Eigenfrequency harmonic oscillators

The quantity eoo represents the optical permittivity, which is determined by the electronic polarizability. The second term represents the ionic crystal lattice as a sum of N classical harmonic oscillators with eigenfrequencies ujj, damping constants Tj and oscillators strengths Sj. In order to fit Equation (5.7) to the observed far infrared spectra, these parameters are used... 

Here F is a constant of the integration and u>q the eigenfrequency of the oscillator. Figure 14.2 shows the simple trajectory of the harmonic oscillator. 

This Bose approximation amounts to adding to each electronic excited state a sequence of equidistant levels, creating a harmonic oscillator that is, it is Thomson s classical model of elastically bound electrons, with this difference that each point dipole presents many eigenfrequencies associated with the transition dipole.23... 

As one can see the experimental set-up allows to register the initial stage of the bubble oscillation. Here a transient regime of non-established harmonic oscillations is clearly observed for larger frequencies and the measured pressure oscillation is a sum of a non-damped oscillation of the externally applied frequency and a damped oscillation with the meniscus eigenfrequency (for details cf. [208]). At larger frequencies the damped oscillation contribution as shown in Fig. 4.46c is practically absent as the damping time is sufficiently small. 

Fig. 4.46 Pressure oscillations for water at 450 Hz a) -total signal b) established harmonic oscillation with the externally applied frequency, c) damped oscillation with the bubble eigenfrequency...

By means of various methods (Ewins 1984), the respective frequency response Xij a)) can be approximated by the superposition of several (n) adapted harmonic oscillators (so-called modal harmonic oscillators). Their parameters, the modal eigenfrequency co , the associated eigenvector as well as the modal damping 6 , correspond to the modal matrices Mm, Dm, and Cm (Gawronski 2004), but they are not comparable with a real mass, damping, or stiffness. Those modal harmonic oscillators only describe an... 

The inhomogeneous equation (2.109a) shows that the induced dipole moment of the atom interacting with a monochromatic radiation field behaves like a driven damped harmonic oscillator with coha = (Et - Ea)/ h for the eigenfrequency and Yab = Ya + Yb) for the damping constant oscillating at the driving field frequency a>. 

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