Continuous frequency spectrum

This type of a pattern of singular points is called a centre - Fig. 2.3. A centre arises in a conservative system indeed, eliminating time from (2.1.28), (2.1.29), one arrives at an equation on the phase plane with separable variables which can be easily integrated. The relevant phase trajectories are closed the model describes the undamped concentration oscillations. Every trajectory has its own period T > 2-k/ujq defined by the initial conditions. It means that the Lotka-Volterra model is able to describe the continuous frequency spectrum oj < u>o, corresponding to the infinite number of periodical trajectories. Unlike the Lotka model (2.1.21), this model is not rough since... 

A single pulse or a step function excitation is the basis of relaxation theory. Power dissipation and temperature rise may for instance impede the use of repeated waveforms, and single pulse excitation is necessary. A single pulse is a pulse waveform with repetition interval oo, it has a continuous frequency spectrum as opposed to a line spectrum. The unit impulse (delta function) waveform is often used as excitation waveform. It is obtained with the pulse width 0 and the pulse amplitude oo, keeping the product = 1. The frequency spectrum consists of equal contributions of all frequencies. In that respect, it is equal to white noise (see the following section). Also, the infinite amplitude of the unit pulse automatically brings the system into the nonlinear region. The unit impulse is a mathematical concept a practical pulse applied for the examination of a system response must have limited amplitude and a certain pulse width. 

A periodic waveform occupies a line spectrum an aperiodic waveform occupies a continuous frequency spectrum. 

Figure 8.9 Top Periodic waveform with a line harmonic frequency spectrum. Bottom Nonperiodic waveform has a continuous frequency spectrum. Line spectrum amplitude [volt]. Continuous spectrum amplitude [volt y s ].

This balance is achieved by introducing a finite cavity which reflects all waves perfectly. As well as excluding any energy loss, the cavity guarantees normalizability of all waves. Furthermore, it splits up the continuous frequency spectrum of the Hertzian multipoles. It is a discrete set of coupled ingoing and outgoing modes, which interacts with the particles under consideration. 

Crystals lack some of the dynamic complexity of solutions, but are still a challenging subject for theoretical modeling. Long-range order and forces in crystals cause their spectrum of vibrational frequencies to appear more like a continuum than a series of discrete modes. Reduced partition function ratios for a continuous vibrational spectrum can be calculated using an integral, rather than the hnite product used in Equation (3) (Kieffer 1982),... 

By now it may have dawned on the reader that the long-time Rouse spectrum (i.e., proportionality of xp to p 2) is to be expected for any chain model in which the correlation lengths for both equilibrium conformations and frictional processes are small compared to the chain dimensions (and thus to the wavelength of the slow normal modes). A possible exception is that of the continuous wormlike chain of invariant contour length, which has been studied by Saito, Takahashi, and Yunoki.33 In this latter case, the low-frequency spectrum makes xp proportional to p A, which resembles our special one-dimensional model in the limit 1 — p 1. 

Vibrational Frequency Spectrum of a Continuous Solid.—To find the specific heat, on the quantum theory, we must superpose Einstein specific heat curves for each natural frequency v1y as in Eq. (1.3). Before we can do this, we must find just what frequencies of vibration are allowed. Let us assume that our solid is of rectangular shape, bounded by the surfaces x = 0, x = X, y — 0, y = F, z = 0, z = Z. The frequencies will depend on the shape and size of the solid, but this does not really affect the specific heat, for it is only the low frequencies that art very sensitive to the geometry of the solid. As a first step in investigating the vibrations, let us consider those particular waves that arc propagated along the x axis. 

Microwaves from part of a continuous electromagnetic spectrum that extends from low frequency alternating currents to cosmic rays shown in Table 1. 

The basic operation of the instrument (3) consists of the Ionization of the sample in the cell by a timed electron beam which Is followed, after a short Interval, by an RF pulse applied to the plates of the cell. This pulse coherently excites all ions In the cell into cyclotron motion. The motion continues after cessation of the pulse, and the resonance is detected by the plates of the cell, amplified and the data stored In the computer. The excltatlon/detection cycle is repeated numerous times and the collected data summed. The data Is then subjected to fourler transformation and the frequency spectrum resulting converted into a mass spectrum. The spectrum Is normalized to the major peak. For quantitative work, the calibration can be based either on peak height or peak area. Here, major considerations will Include the resolution chosen and the relative concentrations of the constituents under Investigation. 

A continuous absorption spectrum extending from about 1000 cm-1 to the Cdiam Raman frequency (1332 cm-1) with a peak at 1290 cm-1 (160meV) is also observed in B-containing diamond at RT. This spectrum shows structures... 

In a pioneering study, Stockmeyer (1985) found that the H2 motion in a sample of natural chabazite, predominantly containing calcium ions, was gas-like at room temperature but solid-like below 200 K with a continuous change between the extremes. At low temperature a broad frequency spectrum was derived containing peaks at 48 and 88cm (6 and 11 meV), thought to... 

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