Kinematic wave equation

Equation 34 has the form of the kinematic wave equation and represents a transition traveling with the wave velocity given by... 

The speed of the adsorption wave can be readily derived by introducing the linear isotherm assumption and the chain mle derivative of q with respect to t. The wave speed results because the assumptions turn Eq. (9.10) into a kinematic wave equation and the wave speed W is instantly recognized as ... 

That this is essentially a kinematic wave is seen by dropping the conductivity term in (24) and writing k = Hf + Hs, the concentration of heat and q = vhfT, the flow of heat. We then recover the kinematic wave equation given by Lighthill Whitham. If thermal equilibrium were instantaneously attained so that... 

By introducing Eq. [17] in Eq. [15] the following kinematic wave equation is obtained ... 

Numerical solutions to the coupled heat and mass balance equations have been obtained for both isothermal and adiabatic two- and three-transition systems but for more complex systems only equilibrium theory solutions have so far been obtained. In the application of equilibrium theory a considerable simplification becomes possible if axial dispersion is neglected and the plug flow assumption has therefore been widely adopted. Under plug flow conditions the differential mass and heat balance equations assume the hyperbolic form of the kinematic wave equations and solutions may be obtained in a straightforward manner by the method of characteristics. In a numerical simulation the inclusion of axial dispersion causes no real problem. Indeed, since axial dispersion tends to smooth the concentration profiles the numerical solution may become somewhat easier when the axial dispersion terra is included. Nevertheless, the great majority of numerical solutions obtained so far have assumed plug flow. 

The mathematical justification for the assumption of coherent concentration fronts proceeds via the inverse of this argument. It has been shown that, in a system governed by the kinematic wave equation [Eq. (9.12)] involving a transition between two constant states (a Riemann problem), if a unique solution exists it is always a function of the combined variable the... 

Often, bridges are in sump areas, or the lowest spot on the roadway profile. This necessitates the interception of most of the flow before reaching the bridge deck. Two overland flow equations are the kinematic wave equation... 

A. Soria. Kinematics Waves and Governing Equations in Bubble Columns and Three Phase Fluidized Beds. PhD thesis. University Western Ontario, Canada, 1991. 

In Reprint C in Chapter 7, the behavior of a tracer pulse in a stream flowing through a packed bed and exchanging heat or matter with the particles is studied. It is shown that the diffusion in the particles makes a contribution to the apparent dispersion coefficient that is proportional to v2 fi/D. The constant of proportionality has one part that is a function of the kinematic wave speed fi, but otherwise only a factor that depends on the shape of the particle (see p. 145 and in equation (42) ignore all except the last term and even the suffixes of this e, being unsuitable as special notation, will be replaced by A. e is defined in the middle of p. 143 of Chapter 7). In this equation, we should not be surprised to find a term of the same form as the Taylor dispersion coefficient, for it is diffusion across streams of different speeds that causes the dispersion in that case just as it is the diffusion into stationary particles that causes the dispersion in this.7 What is surprising is that the isothermal diffusion and reaction equation should come up, for A is defined by... 

The theory of kinematic waves, initiated by Lighthill Whitham, is taken up for the case when the concentration k and flow q are related by a series of linear equations. If the initial disturbance is hump-like it is shown that the resulting kinematic wave can be usually described by the growth of its mean and variance, the former moving with the kinematic wave velocity and the latter increasing proportionally to the distance travelled. Conditions for these moments to be calculated from the Laplace transform of the solution, without the need of inversion, are obtained and it is shown that for a large class of waves, the ultimate wave form is Gaussian. The power of the method is shown in the analysis of a kinematic temperature wave, where the Laplace transform of the solution cannot be inverted. 

A kinematic wave may be called linear if the relationship between the flow and the concentration can be expressed by one or more linear equations, algebraic or differential. The term linear may also be applied when a diffusion term is included in the continuity equation as is done in 3 of Lighthill ... 

Equation (11) simply means that the total disturbance ffi v(x, t) dr is the same at any point x as it is at x = 0. We could without loss of generality put a0 = fio = 1. Equation (12) shows that the mean time of the disturbance at x differs from that at x = 0 by a quantity strictly proportional to x moreover, if the wave velocity is judged by the progress of this mean time the wave moves with a constant velocity 3vq/2, the kinematic wave velocity. Equation (13) shows that the increase of the variance is also proportional to x and the constant of proportionality is a measure of the rate of dispersion. 

Consider dispersion of a linear kinematic wave in dimensionless form.[14] The governing equation and boundary/initial conditions are ... 

There is also numerical evidence of the above given solitonic properties of nonlinear waves. For the particular case of Equation 4.3 and related two-side propagating wave equations, Christov and Velarde have calculated wave prohles and discussed the kinematics of the collisions of solitonic surface waves in two extreme limiting cases. 

In this chapter, we use the results of numerical infiltration experiments in dual porosity media performed with a three-dimensional lattice-gas model to characterize preferential flow as response to rainfall intensity. From the temporal and spatial evolution of the water content during infiltration and drainage, we evaluate the adequacy of a kinematic wave approximation to describe the flow. We also discuss the conceptual basis of the asymptotic kinematic approach to Richards equation in comparison with the macropore kinematic equation. 

As we have noted in the introduction, experimental evidence for a kinematic description of excitable wave fronts is rich. Based on hyperbolic wave equations and the Huygens principle, Wiener and Rosenblueth [81] recommend the eikonal approach of geometric optics waves propagate at a constant normal speed... 

This equation defines the characteristic speed, that is, the kinematic wave speed. Note that for s = constant the characteristic speed drldt= which is analogous to the wave speed in the gravity sedimentation problem except that here is not constant with distance. The analogy also holds with s — s p) if s d sp)/dp is defined as a hindered settling factor G(d>) (Eq. 5.4.18). 

Kinematic waves result from the existence of a functional dependency between some "flux" and the corresponding "density", otherwise related through a balance equation (Whitham, 1974, p. 27). In two-phase flows, the foregoing flux and density are respectively the drift flux (volumetric flow rate of one phase with respect to the mixture) and the void fraction (volumetric concentration of the same phase in the mixture). Kinematic waves convey information on the flow structure and the associated variables such as the void fraction and the drift flux. In particular, they control flow pattern changes and density wave oscillations. 

The general dispersion equation. Equation 16, can be rearranged in terms of the celerity of kinematic wave, C. As indicated by Equation 20, the neutrally stable... 

Note, however, that a pure dynamic wave may be physically realized in inviscid flows. In viscid flow systems, the wave characteristics may be related to those of pure dynamic and kinematic waves by introducing - cj from Equation 36... 

Equations 40 indicate that the locus for which the kinematic wave velocity is equal to that of the dynamic wave, = C, represents neutral stable wave modes. 

These equations are of charcteristic hyperbolic fomi and represent kinematic waves with propagation velocities given by... 

The standard procedure (Rose 1961), for heavy atoms, is to solve the wave equation in a spherical potential with relativistic kinematics. In open-shell systems the charge density is spherically symmetrized by averaging over all azimuthal quantum numbers. The wave equations to be solved in practice are, therefore, the coupled first-order differential equations for the radial components of the Dirac equation (Rose 1961)... 

Discharge can be related to overland flow depth by the St. Venant equations (77). The kinematic wave form is the most commonly used simplification for... 

In agreement with the descriptive terms used in clinical practice, the PDF model accounts for the effects of stiffness (kx), relaxation (cx), and load (x) on E-waves (Equation 28.5). Normal E-wave contours are well fit by underdamped kinematics where damping is low relative to stiffness ... 

This equation describes the propagation of kinematic waves, as may be readily verified from the expression for a travelling wave, eqn (7.18), and by proceeding exactly as illustrated in eqns (7.19)-(7.22) the wave solution to eqn (13.23) travels, without change of amplitude, at the kinematic-wave speed v = mk, <2 = 0. 

The chapters presented by different experts in the field have been structured to develop an intuition for the basic principles by discussing the kinematics of shock compression, first from an extremely fundamental level. These principles include the basic concepts of x-t diagrams, shock-wave interactions, and the continuity equations, which allow the synthesis of material-property data from the measurement of the kinematic properties of shock compression. A good understanding of these principles is prerequisite... 

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