Sommerfeld condition

If we suppose the medium is composed of a known part (the background) identified by the density po and the celerity co, and an unknown part (the perturbation) identified by p and c, the equation that describes acoustic propagation/diffiision phenomena in the medium (including the boundary and Sommerfeld conditions) result from the Pekeris equation and is given, for weak-scattering (pc poCo), by ... 

Equating this action potential to an integer leads directly to the Sommerfeld conditions for periodic systems. 

This equation resembles the Bohr-Sommerfeld condition pdx = nh, but differs from it in that S now is the solution of the quantum HJ equation and not of the classical one as before. 

It is interesting to note that straightforward Bohr-Sommerfeld quantization of the action (6.1.11) yields the exact result (6.1.25) for the bound state energies. In our units the Bohr-Sommerfeld condition results in / = n, n = 1,2,. Inserting this result into (6.1.13) indeed reproduces (6.1.25) exactly. This is the same happy accident which allowed Bohr (1913) to obtain the Balmer formula from a simple solar system model of a one-electron atom. 

The current technique for derivation of experimental potential energy curves, developed in the 1930s but popularized only with the advent of high-speed computers, is based on the Sommerfeld condition [15] for quantization of action in systems undergoing periodic motion. 

Results have shown that the angular location of the position of maximum pressure is independent of film thickness and a function only of radius ratio. In Figure 8 the ratio of this angle to the analytical value determined by Snldle and Archard (4) has been plotted for both the Reynolds and half-Sommerfeld boundary conditions. It can be seen that with the half-Sommerfeld condition the agreement with the Snldle and Archard values is excellent. 

In this section, we intend to show that for a certain type of models the above imposed restrictions become the ordinary well-known Bohr-Sommerfeld quantization conditions [82]. For this purpose, we consider the following non-adiabatic coupling matrix x ... 

Sommerfeld, A. and Welker, H. 1938. Artificial limiting conditions in the Kepler problem. Ann. Phys. 32 56-65. 

These are the same quantized energy levels that arose when the wavefunction boundary conditions were matched at x = 0, x = Lx and y = 0, y = Ly. In this case, one says that the Bohr-Sommerfeld quantization condition ... 

In addition to the general treatments of wavy flow, a number of theories concerning the stability of film flow have been published in these the flow conditions under which waves can appear are determined. The general method of dealing with the problem is to set up the main equations of flow (usually the Navier-Stokes equations or the simplified Nusselt equations), on which small perturbations are imposed, leading to an equation of the Orr-Sommerfeld type, which is then solved by various approximate means to determine the conditions for stability to exist. The various treatments are lengthy, and only the briefest summaiy of the results can be given here. 

As pointed out by Edmonds and Starace,12,13 the atoms are excited near the origin and can only escape in the z directions. The motion in the x,y plane is bound and is most likely to be the source of the quasi Landau resonances. To find the locations of the resonances it is adequate to ignore the z motion entirely and simply compute the energy spectrum of the motion in x,y plane. Applying the Bohr-Sommerfeld quantization condition leads to... 

Bonino brought forward a further contribution to the theory of infrared spectra of organic liquids by incorporating the Bohr-Sommerfeld quantum conditions, including the correspondence principle of Bohr as well. This paved the way toward establishing a correlation between the physical and chemical image of molecules in the study of infrared spectra. From this series of papers on infrared spectroscopy, one can already observe the interdisciplinary character of Bonino s thought. In a lecture delivered some years later, Bonino offered these reflections on his chosen field of research ... 

For the stability analysis of fluid dynamical system whose equilibrium solution is given by a parallel or quasi-parallel flow, one has to solve the Orr-Sommerfeld equation depending on whether the mean flow is two- or three-dimensional. For the two-dimensional problem equivalent boundary conditions are given by... 

There is another reason for our preference in calculating system response by integrating (2.6.10) directly and not use contour integral (2.6.13) and (2.6.17). This is due to the restrictive condition (2.6.12) needed to hold for Jordan s Lemma to be used. We will show here, that the condition of Jordan s Lemma does not hold for the Orr-Sommerfeld equation - a result that has not been used in stability studies of fluid dynamics. 

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