Rayleigh method

In the Rayleigh method of carrying out a dimensional analysis the dependent variable is assumed to be proportional to the product of the independent variables raised to different powers. By equating dimensions, the number of independent dimensionless groups and one set of their possible forms can be obtained. By way of illustration two examples may be considered. 

Using these variables and the Rayleigh method, the resulting equation is... 

Our hypothesized set of variables that is believed to govern particle dynamics in tiunbling blenders is shown in Table 1. Using these variables and the Rayleigh method, we derive the following equation ... 

The same procedure is performed for solving the Equation (8). Damping matrix of the building is also calculated using the well known Rayleigh method. 

The fundamental frequency of the tower is computed by the Rayleigh method. Hence the dynamic deflection is assumed, not to differ appreciably from the static deflection DEL ... 

The Rayleigh method for natural frequencies was chosen for the computation of the fundamental frequency of a tower structure because of its considerable accuracy and ease. However, this method can be modified for the computation of a few of the lower modes frequencies. 

Therefore, it can be concluded that the Rayleigh method is a very accurate method for the determination of the fundamental frequency. Other methods stated in the text can also be applied. They can be expensive and yet very elaborate result wise. 

The method established by iterating (11.5.19) is known as the inverse-iteration method with the Rayleigh quotient or simply as the Rayleigh method [7,9]. As should be clear from our discussion, the Rayleigh method is just Newton s method applied to the eigenvalue problem (11.4.28). 

The Schroedinger equation for this potential can be solved by a number "of very nearly exact approaches. The most convenient appears to be the Rayleigh method. The total wavefunction is expanded in the series... 

Fig. 2 shows the CFRP-sandwich specimen and the transducer mounted on the scanner. Fig. 23 presents a C-scan of the specimen as first interesting result. Only the defects visible from the outside are indicated. The distance between transducer and specimen was smaller than the focal length, so that the angle of incidence at the edge of the sound beam converts the longitudinal waves to Rayleigh-waves in the specimen. These waves provide a very sharp image of the surface. This method opens the possibility for a non-contact acoustic microscope. 

Claverie P 1971 Theory of intermolecular forces. I. On the inadequacy of the usual Rayleigh-Schrddinger perturbation method for the treatment of intermolecular forces Int. J. Quantum Chem. 5 273... 

The first temi results in Rayleigh scattering which is at the same frequency as the exciting radiation. The second temi describes Raman scattering. There will be scattered light at (Vq - and (Vq -i- v ), that is at sum and difference frequencies of the excitation field and the vibrational frequency. Since a. x is about a factor of 10 smaller than a, it is necessary to have a very efficient method for dispersing the scattered light. 

The second method to calculate the scattered intensity or R the Rayleigh ratio) is to square the sum in I... 

MacDonald J K L 1933 Successive approximations by the Rayleigh-Ritz variation method Phys. Rev4Z 830-3... 

Her and Plesset proposed an alternative way to tackle the problem of electron correlation tiler and Plesset 1934], Their method is based upon Rayleigh-Schrddinger perturbation 3ty, in which the true Hamiltonian operator is expressed as the sum of a zeroth-er Hamiltonian (for which a set of molecular orbitals can be obtained) and a turbation, "V ... 

By using vapor-liquid equilibrium data the above integral can be evaluated numerically. A graphical method is also possible, where a plot of l/(y - xj versus Xr is prepared and the area under the curve over the limits between the initial and fmal mole fraction is determined. However, for special cases the integration can be done analytically. If pressure is constant, the temperature change in the still is small, and the vapor-liquid equilibrium values (K-values, defined as K=y/x for each component) are independent from composition, integration of the Rayleigh equation yields ... 

Residual minimization method (RMM-DIIS). Wood and Zunger [27] proposed lo minimize the norm of the residual vector instead of the Rayleigh quotient. This is an unconstrained minimization condition. Each minimization step starts with the evaluation of the preconditioned residual vector K for the approximate eigenstate... 

Klahn, B. and Bingel, W. A. [19 77] The Convergence of the Rayleigh-Ritz Method in Quantum Chemistry , Theoretica Chimica Acta, 44, p. 9. 

What is really difficult is to ascertain whether a given differential equation has a limit cycle solution. In fact, aside from a few well-known equations (of van der Pol, Rayleigh, Li6nard, etc.) we hardly know anything at all about the existence of cycles directly, that is, on the basis of the topological methods with which we are concerned here. 

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... 

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