Expansion of Wave Rays and Fronts in Media with Inhomogeneous Structure

In the article on the principles of Fermet, Huygens obtaine the differential equations in the form of Hamilton, which describe the ray trajectories and wave fronts in inhomogeneous media. Established that the vector of Poynting-Umov’s determining the direction of energy propagation in inhomogeneous medium is coincident with the vector tangent to the ray. In the second part of the article established that the equations of the theory of rays’ propagation in inhomogeneous media have the form of equations of nonlinear dynamics and describe the emergence of deterministic chaos in the geometry of rays for a wide variety of types of heterogeneous structures. In this case, the rays behave randomly and their description you must go to the description based on the theory of random functions and fields. In the third part of the paper is considered a model which is equivalent to the random medium and the calculation of the coordinates of the ray (the mathematical expectation and correlation functions). Understanding of these characteristics gives information about the behavior of the trajectories of the rays for these models of media. The description of the behavior of rays on the basis of the equations of statistical mechanics is discussed in the article for functions of Markov’s type.


Introduction
The purpose of this paper is to present generalizations of the results obtained for the first time in the work of the authors for moveouts of rays and wave fronts. In media where this type of inhomogeneity has place the rays be have randomly and therefore, it is necessary to switch to the probabilistic models and corresponding methods.
In the first of the article are formulated the basic equations which are obtained on the basis of the principles of Fermat, Huygens, and installed that lines of energy (vector of Poynting-Umov) and the rays path coincide. On the basis of the optical-mechanical analogy formulated the basic equations of the theory of wave propagation in the form of Hamilton are used in 2, 3 parts of the human body.
Nonlinear differential equations for ray trajectories describe deterministic chaos in the geometry of rays for wide type of an inhomogeneity. It requires a shift to a probabilistic description of the kinematics of the rays. The third part describes the model of random media described on the basis of moments (mathematical expectation and correlation functions) as well as case governmental functions of Markov type.
Huygens' principle of constructing wave fronts in according to algorithm of a contact transformation is easily implemented if the perturbation region is non-concave  If the emitting area has a concavity, the construction of the wave surface is shown in Fig. 1.1b. Presentation of the wave front in the form of the surface is a mathematical idealization, because in reality, a wave is a volume configuration change of the medium points during the passage of a perturbation from some initial configuration to the final. In a homogeneous isotropic medium all front points have the same speed directed along the normal n , then during the time t ∆ surface points are shifted by the same distance s along the normal with the speed V . The wave surface at time point t t + ∆ , constructed according to Huygens' principle as the concavity of secondary waves coincides with the surface, passing through the points lying on one and the same distance along the normal from the wave surface at time point t . These lines, which are orthogonal to the original radiating surface (in particular, points) in a homogeneous isotropic medium, are the rays along which the radiation energy propagates. According to Newton's corpuscular theory, the energy is transferred along the rays, the construction of which in homogeneous isotropic media is carried out with purely geometrical methods. Approaches of Huygens and Newton are known as the optic and mechanical analogs in analytical mechanics [1][2]. With the approach of Newton is associated the analogy of particle motion by inertia under the influence of the initial pulse and in the absence of any effects during the movement. With the approach of Huygens is associated the analogy of contact transformations in the Hamiltonian mechanics, representing a canonical transformation of generalized coordinates and momenta.
An approach based on the construction of rays is effective in solving problems of the wave kinematics by geometrical methods for homogeneous isotropic media, including the case of transmission and reflection of waves at the interface of two media, also through the lens, etc.

Expansion of Rays and Fronts in Determinate Inhomogeneous Media
In the case of inhomogeneous, anisotropic, nonlinear media Huygens' approach is more difficult to implement and Newton's approach allows solving the problem of wave propagation more effectively, if we use the kinematic principle of Farm, according to which the perturbation of the medium state at the source , , x x x that of inhomogeneous media,  -the distance along the ray.
Considering the ratio of θ α β -an angle which for linear homogeneous medium does not depend on α , because 0 V V const = = . As A depends linearly on α , the rays will be straight, and the fronts will be circles in the plane considered. The law of energy conservation in the integral form is Here j P -vector components of the energy density P of Poynting-Umov, E -the total energy density.
In the differential form of (1.6) we obtain The energy flow is directed along the speed V , and hence along the rays. For an arbitrary ray tube of (1.7) follows the conservation equation where  -the distance along the ray. According to the optical-mechanical analogy the system of equations (1.14) can also be written in the form of Newtonian mechanics for potential forces (in the form of second-order equations) 2 where the role of the forces potential is played by The geometry of the spatial curve (ray) is characterized by the curvature k and torsion  , which are calculated according to the formulas grad n grad n grad n k n n n where 0 2 < < π θ -the angle between the ray (vector sin  θ ) and vector grad n .
The radius of the ray curvature Discussed equations correspond to coordinate method of setting a motion in mechanics. Natural way to set a motion in mechanics corresponds to the consideration of the kinematics of the rays in the ray coordinates, related to the initial position of the wave front [4,7,15].
On the surface of the radiating body curvilinear coordinates can be introduced , ξ η Fig. 1.6.  If the initial emitting surface is plane, then the coordinate system is a Cartesian system, and is connected with the surface The equation of the initial surface can be written as Then the differential equations for the rays have the form [4,7,10] , To close the system (1.31), (1.32) it must be supplemented by equations for the mean and Gaussian curvatures of the surface.
In the particular case when the wave front propagates parallel to itself expressions for the mean and Gaussian curvatures have the form The expression for the eikonal in the coordinates , ,t ξ η is written in the form Coordinates , ξ η on the surface 0 τ identify a ray coming from the surface at the moment 0 t t = . For different , ξ η we obtain a family of rays, therefore, at the initial time from the surface 0 τ comes the bundle of rays, that allows you to build a complex radiation pattern, on the basis of which is determined the wave field structure in the physical space.
In the phase space { } , p r the phase portrait under certain conditions, can also be quite complicated.
In the spatial of generalized coordinates i q , which are not related to the wave surface and which are orthogonal curvilinear coordinates, eikonal can be written as The system of equations for the rays in this case has the form The expression for the eikonal is written as If you enter a pulse components in curvilinear coordinates according to the formulas then the equations (1.36) are written as In many specific problems of ray propagation in inhomogeneous media is convenient to use angular variables, for example, in the case of the spherical symmetry. Assuming that 1 * r dp n n p p ctg j p dt r sin cos * sinˆˆr dp n n p p p p dt r The eikonal equation is written as Expansion of Wave Rays and Fronts in Media with Inhomogeneous Structure Thus, depending on the geometry and mechanics of the particular task you can choose a suitable system of coordinates and shape of the ray equations. The most commonly used are phase coordinates ( ) Here, the parameter t It related to the distance s along the ray (coordinate line), and , ξ η identify the ray on the initial surface . If the wave is emitted by a limited surface area, defined by the relation ( ) 0 0 , r r = ξ η , then the rays form a family of rays emanating from this area.
The conversion from the Cartesian coordinate system to the ray system is defined by Jacobian , , x x x D t ∂ = ∂ ξ η and will be one to one, if D 0 ≠ .
A lot of wavefronts obtained in accordance with the principle of Huygens (contact transformations) form a family of equal phase surfaces, eikonal for each of them is written in the form The family of rays emitted by the limited surface area 0 t S forms the bundle of rays. This means that the rays propagate not independent of each other, but they interfere. Due to the interference of secondary waves a significant contribution to the building of the fronts contribute only those rays, for which the phase difference does not differ by more than The value J Excluding t we obtain

Surfaces, where condition
is called the divergence of rays. From (1.51) follows that on the caustic 0 J = , ie cross-section of the ray tube decreases, energy increases, the rays touch caustics and change the direction. The classification of caustics is considered in the catastrophe theory [16]. On caustics and in their neighborhood classical spatial ray solutions are not applicable. There are methods for caustic rays, allowing to solve a number of tasks for caustics [2].

Determinate Chaos of Rays and Fronts in Inhomogeneous Media with Deterministic Structure
The equations describing the propagation of rays in inhomogeneous media are nonlinear because the refractive index ( ) n r dependence on the spatial coordinates. The consequence is the possibility of deterministic chaos in the radiation pattern, when for a determinate particular dependence ( ) n r is obtained not a concrete realization of the trajectory of the ray, but a set of possible trajectories, in the same way as is the case for random functions. The optical-mechanical analogy allows us to consider this problem with the most common positions within the Hamiltonian mechanics. Issues of construction of ray trajectories for different types of the determinate inhomogeneity are subject to numerous studies especially for 2D layered models. Significantly less works are devoted to the chaotization of ray trajectories in determinate inhomogeneous media [11][12][13][14].
First the possibility of deterministic chaos in inhomogeneous media has been considered in [13].
As is well known, the Poynting-Umov vector of energy density is directed along the trajectory of the ray along the tangent and thus chaos in the flows of energy and information in inhomogeneous media greatly complicates the prediction of information and energy processes.
In accordance with the optical-mechanical analogy of differential equations for the ray trajectories in inhomogeneous media are derived from Fermat's principle [1][2][3][4].
The wave surface is determined by the eikonal equation , , x x x r = , for example, 1 2 , x x we take as coordinates on a locally plane wave, and 3 x directed along the straight ray orthogonal to the plane 1 2 0 , x x Fig where ρ a function of is 3 x and determines the coordinates of the ray point in the tangent plane of the wave front. Impulse p is introduced by the relation ˙2 3 , 1 n p dp Taking into account (2.4), (2.5) the Hamiltonian form of the equations for the ray has the form of 3 3 , dp H dp H dx p dx p The waveguide in a inhomogeneous medium is a linearly extended volume, where takes place a determinate expressed change of the refractive index from its boundaries to the axis in such way, that rays falling or occurring therein are propagating along its axis in general experiencing fluctuations near the waveguide axis, waveguide may be inhomogeneous along the waveguide.
The axis of the waveguide 3 Ox is the perpendicular to the plane 2 , Ox x and an attractor for ray trajectories, if the change n along 1 2 , x x has the character of monotonic increase with a maximum on the axis  In this case rays emitted in the waveguide parallel to the axis 3 Ox and rays, entering into the waveguide at some angles oscillate near axis 3 x . The ray acts like a ball, received an initial impulse moving along the ideal (frictionless) chute along the axis 3 x (in the plane 2 , Ox x is a potential well).
Accounting for the presence of a certain type of inhomogeneity along the axis 3 Ox may result in a deterministic chaos in radiation pattern.
We represent the refractive index ( ) In accordance with the representation (2.7) we can, using classical perturbation theory to write down [1][2][3][4][11][12][13][14] ( ) ( ) where the summands The wave nature of propagation of the rays takes place in the ocean acoustics, geophysics, where the free surface of the Earth has waveguide properties and that provides further propagation of surface waves, in optics, in particular, the propagation of light through fiber optic wires.
We assume that the quasi-cylindrical waveguide has a axially symmetrical nature of the change of the refractive index ( ) n ρ , then at any section of the waveguide along the axis 3 Ox the ray pattern will be the same, so for simplicity we consider the case of 2D, when In the phase plane 1 1 , x p phase trajectories of the ray have the form corresponding to the center The transformation (2.32) describes the motion of the ray between the "points" of the inhomogeneity along the axis 3 x on the length interval T . At 0 T → , 1 0 H ∂ = ∂θ and then we obtain the canonical equations (2.6).
If the perturbation has one harmonic the resonance condition is written as: where m -whole number. The condition of conservation of phase volume has the form of [16,17] ( )   As is well known, in irregular fiber waveguides of the gradient, refractive type, underwater, underground acoustic waveguides are possible phenomenon of rays' swing, swinging of the waveguide channel width, "rashes" of rays from the waveguide [2].
In [11] it was found a decrease in the effective cross waveguide size and flashing rays from it.

Expansion of Rays and Fronts in Stochastically Inhomogeneous Media
Models of randomly inhomogeneous media allow to take into account the fact, that in real-world environments, in principle, it is impossible to determine (measure) accurately, to know the material physical and mechanical properties (coefficients) of the medium in sutu, because there is always the effect of artifacts related to measurements. Due to this experimental data always have a dispersion (scatter) in each point of the medium, i.e. can be modeled as a random field of physical media factors. Accordingly, rays and fronts will represent random trajectories and surfaces [16].
The equations for the rays and the eikonal in general case of an arbitrary dependence of the refractive index on the spatial coordinates cannot be solved analytically in the general case.
The most widely used is the method of successive approximations (a small parameter). According to the method analytical approximations for the coordinates ray and the eikonal sequentially are found, and then calculated numerically.
Imagine the square of the refractive index in the form of  4), we obtain a recurrent system of differential equations in partial derivatives of the first order to find the corrections to the eikonal Consider the account of the transverse ray displacement, which is determined by the solution of equation (3.5). The solution of equation (3.6) in this case has the form As follows from (3.16) an approximation to (3.14) is obtained from (3.16) by neglecting the transverse displacements 1 r . This corresponds to the condition In case the medium is not quasi-homogeneous ( ) n r const ≠ , the method of variation of arbitrary constants is applied which we use in mechanics to solve equations with variable coefficients [1]. Solution of the equations (1.5) at 1 k = , is represented as  characterizes the deflection angles of the perturbed ray from the unperturbed ray. Use of the method of successive approximations (small parameter) is based on the presence of small parameters in the task about the wave propagation in a inhomogeneous medium: 1) the wavelength is much smaller than the scale of inhomogeneity changes (asymptotic behavior of the short waves -ray theory); 2) the maximum amplitude fluctuations of physical and mechanical parameters of the medium from their average values are small (weakly inhomogeneous media).
We consider some of the effects of ray propagation in randomly inhomogeneous media within these approximations.
Then in the representation (3.1) the index of refraction ( ) n r is a random function of the spatial coordinates (random field), which is described by its moments or probability distribution. Since in this case, the coordinate points of the trajectories of rays and eikonals will be random functions, then they are also described by the moments or probability distribution. We consider the momentary way of describing random functions within the correlation approximation, if n  σ ε , where n σ is the mean square deviation of the refractive index, ε is the mean square value of the refractive index. The average value and the correlation function of the refractive index ( ) ( ) ( ) are considered to be given.
It is required to find the coordinates of points of the average ray trajectory, the average deviation of rays from the average the trajectory.
In the case where the initial emitting surface is plane and the initial emitted wave is plane too, will take the Cartesian coordinate system at the initial plane  , . x x .
In the case of isotropic fluctuations of the refractive index is obtained a linear dependence on the distance along the middle ray.  2  3  3  1  2  3  3  1  2  3  3   2  1  2  1  3  3  3  3  3   1  2  2  1  3  3  3  3  3 , , By fixing 1 3 x in the formula (3.36), we obtain, that the longitudinal correlation of the eikonal (phase) extends over a distance of about the path passed the ray 3 II K x . As it follows from (3.38) the eikonal of rays bundle has a Gaussian correlation.
In the case of non-planar initial wave of Fig. 3.3 As it follows from (3.39) correlation of the eikonal does not depend on the type of the wave shape and will be equal to the flat, spherical and cylindrical waves.
In a randomly inhomogeneous medium, which is statistically homogeneity and isotropic, the rays propagating in them are in average direct, orthogonal to the initial surface, and the wave surfaces (phase fronts) retain the semblance of an initial surface on average. We introduce on the unperturbed (average) surface the coordinate system , α β From (3.46) follows, that the angles dispersion of a spherical wave arrival is less three times, than at a plane wave, that is caused by the fact that in the spherical wave rays propagate on average nearer to each other than in the plane.
A ray in a randomly nonuniform statistically homogeneous isotropic medium is a space curve, fluctuating around the middle (undisturbed) ray. The average value of fluctuations is determined by the mean square displacement of the ray from the unperturbed position. In the case of on average plane wave, coordinates of points of a ray trajectory are written as (3.1), then in the first approximation From (3.47) follows, that a ray is displaced only in the transverse direction relative to the direction of the undisturbed ray. In (3.47) it is assumed that the initial surface is plane and the front is flat on average, propagating in the direction 3 x , then 1 r , depends on 1 2 , Calculating the elements of the correlation matrix we obtain In the absence of a temporary (frequency) dispersion ε and  are independent of ω , then from (3.56) is obtained 1 1 0 From (3.57) follows that in this approximation the group path correction 1  coincides with the correction of the phase path (eikonal). This is a consequence of equality of the phase and group velocity in a statistically homogeneous isotropic medium. In particular, their dispersions are equal.

( )
Consider the second approach for the description of the ray propagation and of wave fronts in randomly inhomogeneous media. This approach is based on the theory of Markov' processes and allows us to describe the diffusion of rays.
We introduce on the initial surface the Cartesian coordinate system 1 2 3 , , x x x so that the axis 3 x has as direction vector the unperturbed ray, and 1 2 , x x are located in the tangent plane to 0 t S normal to the ray 0 r .
In an approximation of the diffusion random process the Einstein-Fokker equation (EFE) for the density probability has the form of [16][17] ( )   In the case of diffusion of N rays, forming a ray tube is determined that the average cross-sectional area of the ray tube is retained in the plane The end of the unit vector τ , tangent to the ray performs random walks along the unit sphere, depending on t or s . We connect with the mobile point M on the ray a fixed unit sphere so that the Cartesian coordinate axes 1 2 3 , , x x x with a center in point M are moving parallel to a fixed coordinate system on the initial surface. The Frenet trihedron moving along the ray makes the rotational movements, which can be described in a spherical coordinate system Fig. 3.6