Rayleigh oscillations

The most prominent feature in most V(z) curves, after the central maximum at focus, is the series of oscillations at negative defocus associated with Rayleigh wave excitation. It is perhaps therefore not surprising that the most accurate information in the reconstructions of R 6) concerns the Rayleigh velocity. The period of the Rayleigh oscillations is... 

Hence, by measuring the period of the Rayleigh oscillations, the Rayleigh velocity may be deduced directly. Likewise, the exponential decay of the Rayleigh oscillations is often of the form (Kushibiki et al. 1982)... 

The foregoing discussion shows that complicated facts require a continuous readjustment of the analytical theory. Historically, the theory of nonlinear oscillations progressed precisely in this manner in the hands of the early pioneers—Lord Rayleigh, van der Pol, Appleton, and others. The following sections give a brief account of some of these investigations. 

It is well-known, for instance, that if one applies tension at properly timed intervals to pull a stretched wire or string, it begins to oscillate laterally. Lord Rayleigh performed an experiment of this kind by attaching a stretched wire to a prong of a tuning fork when the latter... 

Rate of change of observables, 477 Ray in Hilbert space, 427 Rayleigh quotient, 69 Reduction from functional to algebraic form, 97 Regula fold method, 80 Reifien, B., 212 Relative motion of particles, 4 Relative velocity coordinate system and gas coordinate system, 10 Relativistic invariance of quantum electrodynamics, 669 Relativistic particle relation between energy and momentum, 496 Relativistic quantum mechanics, 484 Relaxation interval, 385 method of, 62 oscillations, 383 asymptotic theory, 388 discontinuous theory, 385 Reliability, 284... 

Combustion-generated noise is a problem in itself. However, if an acoustic wave can interact with the combustion zone, so that the heat release rate is a function of the acoustic pressure, q = f p ), then Equation 5.1.14 describes a forced oscillator, whose amplitude can potentially reach a high value. The condition for positive feedback was first stated by Rayleigh [23] ... 

U nlike Rayleigh s original example of a collapsing empty cavity, this bubble will reduce to a minimum size, on compression, after which it will expand to Rj and subsequently it will oscillate between the two extremes R and Rf in. Obviously at the two extremes of radii, motion of the bubble wall is zero - i. e. R = 0. To determine these radii it is necessary to integrate Eq. A.25. With Z = (R /R), the integration yields ... 

Rayleigh criterion chem The criterion for spontaneous pressure oscillations to accompany combustion, namely, that combustion progresses more rapidly or efficiently during the compression phase of the pressure oscillation than during the rarefaction phase. ra-le krT,tir-e-3n ... 

There is a limit to the charge that the surface of a droplet can sustain, for the electrical stress generated by the surface charges can balance the surface tension forces. At that point the droplet becomes unstable and breaks up. Lord Rayleigh (1882) analyzed the criterion of instability for a conducting droplet using spherical harmonics to describe the modes of oscillation. The natural frequency of the nth mode of oscillation of a droplet was found to be given by... 

Extensive reviews of active instability suppression techniques are found in McManus et al. [18] and Candel [19]. Also, Zinn and Neumeier [20] provide an overview of research and developmental needs for practical applications. Most of the previous studies have used actuators impractical in liquid-fueled systems, such as loudspeakers that impose acoustic perturbations on gaseous flow. The major emphasis in the present study was to establish active instability suppression using liquid-fuel injection. According to Rayleigh s criterion [21, 22], combustion-acoustics interaction can be used to damp the undesirable oscillations provided that pressure fluctuations p and heat release fluctuations q satisfy the proper phase relation such that... 

Thus, at negative defocus, the amplitude of the Rayleigh ray will vary as exp —2z(ao sec0R — aRtan0R). If the Rayleigh contribution to V(z) is smaller than the geometrical contribution, the amplitude of the oscillations in V(z) will follow the same exponential decay. 

There are two particular discontinuities in R(t) that are of great importance for materials in which Rayleigh waves are excited. The first occurs at t0 = 1 /n, because beyond that value Q(f) changes discontinuously to zero. The second is at fR = cos r/tt, because around the Rayleigh angle 6r there is a phase change of 2n in R 6), cf. Fig. 6.3(b)i, and hence in Q(t). The Fourier relationship gives oscillations in V u) of periodicity... 

There is a direct analogy with the fringe pattern that is seen in a Young s double slit experiment, in which the diffraction pattern from two slits produces periodic fringes whose spacing varies inversely with the separation of the slits. The oscillations can also be interpreted in terms of the distortions of the reflected wavefronts in Fig. 7.2 at the Rayleigh angle (Atalar 1979). 

Both the geometrical contribution and the Rayleigh contribution can thus be expressed in terms of material constants and the geometry of the lens, and therefore be directly compared. The complex summation of (7.39) and (7.42) enables V(z) to be computed and leads immediately to oscillations with period... 

Anyone who has successfully used a microscope to image properties to which it is sensitive will sooner or later find himself wanting to be able to measure those properties with the spatial resolution which that microscope affords. Since an acoustic microscope images the elastic properties of a specimen, it must be possible to use it to measure elastic properties both as a measurement technique in its own right and also in order to interpret quantitatively the contrast in images. It emerged from contrast theory that the form of V(z) could be calculated from the reflectance function of a specimen, and also that the periodicity and decay of oscillations in V(z) can be directly related to the velocity and attenuation of Rayleigh waves. Both of these observations can be inverted in order to deduce elastic properties from measured V(z). 

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