Complex frequency

As was mentioned above, the observed signal is the imaginary part of the sum of and Mg, so equation (B2.4.17)) predicts that the observed signal will be tire sum of two exponentials, evolving at the complex frequencies and X2- This is the free induction decay (FID). In the limit of no exchange, the two frequencies are simply io3 and ici3g, as expected. When Ids non-zero, the situation is more complex. 

The specific form of the dependence of the complex frequency Q on parameters of the problem found by Eq. (11.58), is presented as follows ... 

Equation (11.61) has three roots three real, or one real and two complex, depending on the value of determinant D = q - -Pl (Korn and Korn 1968). Since our aim is to determine the complex frequency D, we will consider the complex solution of Eq. (11.61)only. 

Equation (11.99) shows that the effect of heat flux oscillations is not significant in micro-channels with large diameter when the term 4Nu/cf is small enough. Presenting the complex frequency as... 

At 10Hz in a typical Nd-YAG laser 1000Hz/- /Hz, and the typical finesse asymmetry is of the order of one percent. In order to detect a gw signal the laser frequency noise has to be lowered by six orders of magnitudes (compared to the noise of a free running laser), and the two arms made as identical as possible. In order to achieve this complex frequency stabilization methods are employed in all interferometric detectors, and in order to insure the perfect symmetry of the interferometer, all pairs of Virgo optical components are coated during the same run (both Fabry-Perot input mirrors then both end mirrors are coated simultaneously). 

The rate of Au(ffl) reduction should have a correlation with the cavitation efficiency at these frequencies. Therefore, the result of Fig. 5.8 suggests that maximum amounts of reductants are sonochemically formed at 213 kHz in the presence of 1-propanol. The existence of an optimum frequency in the sonochemical reduction efficiency would be explained as follows. As the frequency is increased, the number of cavitation bubbles can be expected to increase. This would result in an increase in the amount of primary and secondary radicals generated and an increase in the rate of Au(HI) reduction. On the other hand, at higher frequencies there may not be enough time for the accumulation of 1-propanol at the bubble/solution interface and for the evaporation of water and 1 -propanol molecules to occur during the expansion cycle of the bubble. This would result in a decrease in the amount of active radicals. Furthermore, the size of the bubbles also decreases with increasing frequency. These multiple effects would result in a very complex frequency effect. 

Treating the free electrons in a metal as a collection of zero-frequency oscillators gives rise51 to a complex frequency-dependent dielectric constant of 1 - a>2/(co2 - ia>/r), with (op = (47me2/m)l/2 the plasma frequency and r a collision time. For metals like Ag and Au, and with frequencies (o corresponding to visible or ultraviolet light, this simplifies to give a real part... 

The combined effects of the dipolar and exchange interactions produce a complex frequency-dependent EPR spectrum, which can however be analysed by performing numerical simulations of spectra recorded at different microwave frequencies. When centre A is a polynuclear centre, the value of its total spin Sa = S, is determined by the strong exchange coupling between the local spins S, of the various metal sites. In this case, the interactions between A and B consist of the summation of the spin-spin interactions between Sb and all the local spins S, (Scheme II). The quantitative analysis of these interactions can therefore yield the relative arrangement of centres A and B as well as information about the coupling within centre A. 

Figure 4. Complex plane of the variable s. The vertical axis Rei is the axis of the rates or complex frequencies. The horizontal axis Imr is the axis of real frequencies to. The resonances are the poles in the lower half-plane contributing to the forward semigroup. The antiresonances are the poles in the upper half-plane contributing to the backward semigroup. The resonances are mapped onto the antiresonances by time reversal. Complex singularities such as branch cuts are also possible but not depicted here. The spectrum contributing to the unitary group of time evolution is found on the axis Re = 0.

As shown in previous papers [10-13], the regularization of the divergences in the series expansion leads to the new parr of transformations At and A . The regularization involves the analytic continuation of real frequencies appearing in C/t into the complex complex plane. At and A are mutually related to each other by complex conjugation of the complex frequencies [in the present case, Z in Eq. (21) below]. 

We discussed in Section 4.3 the electromagnetic normal modes, or virtual modes, of a sphere, which are resonant when the denominators of the scattering coefficients an and bn are minima (strictly speaking, when they vanish, but they only do so for complex frequencies or, equivalently, complex size parameters). But ext is an infinite series in an and bn, so ripple structure in extinction must be associated with these modes. The coefficient cn (dn) of the internal field has the same denominator as bn(an). Therefore, the energy density, and hence energy absorption, inside the sphere peaks at each resonance there is ripple structure in absorption as well as scattering. 

However, this is basically the same as A, since in complex frequency domain... 

This can also be considered in complex-frequency form for which we find... 

For general complex frequencies (still electrically small), we have... 

Equation (2) is, strictly speaking, not suitable for optical fields, which are rapidly varying in time. The damping of the oscillating dipole, and the resultant phase shift, is then conveniently expressed by treating the hyperpolarizabilities as complex, frequency-dependent quantities. For the cubic hyperpolarizability, the relation between the Fourier components of the electric field and the Fourier amplitude of the oscillation of the electric dipole gives... 

If phase-sensitive detection is used then x Ab can be found from the part of applied field (dispersion) and x" a b can be found from the part of that oscillates 90° out of phase with the applied field (absorption). In practice it is unnecessary to measure both X and x because there exists a theoretical relationship between x and x" so that a determination of one member of the pair uniquely determines the other member of the pair. It follows from this fact that the complex function, %(k, oo) — x"(k, oo), is analytic and vanishes on an infinite semicircle in the lower half of the complex frequency plane (z-plane) and that x a b and X" ab are related through the Kramer s Kronig relations... 

For linear, time-invariant systems a complete characterization is given by the impulse or complex frequency response [Papoulis, 1977], With perceptual interpretation of this characterization one can determine the audio quality of the system under test. If the design goal of the system under test is to be transparent (no audible differences between input and output) then quality evaluation is simple and brakes down to the... 

For systems that are nearly linear or time-variant, the concept of the impulse (complex frequency) response is still applicable. For weakly non-linear systems the characterization can be extended by including measurements of the non-linearity (noise, distortion, clipping point). For time-variant systems the characterization can be extended by including measurements of the time dependency of the impulse response. Some of the additional measurements incorporate knowledge of the human auditory system which lead to system characterizations that have a direct link to the perceived audio quality (e.g. the perceptually weighted signal to noise ratio). 

Dimensionless collision frequency (for electrolyte solutions Normalized collision frequency Normalized complex frequency (for electrolyte solutions)... 

We have introduced here a dimensionless energy h = H/(kBT) and the complex frequency... 

Here we introduce reduced quantities the complex frequency z, related to co by a microscopic time scale r, and the normalized concentration G ... 

The absorption coefficient in the FIR spectral region is proportional to the product xIm[L(z)]. Here the spectral function (SF) L(z) is the linear-response characteristic of the model under consideration, where the dimensionless complex frequency z is related to angular frequency co of radiation and mean lifetime x as follows ... 

Substituting the series (390a) and (390b) into integral (394) and introducing a nondimensional function S of the reduced complex frequency Z, one can get the result for S(Z) in an analytical form [5] ... 

At small complex frequency Z, such that Zd simple formula for the complex conductivity, analogous to Eq. (405) ... 

We account for only the torque proportional to the string s expansion AL, which produces the main effect considered in this work. For calculation we employ the spectral function (SF) Lstr(Z), which is linearly connected with the spectrum of the dipolar ACF (see Section II), with Z x Y being the reduced complex frequency. Its imaginary part Y is in inverse proportion to the lifetime tstr of the dipoles exerting restricted rotation. The dimensionless absorption Astr is related to the SF Lstr as... 

For temporally damped waves, the complex wave vector k is just the real part k as the spatial damping coefficient p is equated to zero. Conversely, for spatially damped waves, the complex frequency co is just the real frequency co0,... 

---