Helmholtz frequency

The derivation of equation (81) is invalid unless the resulting Helmholtz frequency cojj is small compared with the lowest natural frequency of acoustic waves in the chamber, since otherwise an acoustic field with a nonuniform p would have to be considered a related restriction limits the length I of the exit tube. If I is comparable with a characteristic length of the chamber, then a representative acoustic frequency is a/l, and the ratio o)fj/(ci/l) is the square root of the ratio of the volume of the tube to that of the chamber, which is small for small tubes in large chambers—the configuration envisaged in the Helmholtz model. 

In Equation (5.58) the outer summation is over the p points q which are used to sample the Brillouin zone, is the fractional weight associated with each point (related to the volume of Brillouin zone space surrounding q) and vi are the phonon frequencies. In addition to the internal energy due to the vibrational modes it is also possible to calculate the vibrational entropy, and hence the free energy. The Helmholtz free energy at a temperature... 

Resonant Sound Absorbers. Two other types of sound-absorbing treatments, resonant panel absorbers and resonant cavity absorbers (Helmholtz resonators), are used in special appHcations, usually to absorb low frequency sounds in a narrow range of frequencies. Resonant panel absorbers consist of thin plywood or other membrane-like materials installed over a sealed airspace. These absorbers are tuned to specific frequencies, which are a function of the mass of the membrane and the depth of the airspace behind it. Resonant cavity absorbers consist of a volume of air with a restricted aperture to the sound field. They are tuned to specific frequencies, which are a function of the volume of the cavity and the size and geometry of the aperture. 

The simplest resonant absorber is known as the Helmholtz resonator. This consists of a chamber connected to the duct (or whatever area is to be controlled) by a narrow neck. The volume of air in the chamber will resonate at a frequency determined by the volume of the chamber, the length of the neck and the cross-sectional area of the neck ... 

The degree of attenuation at the critical frequency can be very large, but this type of silencer has a very narrow bandwidth. This device may be suitable when the machine being dealt with emits sound predominantly of a single wavelength. Lining the chamber with absorbers can expand the absorber bandwidth of a Helmholtz resonator, but this has the effect of reducing the efficiency. The perforated absorber, which forms the basis of many acoustic enclosures and silencers, is a development of the resonator principle. 

The A defined in Equation 5.30 is not to be confused with the Helmholtz free energy. Should the A frequencies be limited to the external hindered translations and rotations, vi g = vi g = 0, and this is an additional simplification. In some molecules, however, there are isotope sensitive low lying internal modes (often internal rotations or skeletal bends). In that cases both terms in Equation 5.30 contribute. 

An electrode bears a layer of indium-tin oxide (ITO) having an impedance of 25 Q, on which is a layer of adsorbed chromophore having an impedance of 1.0 (all values of Z being cited at fixed frequency). In addition, between the chromophore layer and the bulk electrolyte is the Helmholtz double-layer (see Section 5.1.2), which has an impedance of 120 By assuming that these three layers act as impedances in parallel, calculate the total impedance, Ziotai-... 

One complication which may be present, when the Helmholtz model is in other respects appropriate, is that of specific adsorption. If one of the mobile species is to some extent chemically bound rather than being simply electrostatically bound to the metal electrode, Cji may show a dependence on the dc bias potential. Indeed this is the normal method of inferring specific adsorption. Another possibility in this case is that dl exhibits different high frequency and low frequency limits because at high frequencies the specific adsorption being an activated process is too slow to follow changes in interface potential. A further complication which is often present in real systems is the presence of an oxide layer on the surface of the metal electrode. Such an oxide layer can generate a potential... 

The interface impedance for a case such as Ag/Ag4Rbl5 will consist of a capacitance (derived from the Helmholtz formula) in parallel with i et so that in the complex plane impedance a semi-circle will be found from which Qi and can be evaluated. Rq will cause this semicircle to be offset from the origin by a high frequency semicircle due to the bulk impedance between the interface and the reference electrode (Fig. 10.12). There can be no Warburg impedance (a line at 45° to the real axis generally due to diffusion effects) in this case. 

Based on equation 11.41, the difference between the Helmholtz free energies of formation of two isotopic molecules with respect to their gaseous atoms depends on the shift of vibrational frequencies between heavy and light isotope-bearing compounds—i.e., according to Bigeleisen and Mayer (1947),... 

At low temperatures and light frequencies, the separative effect per unit shift (the terms in brackets in eq. 11.44 and 11.45) approaches (figure 11.9) and the Helmholtz free energy difference approaches the differences in zero-point energies. At high T (low frequencies), the separative effect per unit shift approaches zero and the total separative effect sis ")/approaches 1, so that no isotopic fractionation is observed. 

To determine the origin of the instability frequency acoustic analysis was performed which revealed that both the quarter-wave mode of the inlet and the Helmholtz mode of the combustor-inlet system occurred at 35 Hz. The phase... 

Table 29.3 lists the new cavity dimensions that resulted in well-organized oscillations. The fundamental resonance frequencies for these cavities ranged between 20 and 40 kHz for the coupled convective-acoustic mode and 6 and 9 kHz for the Helmholtz mode. As before, the dominant frequencies included not only the fundamental mode frequencies, but also many higher harmonics and overtones. 

The difference between the magnetron and other vacuum tubes is that the electron flow passes along a spiral this route is created by an external magnetic field B (Fig. 3.4). The electron cloud produces resonance cavities several times in its trip to the anode. These cavities work as Helmholtz resonators and produce oscillations of fixed frequency, determined by the cavity dimensions small cavities produce higher frequencies, large cavities smaller frequencies. The antenna in the right zone collects the oscillations. 

Here, ez roc is the zero-point energy of the two soft rocking vibrations mentioned above cot the frequency of the normal mode of AB/s/ corresponding to the reaction coordinate a, aa/s/> and aB the Helmholtz free energies of the internal degrees of freedom of AB/s/, A/s/, and B amc the Helmholtz free energy of the two soft rocking vibrations mentioned above. 

To date, two arrangements of NMR coils, the solenoidal radio frequency (RF) coil (Figure 2A) and the saddle-type (Helmholtz) RF coil (Figure 2B) have been employed as on-line NMR detectors with CE and CEC. Theoretical studies have shown that reduction of the diameter of the RF coils increases the coil sensitivity [32], The miniaturized versions of saddle types are commonly used in commercial probes. As a major development, a saddle coil which houses 1.7-mm-diameter sample tubes has been introduced by Varian. Another significant contribution is the designing of an inverse coil to accommodate 3-mm-diameter sample tubes with a detection volume of 60 pL and a total volume of 140 pL [33], Fabrication procedures hinder further reduction of diameter of saddle-type coils that are optimized for sample volumes smaller than —1 pL. 

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