Small-angle approximation

The convergence power spectrum in the small angle approximation... 

We can now calculate the deflection angle to the first diffraction order in the small angle approximation as the ratio of the wavelength and the grating constant,... 

Nonlinearity makes the pendulum equation very difficult to solve analytically. The usual way around this is to fudge, by invoking the small angle approximation sin X X for x 1. This converts the problem to a linear one, which can then be solved easily. But by restricting to small x, we re throwing out some of the physics, like motions where the pendulum whirls over the top. Is it really necessary to make such drastic approximations ... 

Do you remember the first nonlinear system you ever studied in school It was probably the pendulum. But in elementary courses, the pendulum s essential nonlinearity is sidestepped by the small-angle approximation sin 6 d. Enough of that In this section we use phase plane methods to analyze the pendulum, even in the dreaded large-angle regime where the pendulum whirls over the top. 

Q.28.3 The small angle approximation states that for small angles of incidence, the sine of H can be replaced by 6 itself. Calculate and define for which small angles this statement is a reasonable simplification. 

However, as the calculations show, the contribution to the value of introduced by the additional terms involving a and / is less than -2°. Furthermore, the values of must be increased by °-1° with respect to those calculated from Eq. (4.2) in any case, because of limited accuracy in the adjustment of the wedge in the sample compartment of the spectrometer. Therefore, to calculate the angle Eq. (4.1) is recommended, and in the small-angle approximation (sin a Pa a, cos a 1), it may be rewritten as... 

Paraxial approximation The approximation that all rays are so nearly parallel to the axis that the small-angle approximation may be used. In geometric optics, imaging is perfect in the paraxial approximation. 

The effective chi-parameter is given by the k-dependent generalization d Eq. (6.17). If a small angle approximation where )CiNc(k) is constant is invdred, then Eq. (9.S) is identical in form with Leibler s RPA result [87] and the spinodal condition is far simpler than Eq. (9.4). However, since the true chi- Kiranieter is a wavevector-dependent correlation function, not a phenomenological number, it is functionally related to all the other intramolecular and intermolecular pair correlations in the system. This non-mean-field feature has many important consequences such as the fact that k" is influenced by many chain correlations [86,88]. It must be emphasized that although Eq. (9.5) should be accurate for the hypothetical symmetric block copolymer model, since it does not properly treat compressibility effects it is expected to be inadequate for most real copolymer systems. 

Returning to the definition of the force [equation (6.1)] and in the small angle approximation (cos0 1 — 6 /2), we have... 

Even if you re not impressed by that, notice that these equations prove the small angle approximations... 

On the other hand, cyclization to TM 7.8 requires the approach of the nucleophile under a small angle, approximately 90°, and the endb-position of the double bond, both stereoelectronically unfavorable. Generally, the Baldwin rules state that cyclization to a five-membered ring is favored when the reacting double bond on the trigonal C-atom is exo and disfavored when it is endo. In Table 7.1, the Baldwin... 

The mathematical treatment of the dynamics of the Williams domains is a difficult problem. However, reasonable assumptions (small angle approximation, exponential time response to a stepwise change in voltage) lead to expressions for the time constants which are in quaUtative agreement with experiment ... 

Thus, director fluctuations in the small angle approximation can contribute to the spectral density Ji uj) through the Qi q (r) term where Sq = refers to ordering of the molecule with respect to the local director. This small angle approximation is probably not warranted for low orientational orders, where might be as large as 40°. While g o( )... 

Eq. 20 is derived from Eq. 8. Following the small-angle approximation theory , the photons scattered into the angles outside the diffraction peak were considered to be absorbed (the effective single- scattering albedo A - 0.5). 

The evaluation of the integral over the trajectory is done by breaking the trajectory into two identical contributions, up to the point of closest approach (which, to be consistent with the small angle approximation, we take to be at = Zt) and from that point on. The integral / At AV /di )x I can be evaluated exactly for the long-range part of the potential, —Cs/R. Its approximate value is V b). Putting E-y = /u.v /2 we have the simple final result... 

Such theoretical methods for estimating the selfnoise of an instrument have been applied to a number of seismometers in conjunctirm with their digitizers (Rodgers 1994). However, when such methods are used, a number of assumptions must be made (e.g., that the pendulum obeys the small-angle approximation and all electronic components perfectly match their specified noise levels). 

For small angle approximation, the above equations can be linearized to yield... 

The inequality (6.35) 1 was first derived by Carlsson et al [32] for the special case when the smectic tilt angle 0 is assumed small (see the small angle approximations... 

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