Airy equation

The Airy equation is very often encountered in the resolution of electrochemical mass transport problems in dimensionless variables, and has the form... 

The refracted component is further attenuated according to Beer s law when travelling through the second medium. In a slab of limited thickness d, radiation reflected at the back surface returns to the front surface where it is refracted out of the slab. This radiation might interfere with externally reflected radiation (see Fig. 6.4-8). Including further reflections results in a multibeam interference pattern which is described by the general Airy equation... 

Calculation of the suppression factor for a Morse potential, S((k)=S (k), allows one to see the effect of the potential well in both quasiclassical (large D) and non-quasiclassical (small D) conditions. Of course, in the limit of high energies E D, ka V) the linear approximation to the Morse potential close to the turning point can be used, and the Morse comparison equation goes over to the Airy equation. 

The solution to the Airy equation may be written in terms of the oscillatory integral (see Section 3.4.5)... 

This is the well-known Airy equation, which will be discussed in Chapter 3. Its solution is composed of Airy functions, which are also tabulated. 

We applied the Ricatti transformation to a nonlinear equation in Example 2.10 and arrived at the linear Airy equation... 

The abbreviation F = AR/ — R) is often used, which allows the Airy equations to be written in the form... 

When an external electric field is applied along the periodicity axis of the polymer, the potential becomes non periodic (Fig. 2), Bloch s theorem is no longer applicable and the monoelectronic wavefunctions can not be represented under the form of crystalline orbitals. In the simple case of the free electron in a one-dimensional box with an external electric field, the solutions of the Schrddinger equation are given as combinations of the first- and second-species Airy functions and do not show any periodicity [12-16],... 

The resolution or "resolving power" of a light microscope is usually specified as the minimum distance between two lines or points in the imaged object, at which they will be perceived as separated by the observer. The Rayleigh criterion [42] is extensively used in optical microscopy for determining the resolution of light microscopes. It imposes a resolution limit. The criterion is satisfied, when the centre of the Airy disc for the first object occurs at the first minimum of the Airy disc of the second. This minimum distance r can then be calculated by Equation (3). 

This eigenvalue problem is solved by inspection - this is just Airy s equation. The properly normalized stationary (A = 0) solution is... 

In the vicinity of x1 a shift in energy means only a redefinition of x, so one may suppose without the loss of generality that E = 0. Then the solution of the Schrodinger equation, near the right turning point is expressed via the Airy functions ... 

In the time-independent formulation, the absorption cross section is proportional to (4>/(.R .E) i(R] E )) 2. Approximate expressions may be derived in several ways. One possibility is to employ the semiclas-sical WKB approximation of the continuum wavefunction (Child 1980 Tellinghuisen 1985 Child 1991 ch.5). Alternatively, one may linearly approximate the excited-state potential around the turning point and solve the Schrodinger equation for the continuum wavefunction in terms of Airy functions (Freed and Band 1977). Both approaches yield rather accurate but quite involved expressions for bound-free transition matrix elements. Therefore, we confine the subsequent discussion to a merely qualitative illustration as depicted in Figure 6.2. 

Close-coupling equations were solved using the hybrid modified log-derivative / Airy propagator of Alexander and Manolopulos [67]. The agreement... 

The radius, r, of the Airy Disc is given by the Rayleigh criterion equation which is shown in Eq. 1. ... 

The Airy functions, Ai(z) and Bi(z), are independent solutions of this equation, where... 

Equations (8-23) and (8-24) can be multiplied to give the final transfer function relating changes in h to changes in/, as shown in Eig. 8-13. This is an example of a second-order transfer function. This transfer function has a gain R2 and two time constants AiRi and A2R2. For two tanks with equal areas, a step change in / produces the S-shaped response in level in the second tank shown in Fig. 8-14. 

Thus the same result is found for adiabatic processes as for direct heat transfer AStotai is always positive, approaclring zero as a limit when tire process becomes reversible. This same conclnsion can be denronstratedfor airy process wlratever, leading to the general equation ... 

Many authors have given analytic solutions with differing degrees of accuracy. Deslouis et al. developed a method that, after an approximation, reduces the problem to the canonical equation for Airy functions. Tribollet and Newman gave a solution under the form of two series one for K < 10 and one for K > 10. The two series overlapped well. 

Let us consider a more general equation, the one which is called the reduced velocity gauge or the Airy-Gordon-Volkov wave equation [11],... 

The method described for example 3.4 can be used to solve boundary value problems. Consider the boundary value problem given by the Airy differential equation ... 

In order to remedy the recognized deficiencies of equation (4) for the scattering cross section, such as unphysical discontinuities at 6 = 0, the so-called glory angle [19], and at angles where d3/db = 0, called rainbow angles [19], as well as the lack of the interference between the various trajectories in the sum of equation (4), semiclassical corrections such as the uniform Airy or Schiff [20] approximations can be included. 

Airy Stress Eunction and the Biharmonic Equation The biharmonic equation in many instances has an analogous role in continuum mechanics to that of Laplace s equation in electrostatics. In the context of two-dimensional continuum mechanics, the biharmonic equation arises after introduction of a scalar potential known as the Airy stress function f such that... 

Given this definition of the Airy stress function, show that the equilibrium equations are satisfied. 

Compatibility conditions address the fact that the various components of the strain tensor may not be stated independently since they are all related through the fact that they are derived as gradients of the displacement fields. Using the statement of compatibility given above, the constitutive equations for an isotropic linear elastic solid and the definition of the Airy stress function show that the compatibility condition given above, when written in terms of the Airy stress function, results in the biharmonic equation = 0. 

Thus, the solution of two-dimensional elastostatic problems reduces to the integration of the equations of equilibrium together with the compatibility equation, and to satisfy the boundary conditions. The usual method of solution is to introduce a new function (commonly known as Airy s stress function), and is outlined in the next subsections. 

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