Traveling wave theory

Remarks on the traveling wave theory of dynamic phase transitions... 

Remarks on the Traveling Wave Theory of Dynamic Phase Transitions... 

The traveling wave theory of phase transitions attempts to determine when two homogeneous phases (y, u, ), (V", u, 0) may be joined by a traveling wave solution... 

The above referenced experiments (adiabatic) occur in flow geometries much more complicated than the simple (isothermal) planar Riemann (shock structure) problems discussed above. Furthermore, the interesting results on shock splitting given in [27], [41-44] all involved piston and not Riemann initial value problems. Hence a direct comparison with the local shock structure theory given here may be difficult. Nonetheless for completeness we record some obvious consistencies and discrepancies between the experimental results and traveling wave theory. 

With respect to the fact that traveling wave theory given above was isothermal while the experiments axe not we recall that the papers [40], [21] show that the same qualita-... 

Discrepancy A major discrepancy between the traveling wave theory and the shock splitting results of [41-44], [27] is the fact that phase transition in these experiments goes from vapor —> a liquid, vapor mixture. Several reasons could be suggested for this discrepancy. Of course there is an obvious one, i.e., the van der Waals equation of state may be a poor model of the material under consideration. On the other hand one must in fairness recall... 

To analyze a transient in a distributed-parameter line, a traveling-wave theory is explained for both single- and multiconductor systems. A method to introduce velocity difference and attenuation in the multiconductor system is described together with field test results. Impedance and admittance formulas of unusual conductors, such as finite-length and vertical conductors, are also explained. 

In this section, examples of hand calculations of transients with a pocket calculator are explained by adopting (1) the traveling-wave theory described in Section 1.6 and (2) the Laplace transform by using a lumped parameter circuit equivalent to the distributed line [2]. These two approaches are the most powerful to analyze a transient theoretically by hand and they also correspond to the following representative simulation meth-ods ... 

The transient voltage at a tower top, at which lightning strikes, is easily calculated by applying traveling wave theory. An example has been explained in Section 1.7 (see Problem 1.19). 

Quantum theory considers radiation as a stream of energy packets - photons or quanta - travelling through space at a constant velocity (c when in a vacuum). The energy of a photon is related to the frequency of the radiation, as defined in wave theory, by the expression... 

I he notation 0e indicates that this is the dielectric function at frequencies low i ompared with electronic excitation frequencies. We have also replaced co0 with l (, the frequency of the transverse optical mode in an ionic crystal microscopic theory shows that only this type of traveling wave will be readily excited bv a photon. Note that co2 in (9.20) corresponds to 01 e2/me0 for the lattice vibrations (ionic oscillators) rather than for the electrons. The mass of an electron is some thousands of times less than that of an ion thus, the plasma liequency for lattice vibrations is correspondingly reduced compared with that lor electrons. 

Tyson, J. J. and Keener, J. P. (1988). Singular perturbation theory of traveling waves in excitable media (a review). Physica, D 32, 327-61. (December)... 

As we have seen, when X-rays are passed through a gas, only a very few of the electrons in the atoms are knocked out. This is difficult to explain on the wave theory, but follows at once from the particle theory. On the wave theory we should expect all the electrons to be affected by the waves because the waves are continuous and pass over all the electrons present. On the particle theory a photon traveling through the gas goes on between the electrons until it hits one. The small number hit can be attributed to the extremely small size of the electrons and photons. 

In the apparatus shown in Fig. 4 a narrow stream of electrons travels along the tube until it hits the glass. This stream of particles can be deflected by means of electric or magnetic forces as we have seen. On the wave theory the stream of electrons is to be thought of as a narrow beam or ray of waves,... 

The hydrogen electron visualized as a standing wave around the nucleus. The circumference of a particular circular orbit has to correspond to a whole number of wavelengths, as shown in (a) and (b), or else destructive interference occurs, as shown in (c). This model is consistent with the fact that only certain electron energies are allowed the atom is quantized. (Although this idea has encouraged scientists to use a wave theory, it does not mean that the electron really travels in circular orbits.)... 

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