Rayleigh’s theorem

The previous section dealt with a horizontal scale change in one domain resulting in both horizontal and vertical changes in the other domain. Rayleigh s theorem, on the other hand, makes a statement about the area under the squared modulus. This integral, in fact, has the same value in both domains ... 

U = 0, where y is the wall-normal co-ordinate inside the shear layer. A stronger version of the Rayleigh s theorem was given later by Fj rtoft (1950). 

From each bulk band, at least one surface mode originates, as shown in Figure 9.46. According to Rayleigh s theorem [82], the number of localized surface modes from each bulk band is given by the number of degrees of freedom, which are affected by the surface perturbation. For an ideal bulk-terminated surface, as in the model calculation of Figure 9.46, no more than one surface phonon mode is expected for each bulk band. 

Rayleigh s Inflection Point Theorem A necessary condition for instability is that the basic velocity profile should have an inflection point. 

Thus, if the velocity profile is a monotonically growing function of its argument with a single inflection point then the above necessary condition for instability can be written for this velocity profile as U" U — Ug) < 0 for the range of integration, with equality only at y = y. Both the Rayleigh s and Fj rtoft s theorem are necessary condition and they do not provide a sufficient condition for instability. 

The last section in this chapter is a brief introduction to stability of parallel shear flows. We consider three basic issues (i) Rayleigh s equation for inviscid flows, (ii) Rayleigh s necessary condition on an inflection point for inviscid instability, and (iii) a derivation of the Orr-Sommerfeld equation and Squire s theorem. 

Observables calculated from approximate wavefunctions as in Equation 1.13 are called expectation values, an expression used in probability theory. In practice, we will always have to be satisfied with approximate wavefunctions. How can we choose between different approximations And if our trial wavefunction has adjustable parameters (such as the coefficients of atomic orbitals in molecular orbitals see Section 4.1), how can we choose the adjustable parameters best values Here, Rayleigh s variation theorem is of great value. It tells us that the expectation value for the ground state energy E, (E ), calculated from an approximate wavefunction (P is always larger than the true energy E (Equation 1.14). Proof of the variation theorem is given in textbooks on quantum mechanics.18... 

The dimensionless equation describing the transfer phenomena may be obtained either by direct reference to the ratios of the physical quantities or by recourse to the classical techniques of dimensional analysis, i.e., the Buckingham n Theorem or Rayleigh s method of indices. In addition, the basic differential equations governing the process may be reduced to dimensionless form and the coefficients identified. In general, the dimensionless equation for heat transfer through the combined film is... 

There are two classical methods in dimensional analysis, Buckingham s pi theorem and the method of indices by Lord Rayleigh. Here we will briefly explain the more common of the two Buckingham s theorem. 

Rayleigh, 1899. On James Bernoulli s theorem in probabilities, Phil. Mag., 47, 246-251. 

The Fourier integral transformation as formulated in Eqs. 1 and 2 has the mathematical property (known as Rayleigh s or Parseval s theorem)... 

Brief reflection on the sampling theorem (Chapter 1, Section IV.C) with the aid of the Fourier transform directory (Chapter 1, Fig. 2) leads to the conclusion that the Rayleigh distance is precisely two times the Nyquist interval. We may therefore easily specify the sample density required to recover all the information in a spectrum obtained from a band-limiting instrument with a sine-squared spread function evenly spaced samples must be selected so that four data points would cover the interval between the first zeros on either side of the spread function s central maximum. In practice, it is often advantageous to place samples somewhat closer together. 

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