Kirchhoff transformation

The above equation can be reduced to a linear differential equation by the introduction of a new temperature 0 related to the temperature T by the Kirchhoff transformation [4,11] ... 

According to the superposition theorem of system theory for linear responses, this response to a step-function in the current can be employed to deduce the impedance behavior. As regards a qualitative discussion, one can adopt the above description by just replacing short/long times by high/small frequencies. Quantitatively the impedance is given by a Laplace transformation of Eq. (64) (or equivalently by applying Kirchhoff s laws to the equivalent circuit (Eq. (63))) with the result... 

Hence, the extremum L(x) is the point of minimum. Thus, the problem of entropy maximization is transformed into the problem of heat minimization and the Kirchhoff and Prigogine theorems result from the extension of the second law to the passive isothermal circuits. The graphical interpretation of problem (21) is given in Figure 3b. 

If the transformation pathway cannot be reduced to monomolecular reactions, nonunit stoichiometric coefficients may appear at some junction points of the kinetic resistors. In terms of electric circuits, this means that the absence of the balance of the current inflow and outflow at this June tion point may cause norJinearity and deviations from the canonical form of the KirchhofF equation. 

Lieberman, H., Kirchhoff, H., Gliksman, W., Loewy, L., Gruhn, A., Hammerich, T. H., Anitschkoff, N. and Schulze, B. (1935). Transformation products of succinyl-succinic esters. VI. Formation of quinacridones from p-diarylaminoterephthalic acids. Annalen, 518, 245-59. [259]... 

A similar result holds true for the case shown in Figure 13-4b. Applying the inverse Fourier transform to both sides of the last expression, we cirrive at the Kirchhoff integral formula for an unbounded domain ... 

Applying the Fourier transform to both sides of this equation, we can obtain the corresponding Kirchhoff formula in the frequency domain. According to the convolution theorem (Arfken and Weber, 1995), the Fourier transform of the convolution of two functions is equal to the product of spectra of these functions. Therefore we obtain ... 

Therefore, the auxiliary field (r,f ) is the back-propagated field, and the adjoint Kirchhoff operator K for a residual field can be calculated by back propagation of the residual field. We will show in the next sections that this procedure is similar to Stolt s Fourier-based migration transformation. Thus, this result demonstrates that migration is similar to applying an adjoint Kirchhoff operator to observed scattered wavefield data. 

Another method of practical realization of the Kirchhoff type reverse-time migration is based on the Rayleigh integral formula in the frequency domain (15.200). Applying an inverse Fourier transform to both sides of the Rayleigh formula, we obtain for f = 0... 

Figure 6a shows the transmission hne representing a viscoelastic layer [64]. Every layer is represented by a T . The apphcation of the Kirchhoff laws to the Ts reproduces the wave equation and the continuity of stress and strain. The detailed proof is provided in [4]. To the left and to the right of the circuit are open interfaces (ports). These can be exposed to external shear waves. They can also be connected to the ports of neighboring layers (Fig. 6b). Alternatively, they may just be short-circuited, in case there is no stress acting on this surface (left-hand side in Fig. 6c). Finally, if the stress-speed ratio Zl (the load impedance, see below) of the sample is known, the port can be short-circuited across an element of the form AZl, where A is the active area (right-hand side in Fig. 6c). Figure 6c shows a viscoelastic layer which is also piezoelectric. This equivalent circuit was first derived by Mason [4,55]. We term it the Mason circuit. The capacitance, Co, is the electric capacitance between the electrodes. The port to the right-hand side of the transformer is the electrical port. The series resonance frequency is given by the condition that the impedance of the acoustic part (the stress-speed ratio, aju) be zero, where the acoustic part comprises all elements connected to the left-hand side of the transformer.

When applying the Kirchhoff laws to such a network, one finds the same resonance conditions as with the mathematical and the optical approaches. Why should one bother going through these transformations if the results are the same There are important benefits tied to the use of equivalent circuits ... 

Spatial information, which is described by an electric field f(x, y, 0) at z = 0 in Cartesian coordinates, is transformed while propagating in free space by a relation obeying the well-known Fresnel-Kirchhoff integral. 

The Fresnel-Kirchhoff (FK) integral for transformation of a light beam through free space and a GI medium is discussed. Here we shall describe how a Gaussian beam changes when propagating in free space and in a GI medium. 

In 1833, Payen (with Persoz) found that the transformation of starch into dextrin and sugar by malt (discovered by Kirchhoff) was due to a substance, diastase, extractable with water from germinated barley. They purified the material by repeated precipitation with ethanol and found that the activity was destroyed at lOO C (3, 39). 

Linear differential equation of first order called the heat balance equation of a simple body, has found wide application in calorimetry and thermal analysis as mathematical models used to elaborate various methods for the determination of heat effects. It is important to define the conditions for correct use of this equation, indicating all simplifications and limitations. They can easily be recognized from the assumption made to transform the Fourier-Kirchhoff equation into the heat balance equation of a simple body. 

Since the first Piola-Kirchhoff stress II is not symmetric as understood by (2.110), we introduce a symmetrized tensor T, called the second Piola-Kirchhoff stress, and the Euler stress t, which is the transformed tensor of T, into the deformed body using the rotation tensor R ... 

Variations in the heat of transformation at constant pressure with changing temperature - Kirchhoff reiation... 

This relation constitutes what we call the Kirchhoff relation. An equivalent relation would give the variation of the heat of transformation at constant volume with the temperature as a function of the molar specific heat capacity at constant volume associated with the transformation. 

A physical Lagrangian stress tensor is defined and established by applying vector transformation to the second Piola Kirchhoff stress tensor 11 components using equation (27), such that ... 

---