Wave, standing

There are two trigonometric terms in this expression. The first term, cos kx, is a function of a point s coordinates only and can be considered as a variable amplitude of the standing wave changing from point to point, i.e.. 

The antinodes coordinates can be found from eq. (2.9.4). Indeed, the point s coordinate in which oscillation acquires the maximum displacement satisfies equation kx= nn with n = 0, 1, 2,. .. So we can obtain the antinodes coordinates  

The distance between adjacent nodes (or antinodes) is referred to as the standing wave wavelength. It can be seen from eqs. (2.9.5) and (2.9.6) that this length is 2/2, i.e.. 

The Lotka-Volterra equations written in the dimensionless parameters contain only several control parameters birth and death rates a, (3 and the ratio of diffusion coefficients n, = Da/ Da + D ), 0 /t 1, i.e., Da = 2k, Db = 2(1 -k) whereas their sum is constant, Da + Db = 2. Lastly, it is also the space dimension d determining the functionals J[Z], equations (5.1.36) to (5.1.38), the Laplace operator (3.2.8) as well as the boundary condition (8.2.21) for the correlation functions of similar particles. Before discussing the results of the joint solution of the complete set of the kinetic equations, let us consider first the following statements. 

Statement 1. Provided K t) = K = const, i.e., neglecting change in time of the correlation functions, equations (8.2.12) and (8.2.13) of the concentration dynamics describe undamped concentration oscillations with the frequencies uj uq = /a, dependent on the initial conditions. The dependence u = uj K) is weak. This statement is based on the analysis of the Lotka-Volterra model by both topological and analytical methods (see Section 2.1.1). 

Statement 2. Substitution into the concentration dynamics (equations (8.2.12) and (8.2.13)) of the reaction rate K = K N, N, ), dependent on the current concentrations, changes the nature of the singular point. In particular, a centre (neutral stability) could be replaced by stable or unstable focus. This conclusion comes easily from the topological analysis its illustrations are well-developed in biophysics (see, e.g., a book by Bazikin [30]). 

Statement 3. If the concentrations are fixed, = const, = const, the set of kinetic equations (8.2.17), (8.2.22) and (8.2.23) as functions of the control parameter demonstrates two kinds of motions for /t k the stationary (quasi-steady-state) solution holds, whereas for k /t a regular (quasi-regular) oscillations in the correlation functions like standing waves 

This statement is not self-evident and needs some comments. A role of concentration degrees of freedom in terms of the formally-kinetic description was discussed in Section 2.1.1. Stochastic approach adds here a set of equations for the correlation dynamics where the correlation functions are field-type values. Due to very complicated form of the complete set of these equations, the analytical analysis of the stationary point stability is hardly possible. In its turn, a numerical study of stability was carried out independently for the correlation dynamics with the fixed particle concentrations. 

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The second model is a quantum mechanical one where free electrons are contained in a box whose sides correspond to the surfaces of the metal. The wave functions for the standing waves inside the box yield permissible states essentially independent of the lattice type. The kinetic energy corresponding to the rejected states leads to the surface energy in fair agreement with experimental estimates [86, 87],... 

Figure Bl.19.6. Constant current 50 mn x 50 mn image of a Cu(l 11) surface held at 4 K. Tliree monatomic steps and numerous point defects are visible. Spatial oscillations (electronic standing waves) with a...

Crommie M F, Lutz C P and Eigler D M 1993 Imaging standing waves in a two-dimensional electron gas Nature 363 524... 

Hasegawa Y and Avouris Ph 1993 Direct observation of standing wave formation at surface steps using scanning tunnelling spectroscopy Rhys. Rev. Lett. 71 1071... 

Cowan P L, Golovchenko J L and Robbins M F 1980 X-ray standing waves at crystal surfaces Rhys. Rev.L44 1680-3... 

Woodruff D P, Cowie B C C and Ettem a A R H F 1994 Surface structure determination using x-ray standing waves a simple view J. Rhys.. Condens. 6 10 633—45... 

Figure Cl.4.3. Schematic diagram of the Tin-periD-lin configuration showing spatial dependence of the polarization in the standing-wave field (after 1171).

Figure Cl.4.4. Schematic diagram showing how the two 2 levels of the ground state couple to the spatially varying polarization of the Tin-periD-iin standing wave light field (after 1171).

Figure Cl.4.5. Population modulation as the atom moves through the standing wave in the Tin-periD-lin one dimensional optical molasses. The population lags the light shift such that kinetic is converted to potential energy then dissipated into the empty modes of the radiation field by spontaneous emission (after 1171).

Fig. 3.17 The two possible sets of standing waves at the Brillouin zone boundary. Standing wave A concentrates electron density at the nuclei, whereas wave B concentrates electron density between the nuclei. Wave A thus has a lower energy than wave B.

For ultrasonic nebulizers, the liquid is fragmented into droplets by an acoustic standing wave, usually produced by a piezoelectric transducer. 

The picture of the electron in an orbit as a standing wave does, however, pose the important question of where the electron, regarded as a particle, is. We shall consider the answer to this for the case of an electron travelling with constant velocity in a direction x. The de Broglie picture of this is of a wave with a specific wavelength travelling in the x direction as in Figure 1.4(a), and it is clear that we cannot specify where the electron is. 

Figure 1.3 (a) A standing wave for an electron in an orbit with n = 6. (b) A travelling wave, resulting... 

Optical trapping can also be used as a hthographic tool (90). For example, a combination of optical molasses and an optical standing wave have been used to focus a beam of neutral sodium atoms and deposit them in the desired pattern on a suitable substrate (eg, siUcon). Pattern resolutions of the order of 40 nm with good contrast (up to 10 1 between the intended features and the surrounding unpattemed areas) and deposition rates of about 20 nm /min were obtained (90). 

Impedance Tube Test Methods. There are two impedance tube test methods ASTM C384-90a (3) and ASTM E1050-90 (4). Test method C384-90a makes use of a tube with a test specimen at one end, a loudspeaker at the other, and a probe microphone that can be moved inside the tube. Sound emitted from the loudspeaker propagates down the tube and is reflected back by the specimen. A standing wave pattern develops inside the tube. 

ASTM E1050-90 also makes use of a tube with a test specimen at one end and a loudspeaker at the other end, but iastead of a single movable microphone there are two microphones at fixed locations ia the tube. The signals from these microphones are processed by a digital frequency analysis system which calculates the standing wave pattern and the normal iacidence sound-absorption coefficients. 

Another problem, prevalent ia areas where severe icing conditions are met, is referred to as galloping of power lines. When ice forms on a power line, there is frequently a prevailing wiad which causes the ice to take a teardrop or airfoil shape. This foil provides an aerodynamic lift to the conductors and under certain conditions the conductors can go iato a resonant vibration such that large standing waves are created that exert enormous forces on the system. Miles of power lines and the towers along them have been destroyed by this phenomenon. 

Acoustic Coupling When the shell-side fluid is a low-density gas, acoustic resonance or coupling develops when the standing waves in the shell are in phase with vortex shedding from the tubes. The standing waves are perpendicular to the axis of the tubes and to the direction of cross-flow. Damage to the tubes is rare. However, the noise can be extremely painful. 

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