional wave numbers. We contrast this quantity with classical climatological means of the vertical wave number. Introducing a membership value function (MVF) based on fuzzy logic, we objectively generate a modified set of PDFs (mPDFs) and demonstrate their superior performance compared to the climatological mean of vertical wave number and the original PDFs. We argue that mPDFs allow an even better understanding of how background conditions impact wave propagation in a climatological sense. As expected, probabilities are decreasing with increasing zonal wave numbers. In addition we discuss the meridional wave number dependency of the PDFs which is usually neglected, highlighting the contribution of meridional wave numbers 2 and 3 in the stratosphere. We also describe how mPDFs change in response to strong vortex regime (SVR) and weak vortex regime (WVR) conditions, with increased probabilities of the wave propagation during WVR than SVR in the stratosphere. We conclude that the mPDFs are a convenient way to summarize climatological information about planetary wave propagation in reanalysis and climate model data.

Waves

2019 ◽  
pp. 1-19
Author(s):  
B. D. Guenther
Keyword(s):  
General Theory ◽  
Plane Wave ◽  
Phase Velocity ◽  
Harmonic Wave ◽  
Simple Wave ◽  
Wave Number ◽  
Plane Harmonic Wave ◽  
Frequency Wave

A description of the solution of the wace equation is described as a wave that propagates without change. The set of parameters needed to describe a wave are: period, frequency, wave number, wavelength, and phase velocity. We will use a plane harmonic wave In 3 dimensions in all of our discussions and use complex notation to make the math simplier. We show we are justified in using such a simple wave by the fact that Foourier Theory allows us to construct any wave as a series of hamonics of the plane wave. The theory also will be needed in our discussion of defraction and imaging. There are a few topics that are more difficult and are marked. Their discussion can be skipped without loss of understanding of the general theory of optics.

scholarly journals On the climatological probability of the vertical propagation of stationary planetary waves

2015 ◽  
Vol 15 (22) ◽  
pp. 32289-32321
Author(s):  
K. Karami ◽  
P. Braesicke ◽  
M. Sinnhuber ◽  
S. Versick
Keyword(s):  
Wave Propagation ◽  
Climate Model ◽  
Reanalysis Data ◽  
Planetary Waves ◽  
Superior Performance ◽  
Wave Number ◽  
Refractive Indices ◽  
Wave Numbers ◽  
Membership Value ◽  
Strong Vortex

Abstract. We introduce a diagnostic tool to assess in a climatological framework the optimal propagation conditions for stationary planetary waves. Analyzing 50 winters using NCEP/NCAR reanalysis data we derive probability density functions (PDFs) of positive refractive indices as a function of zonal and meridional wave numbers. We contrast this quantity with classical climatological means of the refractive index. Introducing a Membership Value Function (MVF) based on fuzzy logic, we objectively generate a modified set of PDFs (mPDFs) and demonstrate their superior performance compared to the climatological mean of refractive indices and the original PDFs. We argue that mPDFs allow an even better understanding of how background conditions impact wave propagation in a climatological sense. As expected, probabilities are decreasing with increasing zonal wave numbers. In addition we discuss the meridional wave number dependency of the PDFs which is usually neglected, highlighting the contribution of meridional wave numbers 2 and 3 in the stratosphere. We also describe how mPDFs change in response to strong vortex regime (SVR) and weak vortex regime (WVR) conditions, with increased probabilities during WVR than SVR in the stratosphere. We conclude that the mPDFs are a convenient way to summarize climatological information about planetary wave propagation in reanalysis and climate model data.

MODELLING OF ACOUSTIC EMISSION SOURCE AND WAVE RESPONSE IN LAYERED MATERIALS

2014 ◽  
Vol 44 (1) ◽  
pp. 21-44 ◽  
Author(s):  
A. Alamin ◽  
R. Zhang
Keyword(s):  
Acoustic Emission ◽  
Wave Propagation ◽  
Integral Transformation ◽  
Layered Media ◽  
Transformation Method ◽  
Wave Number ◽  
Dislocation Source ◽  
Frequency Wave ◽  
Wave Response ◽  
Time Space

Abstract This study proposes a model of wave propagation in layered media for the use in acoustic emission (AE) studies. This model aims to find an AE response at a free surface to the propagating waves originating at a dislocation source either in one layer medium or a layer-to-layer interface. Each of the layered media is assumed to be homogenous, linear elastic and isotropic. An integral transformation method has been applied to determine the wave response in frequency-wave number domain, which is then converted to time-space domain. In the numerical examples, we first select truncated values with the finite integral transformation, so that no wave interference happens in the responses from wave reflection at truncated boundaries. Next, we simulate wave propagation in an elastic half space, and compare results obtained with that from other kind bottom boundary. Next, we introduce a dis- location source in interface and compare a simulated AE wave response obtained with that computed in the layered medium to demonstrate the performance of the model. In each simulation, the results show good agreement with the reference solutions.

Infrared spectra and substituent effects in 2-(3-, and 4-substituted phenylmethylene)-1,3-cycloheptanediones

1983 ◽  
Vol 48 (2) ◽  
pp. 586-595 ◽  
Author(s):  
Alexander Perjéssy ◽  
Pavol Hrnčiar ◽  
Ján Šraga
Keyword(s):  
Substituent Effects ◽  
Phenyl Group ◽  
Wave Number ◽  
Vibrational Coupling ◽  
Wave Numbers ◽  
Stretching Vibration ◽  
Stretching Vibrations ◽  
The Relationship ◽  
Derivatives Of ◽  
Effect Of Structure

The wave numbers of the fundamental C=O and C=C stretching vibrations, as well as that of the first overtone of C=O stretching vibration of 2-(3-, and 4-substituted phenylmethylene)-1,3-cycloheptanediones and 1,3-cycloheptanedione were measured in tetrachloromethane and chloroform. The spectral data were correlated with σ+ constants of substituents attached to phenyl group and with wave number shifts of the C=O stretching vibration of substituted acetophenones. The slope of the linear dependence ν vs ν+ of the C=C stretching vibration of the ethylenic group was found to be more than two times higher than that of the analogous correlation of the C=O stretching vibration. Positive values of anharmonicity for asymmetric C=O stretching vibration can be considered as an evidence of the vibrational coupling in a cyclic 1,3-dicarbonyl system similarly, as with derivatives of 1,3-indanedione. The relationship between the wave numbers of the symmetric and asymmetric C=O stretching vibrations indicates that the effect of structure upon both vibrations is symmetric. The vibrational coupling in 1,3-cycloheptanediones and the application of Seth-Paul-Van-Duyse equation is discussed in relation to analogous results obtained for other cyclic 1,3-dicarbonyl compounds.

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