Nodal lines

Knoten-linie, /. nodal line, -punkt, m. nodal point junction, -wurz,/. figwort, Pharm.) scrophularia. -zahl, /. nodal number number of nodes. 

Keywords distribution functions, transmission, nodal lines... 

Figure 3. The complexity of nodal lines, nodal points and saddles for the transmission through chaotic (Sinai) (left) and regular billiard (right). 

Instead of nodal lines in closed systems we are interested in the statistics of NPs for open chaotic billiards since they form vortex centers and thereby shape the entire flow pattern (K.-F. Berggren et.al., 1999). Thus we will focus on nodal points and their spatial distributions and try to characterize chaos in terms of such distributions. The question we wish to ask is simply if one can find a distinct difference between the distributions for nominally regular and irregular billiards. The answer to this question is clearly positive as it is seen from fig. 3. As shown qualitatively NPs and saddles are both spaced less regularly in chaotic billiard in comparison to the integrable billiard. The mean density of NPs for a complex RGF (9) equals to k2/A-k (M.V. Berry et.al., 1986). This formula is satisfied with good accuracy in both chaotic and integrable billiards. 

The vibratory mode most frequently encountered is of the plate type and involves either the shroud or the disc. Fatigue failure generally originates at the impeller outside diameter, adjacent to a vane often dne to the vibratory motion of the shroud or disc. The fatigue crack propagates inward along the nodal line, and finally a section of the shrond or disc tears out. 

The spherical harmonics in real form have explicit nodal lines on the unit sphere. Morse and Feshbach (1953) have given a detailed description of those real spherical harmonics, and gave them special names. Here we list those real spherical harmonics in normalized form. In other words, we require... 

The first of the real spherical harmonics is a constant, which does not have any nodal line ... 

The ones with / 4 0 and m = 0 have nodal lines dividing the sphere into horizontal zones, which are called zonal harmonics. The first two are... 

The ones with m = I divide the sphere into sections with vertical nodal lines, which are called sectoral harmonics. Those are ... 

The rest are called tesseral harmonics, which have both vertical and horizontal nodal lines on the unit sphere ... 

Fig. A.I. Real spherical harmonics. The first one, Y , is a constant. The coordinate system attached to the unit sphere is shown. The two zonal harmonics, IT and II, section the unit sphere into vertical zones. The unshaded area indicates a positive value for the harmonics, and the shaded area indicates a negative value. The four sectoral harmonics are sectioned horizontally. The two tesserai harmonics have both vertical and horizontal nodal lines on the unit sphere. The corresponding "chemists notations," such as (3z — r ), are also marked.

The transform concept has been illuminating in many ways. One more example will be mentioned. In the transform of a centrosymmetric group of atoms, the phase angle is necessarily either 0 or n, and regions of opposite sign are separated by nodal lines of zero intensity in the transform of a non-centrosymmetric group there is no such limitation of phase angles, so that the intensity does not have to go... 

The orbitals q)2 and q)3 have one nodal line each, passing either through two atoms or through two bonds but since these orbitals have the same number of nodes, it follows that they must have the same energy (they are degenerate). The same remark applies to the first antibonding orbitals cpA and (p5, which have two nodal lines each. 

Fig. 7. (a) Wave pattern generated on a circle with scaled radius kr = 1.82. The pattern is the product of a radial part, J,(kr) [the first-order Bessel function] and an angular part, cos ( >- The dashed nodal line of zero (i.e., steady state) concentration runs along diameter of the circle from < > = 90° to < > = 270°. The dotted circle outlines the circular radius. (b) Wave pattern, JAkr) cos 2dashed lines are crossed nodal lines on two perpendicular diameters, (c) Pattern generated at a scaled radius of 3.8, where the zero in the derivative of JAkr) matches the radial boundary condition. The pattern is JAkr) cos 0c >, which has no angular variation. The nodal line is concentric with the outer radius. 

Reaction-diffusion systems provide a means to subdivide successively a domain at a sequence of critical parameter values due to size, shape, diffusion constants, or other parameters. The chemical patterns that arise are the eigenfunctions of the Laplacian operator on that geometry. The succession of eigenfunctions on geometries close to the wing, leg, haltere, and genital discs yield sequential nodal lines reasonably similar to the observed sequence and symmetries and geometries of the observed com-... 

Region I is separated from region II by the nodal line... 

Fig. 2.14. They - x phase plane in the late diffusional stage of formation of two compound layers. The nodal lines 1 and 2 separate the phase plane into three regions. In regions I and HI the thickness of one of the layers increases, while that of the other decreases. In region II both layers grow simultaneously. The arrows at phase trajectories indicate the direction of variation of the layer thicknesses with increasing time.

The nodal line between the regions II and III is the straight line... 

The two liquids thus formed are immiscible, but in thermodynamic equilibrium. Therefore, we may speak of a dynamic system of two immiscible phases. Figure 3.10 shows an example of a practical system applied to create a dynamic LLC phase system. A practical phase system can be created by pumping a mobile phase through a column, the composition of which corrresponds to a ternary mixture that is in dynamic equilibrium with another mixture (the two mixtures can be connected by a nodal line). If the mobile phase is the more polar one of the two ternary mixtures in equilibrium, then a non-polar (hydrophobic) solid support must be used and a reversed phase system can be generated. If the mobile phase is the less polar of the two mixtures in equilibrium, a polar support is required. 

Figure 3.10 Example of a phase diagram for a ternary system used to create a dynamic LLC system. Components Ethanol (EtOH), Acetonitrile (ACN) and Iso-octane (2,2,4-trimethylpentane TMP). I — V nodal lines. Circles compositions determined experimentally by titration (full circles) and GC (open circles). Figure taken from ref. [315]. Reprinted with permission.

Seiche sea level oscillations. The level of any basin, being turned out of its equilibrium state by a certain force, returns to its initial position performing decaying oscillations with respect to one or several horizontal lines (nodal lines) until their energy is expended for bottom and coastal friction. These free oscillations are known as seiches (uninodal or multinodal depending on... 

A simple finite-difference solution to the above set of equations will be discussed here. A series of nodal lines in the Z- and / -directions are introduced. 

A series of evenly spaced nodal lines as shown in Fig. 8.25 will be used. The grid spacings in both the X- and F-directions are thus constant and equal to AX and AY respectively as indicated in Fig. 8.25. 

A forward-marching implicit finite-difference solution of the energy equation will again be considered. In order to obtain this solution, a series of nodal lines running parallel to the x and y-axes are again introduced as shown in Fig. 10.15. 

Nodal lines used in numerically solving the energy equation in the boundary layer. 

Nodal lines used in deriving finite-difference approximations. 

Figure 9. Traces of the nodal surfaces in the xy-, xz- and yz- planes for the sp2dx2 y2 hybrid. It can be seen that while the basic features in two of the planes (xy and yz) are rather similar to the one observed in spazdbz2 type hybrids, the situation in the xz-plane is completely different. The lower right plot shows the nodal lines that are the intersections of various surfaces displayed in Fig. (8).

Fig. (10) shows a surface plot of a section of the momentum density in the xt/-plane, where the density is accumulated (left). The seemingly monotonous distribution exhibits on closer inspection a good deal of fine structure first, there are the aforementioned holes in the vicinity of the nodal lines secondly, the apparent maximum reveals itself on an enhanced scale (right plot), to be a saddle point that is minimal in the x-direction. 

For given t and me / 0, the wavefronts S — constant, are planes parallel to, and ending on the z-axis with starting values ro0o(j)o as in figure 5. (z is a nodal line of tr when me 0). 

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