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.

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.

The leaky-mode period equation—a plane-wave approach

1970 ◽  
Vol 60 (6) ◽  
pp. 1989-1998 ◽  
Author(s):  
L. E. Alsop
Keyword(s):  
Plane Wave ◽  
Phase Velocity ◽  
Half Space ◽  
Leaky Mode ◽  
Wave Number ◽  
Complex Frequency ◽  
The Real ◽  
Leaky Modes ◽  
Phase Velocity Dispersion ◽  
Period Equation

Abstract It is shown that the plane-wave picture of a leaky mode proposed by Burg, Ewing, Press and Stulkin (1951) yields the accepted period equation for leaky modes in a water layer a half-space. The resultant mode is formed by an inhomogeneous wave with real frequency and complex wave number and phase velocity. Another form of mode considered is that formed by a homogeneous wave in the guide with real phase velocity and complex frequency and wave number. The phase-velocity dispersion curve for this case is appropriate for determining shear-wave coupling to PL waves. The procedures of the article could be readily extended to the more complicated case of a solid layer over a half-space. It is also demonstrated that the derivative of the real part of angular frequency with respect to the real part of the wave number is a good approximation to the group velocity for leaky modes with low losses.

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.

scholarly journals Evaluation Study of Boundary and Depth of the Soil Structure for Geotechnical Site Investigation using MASW

2017 ◽  
Vol 2 (1) ◽  
pp. 31
Author(s):  
A. Arisona ◽  
Mohd Nawawi ◽  
Amin E. Khalil ◽  
U K Nuraddeen ◽  
Mohd Hariri ◽  
...  
Keyword(s):  
Phase Velocity ◽  
Soil Structure ◽  
Subsurface Layer ◽  
Saturated Soil ◽  
Wave Number ◽  
Velocity Versus ◽  
Frequency Wave ◽  
Soil Zone ◽  
Clayey Silt ◽  
Best Fit

This study reviews the correlation between the experimental Rayleigh dispersion curve and the Vp & Vs ground model versus depth. Six samples of stations A , B , C , D ,  E  and  F  were used in the experiment.The geophone spacing used was set 1 m and total length of each line was 23 m. The result shows positive significance (best fit) of R2 that ranges from 0.80 to 0.90. The fk (frequency-wave number method) dispersion curves analysis confirmed that the soil structure investigated is divided into three zones: (1) Unsaturated soil zone (clay soil), in which the layer is dominated by soil with typically alluvial clayey silt and sand. The Vp ranges from 240 m/s to 255 m/s at a depth of 2 to 8 m. (2) The intermediate zone (stiff soil), in which the layer is dominated by sand, silt, clayey sand, sandy clay and clay of low plasticity. This structure is interpreted as partially saturated soil zone, the soil is typically very dense. It contains soft rock typically fill with cobble, sand, slight gravel and highly weathered at depth of 18 to 30 m with Vp of  255 to 300 m/s. (3) Saturated soil zone at a depth of  8 to 18 m with Vp of 300 to 390 m/s. There is a very good agreement between wave-number (k) and phase velocity (Vw)  produced. Both the two parameters shows similar pattern in the topsoil and subsurface layer, which constitute boundary field of soil structure. Moreover, relationship between phase velocity versus wave-length shows best fit of model from inversion with measured value (observed) in  implementation of the boundary and depth of each layer.

Analysis and Interpretation of Longitudinal Waves in Periodic Multiphase Rods Using the Method of Reverberation-Ray Matrix Combined With the Floquet-Bloch Theorem

2013 ◽  
Vol 136 (1) ◽  
Author(s):  
Y. Q. Guo ◽  
D. N. Fang
Keyword(s):  
Phase Velocity ◽  
Longitudinal Wave ◽  
Vibration Isolation ◽  
Periodic Structures ◽  
Wave Number ◽  
Wave Band ◽  
Frequency Wave ◽  
Longitudinal Waves ◽  
Bloch Theorem ◽  
Velocity Spectra

The method of reverberation-ray matrix (MRRM) combined with the Floquet–Bloch theorem, which serves as an alternative method, is presented for accurately analyzing longitudinal waves in general periodic multiphase rods. Closed-form dispersion relation of periodic quaternary rods is derived. Based on this relation, the functions of constituent-rod parameters in the formation of longitudinal-wave band structures are analytically revealed. Numerical examples validate the proposed method and indicate the characteristics/applications of all kinds of dispersion curves that include the frequency-wave number spectra, the frequency-wavelength spectra, the frequency-phase velocity spectra, the wave number-phase velocity spectra and the wavelength-phase velocity spectra. The effect of unit-cell layout on the frequency band properties and the functions of constituent-rod parameters in the band structure formation are also illustrated numerically. The analysis and interpretation of longitudinal waves in periodic multiphase rods given in this paper will push forward the design of periodic structures for longitudinal wave filtering/guiding and vibration isolation/control applications.

A Dispersive Nonlocal Model for In-Plane Wave Propagation in Laminated Composites With Periodic Structures

2015 ◽  
Vol 82 (3) ◽  
Author(s):  
H. Brito-Santana ◽  
Yue-Sheng Wang ◽  
R. Rodríguez-Ramos ◽  
J. Bravo-Castillero ◽  
R. Guinovart-Díaz ◽  
...  
Keyword(s):  
Wave Propagation ◽  
Plane Wave ◽  
Phase Velocity ◽  
Incident Angle ◽  
Unit Cell ◽  
Wave Number ◽  
Periodic Distribution ◽  
Phase And Group Velocities ◽  
Group Velocities ◽  
Plane Wave Propagation

In this paper, the problem of in-plane wave propagation with oblique incidence of the wave in an isotropic bilaminated composite under perfect contact between the layers and periodic distribution between them is studied. Based on an asymptotic dispersive method for the description of the dynamic processes, the dispersion equations were derived analytically from the average model. Numerical examples show that the dispersion curves obtained from the present model agree with the exact solutions for a range of wavelengths. Detailed numerical simulations are provided to illustrate graphically the phase and group velocities. Such illustrations allow the identification and comparison of the effects of the unit cell size, wave number and incident angle. It was observed that, as the incident angle increases, the dimensionless quasi-longitudinal phase velocity increases, and the dimensionless quasi-shear phase velocity decreases. In addition, the phase and group velocities decrease as the size of the unit cell increases. The frequency band structure, as a function of the wave-vector components is calculated.

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.

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