Dispersive wave propagation

Kinra, V. K., "Dispersive Wave Propagation in Random Particulate Composites, Recent Advances in Composites in the U.S. and Japan," ASTM STP 864. 1985, pp. 309-325. 

In a dispersive wave propagation system, a change in pulse shape occurs as an ultrasonic wave travels in a structure that is independent of any absorption of ultrasonic energy. In... 

We used the concept of sound velocity dispersion for explanation of the shift of pulse energy spectrum maximum, transmitted through the medium, and correlation of the shift value with function of medium heterogeneity. This approach gives the possibility of mathematical simulation of the influence of both medium parameters and ultrasonic field parameters on the nature of acoustic waves propagation in a given medium. 

Cheng, L. Y, D. A. Drew, and R. T. Lahey, Jr., 1983, An Analysis of Wave Dispersion, Sonic Velocity and Critical Flow in Two-Phase Mixtures, NUREG-CR-3372, US NRC, Washington, DC. (3) Cheng, L. Y., D. A. Drew, and R. T. Lahey, Jr., 1985, An Analysis of Wave Propagation in Bubbly, Two-Component, Two Phase Flow, Trans. ASMEJ. of Heat Transfer, 107 402-408. (3)... 

Wave propagation in periodic structures can be effieiently modeled using the eoncept of Bloeh (or Floquet-Bloch) modes . This approach is also applicable for the ealeulation of band diagrams of 1 -D and 2-D photonic crystals . Contrary to classical methods like the plane-wave expansion , the material dispersion ean be fully taken into aeeount without any additional effort. For brevity we present here only the basie prineiples of the method. 

The motion of atoms in the lattice can be depicted as a wave propagation (phonon). By dispersion we mean the variation in the wave frequency as reciprocal space is traversed. The propagation of sound waves is similar to the translation of all atoms of the unit cell in the same direction hence the set of translational modes is commonly defined as an acoustic branch. The remaining vibrational modes are defined as optical branches, because they are capable of interaction with light (see McMillan, 1985, and Tossell and Vaughan, 1992, for more exhaustive explanations). 

The matter wave function is formed as a coherent superposition of states or a state ensemble, a wave packet. As the phase relationships change the wave packet moves, and spreads, not necessarily in only one direction the localized launch configuration disperses or propagates with the wave packet. The initially localized wave packets evolve like single-molecule trajectories. 

The other two local equilibrium states (a2 and as) are coupled and gives sound wave propagation. When the sound wave damping is small compared to its rate of oscillation, the sound wave obeys a dispersion relation,... 

In Maxwell s theory, this dispersion of energy is considered to be negligible, and no damping occurs during the propagation of an electromagnetic wave. Let us consider the plane waves propagating in the z direction ... 

A difficulty arises in the production of second harmonic radiation because of dispersion , that is the dependence of wave velocity on frequency. We consider the primary wave (angular frequency co) interacting with a bound electron and producing a second harmonic (2co) wave. At this point the two waves are in phase. As the co-wave propagates it generates new second harmonic, 2co-waves and, because of dispersion, the co-wave will, in general, travel with a different phase velocity than that of the 2co-wave . As a result a new 2co-wave will interfere with those generated earlier, and only constructively if they have the required phase relationship. 

D. L. Hovhannisyan, Analytic solution of the wave equation describing dispersion-free propagation of a femtosecond laser pulse in a medium with cubic and fifth-order nonlinearity, Optics Commun. 196, 103 (2001)... 

Let us explain the implication of dispersion relation through a simple example of one-dimensional wave propagation whose governing equation is given by. 

Consider a dilute suspension of Np spherical soft particles moving with a velocity U exp(—/fflf) in a symmetrical electrolyte solution of viscosity r] and relative permittivity r in an applied oscillating pressure gradient field Vp exp(—imt) due to a sound wave propagating in the suspension, where m is the angular frequency 2n times frequency) and t is time. We treat the case in which m is low such that the dispersion of r can be neglected. We assume that the particle core of radius a is coated... 

The constraint of a collision in a given sequence in our simple chain model means that there is a shock front propagating through the system, a front which reverses its direction every time an end atom collides with the hard walls. When a perfectly ordered crystal hits a hard wall, one can understand how a dispersion-free propagation of a shock wave is possible. The new feature is that such a shock front was seen in full MD simulations of impact heated clusters, using realistic forces, and has been recently studied in more detail. ... 

When the shear waves propagate through the elastic layer, or the elastic plate and reach the steady state, the type of the wave, SH wave for example, and it s dispersion relation are determined by the boundary conditions at the plate surfaces [7]. We have assumed that the sound waves modulate the stress fields at the tip of the crack, and then solved the wave equations with the boundary conditions at the surfaces of the crack and the plate. If the analysis is extended to derive the higher order fields and the dispersion relation of the wave is then obtained, such a wave do exist in the steady state. In this case we could confirm the existence of such "new wave" associated with the crack. Much algebra is required to obtain the higher order fields, however, it is not difficult to see the structure of the fields with the boundary condition at the plate surfaces. We find the boundary conditions at the plate surfaces for the second order stress fields are satisfied by the factor, cos /5 z, in the similar manner to Eq. 

When voltage U2 is applied at the transducer, a sound wave propagates into the colloid. If the densities of the dispersed and continuous phases differ, relative motion between the colloidal particles and their double layer will result. The combined relative motion will generate an electric field, which is detected as voltage Ui between the electrodes. The measured signals are proportional to the high-frequency electrophoretic mobility As derived by Babchin et al. (28), the frequency-dependent electrophoretic mobility, ix a)), for the case of low potentials, can be expressed by... 

For the evaluation of coefficients a, 02, and b, in (3.52), the Fourier analysis is selected. Assume that f represents a plane wave propagating toward the -axis, namely f has the exponential form of ejdx-a>t) Application of (3.52) to this function gives the dispersion relation of... 

The simulation of dispersive media is based on the auxiliary differential equation technique [6], while the frequency-dependent constitutive relation D(r, co) = s(ry)E(r, co) governs wave propagation in their interior. Let us now investigate three different cases of such materials characterized by diverse s(co). 

Km and Ke being the wavevectors along m and axes, respectively. Thus, we conclude that in calculations of dispersion laws of waves propagating in superlattices with plane interfaces, it is sufficient to consider first the one-dimensional models with the axis directed perpendicular to the interfaces. Then the general formulas for dispersion laws can be obtained by means of replacements (9.54) or (9.55). So we reduce 2D or 3D problems to ID problem with only one coordinate directed perpendicular to interfaces. 

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