Traveling wave transformation

Let us discuss modal current (voltage) distribution on a completely transposed line. It should be noted that modal voltage distribution is the same as that of current in the completely transposed line case because the impedance and admittance matrices are completely symmetrical. Assume that the phase currents are 1, and 1 as illustrated in Figure 1.20. Using traveling-wave transformation, we obtain the following relation ... 

It is observed that the frequency dependence of A2 in Table 1.4 is less than 10% for the range of frequencies from 100 Hz to 1 MHz. The change is small compared with the parameters explained earlier , thus, the frequency-dependent effect of the transformation matrix in the case of an untransposed horizontal line can be neglected. Then, the following approximation is convenient because it agrees with the traveling-wave transformation of Equation 1.179, explained in Sections 1.4.4.1 and 1.4.4.2 ... 

In the case of a completely transposed three-phase line, any of the transformation matrices explained in Section 1.4.4.1 can be used. The current transformation matrix is the same as the voltage transformation matrix. Let us apply the traveling-wave transformation ... 

Stationary, traveling wave solutions are expected to exist in a reference frame attached to the combustion front. In such a frame, the time derivatives in the set of equations disappear. Instead, convective terms appear for transport of the solid fuel, containing the unknown front velocity, us. The solutions of the transformed set of equations exist as spatial profiles for the temperature, porosity and mass fraction of oxygen for a given gas velocity. In addition, the front velocity (which can be regarded as an eigenvalue of the set of equations) is a result from the calculation. The front velocity and the gas velocity can be used to calculate the solid mass flux and gas mass flux into the reaction zone, i.e., msu = ps(l — e)us and... 

Note that a travelling wave solution is related to a similarity solution via the following known transformation ... 

It seems curious to ask what sort of a travelling wave is obtained when a transformation inverse to (3.2.18) is applied for m > 1 in particular, we ask what is the wave parallel of the analogue of compact support.)... 

DPPH = 2,2-diphenyl-1-picrylhydrazyl ENDOR= electron-nuclear double resonance EPR = electron paramagnetic resonance ESE = electron spin echoes ESEEM = electron spin echo envelope modulation EFT = fast fourier transformations FWHM = fidl width at half maximum HYSCORE = hyperfine sublevel correlation nqi = nuclear quadrupole interaction TauD = taurme/aKG dioxygenase TWTA = traveling wave tube amphfier ZFS = zero field sphtting. 

In this chapter, we describe the technique of Fourier transform microwave spectroscopy. We distinguish here two rather different types of sample absorption cells which require somewhat different theoretical descriptions. First, we describe the theory for the relatively broad-band waveguide absorption cell in which the radiation is described as a traveling wave. Second, we describe the narrow-band Fabry-Perot cavity absorption cell in which the radiation is described as a standing wave. 

In this section, we replace the broadband waveguide absorption cell with a narrow-band Fabry-Perot cavity. The traveling wave is then replaced with a standing wave. We consider a static gas polarization and subsequent coherent emission in the Fabry-Perot cavity.7.8 However, the use of a Fabry-Perot cavity and the pulsed Fourier transform microwave method is also well-suited for the measurement of the resonant transitions of transient or otherwise short-lived species. 

Structural characterization of proteins and peptides using quadrupole ion trap mass spectrometry, Fourier transform-ion cyclotron resonance (FT-ICR) mass spectrometry, and traveling wave ion mobility mass spectrometry... 

Part 2. Ion Conformation and Structure presents discussions of structural characterization of proteins and peptides using quadrupole ion trap mass spectrometry, Fourier transform ion cyclotron resonance mass spectrometry, and the novel method known as traveling wave ion mobility mass spectrometry. In addition to the observation of collective fluctuations of the molecular substructures within biomolecules, the organization of atoms in small ion clusters is investigated using electron diffraction. 

One of the ways to get an analytical solution of the Stnoluchowski equation is to use some transformations, which reduce the Smoluchowski equation into Pick s equation [34,35], The interesting fact is that for sufficiently large V , the solution of the Smoluchowski equation behaves like a travelling wave , i.e. it starts to behave like a solution of the following equation ... 

Although the basic principles of ESR and NMR are similar, practical difficulties mean that pulse methods are less useful in ESR. This is because the pulse power required to produce the frequency span of a typical ESR spectrum would be several kilowatts and the pulse very short (nanoseconds). The pulse or FT (Fourier transform) ESR spectrometer is usually based on a standard CW instrument because it is often useful to record a standard ESR spectrum before carrying out pulse experiments. The pulse microwave source is usually a travelling wave tube other essentials... 

Travelling waves with a constant velocity u on an unbounded interval can be studied upon coordinate transformation — mt which brings the partial differential equations (1)... 

We have shown that multiple travelling front waves can occur in a reaction-diffusion-convection system. These waves can be studied in an unbounded system by using a wave transformation and solving a special boundary value problem with the use of continuation methods. These results provide various parameter dependences of the velocity of the wave. Moreover, in a bounded system the waves move back and forth through the system and form remarkable zig-zag patterns. 

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