# Nonlinear Rays and Fronts Dynamics in Stochastically Inhomogeneous Media and in Media with Deterministic Structure

**DOI:**https://doi.org/10.1007/978-3-662-53605-6_107-1

## Synonyms

## Definitions

The behavior of rays’ trajectories in stochastically inhomogeneous media is governed by nonlinear dynamic equations which describe the emergence of deterministic chaos in the geometry of rays and fronts for a wide variety of types of heterogeneous structures.

## Preliminary Remarks

In the entry “Rays Propagation in Inhomogeneous Media”, it has been shown that the approach based on the construction of rays is effective for solving the problems of the wave kinematics by geometrical methods for inhomogeneous isotropic media, including the cases of transmission and reflection of waves at the interface of two media, through lens, etc. Below the peculiarities of rays and fronts propagation in the stochastically inhomogeneous media and media with deterministic structure will be discussed.

## Rays’ Propagation in Stochastically Inhomogeneous Media

Models of randomly inhomogeneous media allow one to take into account the fact that in real-world environments, in principle, it is impossible to measure accurately the material physical-and-mechanical properties (coefficients) of a medium in situ, because there is always the effect of artifacts related to measurements. Therefore experimental data always have dispersion (scatter) in each point of the medium, i.e., it could be modeled as a random field of physical factors. Accordingly, rays and fronts will represent random trajectories and surfaces (Chigarev 2000).

It is known that equations for the rays and the eikonal in the case of an arbitrary dependence of the refractive index on the spatial coordinates cannot be solved analytically in the general case. Among the approximate approaches, the method of successive approximations (a small parameter technique) is the most widely used, according to which analytical approximations for the coordinates of the ray and the eikonal sequentially could be found and then calculated numerically.

*k*≥ 1) give the corrections arising from the inhomogeneity.

*F*

_{n}depend only on prior approximations, i.e., they are known functions.

*ε*〉 = const, and the rays will be in average straight

*t*

_{0}= 0, \( {\overline{r}}^{(0)}=0 \), and \( {{\overline{p}}_1}^{(0)}={V}^{(0)}\overline{N}/2{p}_N^{(0)} \), where \( \overline{N} \) is the normal to the primary surface

*Q*,

*V*

^{(0)}is the refractive index at

*Q*, and \( {p}_N^{(0)} \) is the initial velocity of the normal ray on

*Q*, the trajectory of the ray could be found in a parametric form.

The eikonal corrections due to the refractive index fluctuations could be considered for two cases, namely, with and without taking the lateral displacement of the ray into account.

*n*= 2 is reduced to

Note that these corrections (14) and (15) are applicable, if the displacement corrections of the perturbed ray from the unperturbed ray are small compared with the characteristic cross-sectional dimension of the perturbation *ℓ*_{⊥}.

From (16) it follows that an approximation to (14) is obtained from (16) by neglecting the transverse displacements *r*_{1}. This corresponds to the condition \( \left|{\overline{r}}_1\right|\ll {{\ell}}_{\perp } \).

*k*= 1 is represented as

*c*

_{k}(

*t*) are coefficients to be determined and \( {\overline{\rho}}_k(t) \) are the fundamental solutions of the system of uniform linear equations

*c*

_{k}(

*t*) could be obtained

*c*

_{k}(0) should be found from the initial conditions.

Substituting (22) in (17) results in the expressions for ray correction in inhomogeneous media.

The utilization of the method of successive approximations is based on the presence of a small parameter, and the following values could be used as small parameters in the problems of wave propagation in a inhomogeneous medium, the wavelength, since it is much smaller than the scale of inhomogeneity changes (asymptotic behavior of the short waves within the ray theory) or the maximum amplitude fluctuations of physical-and-mechanical parameters of the medium from their average values, which are small in weakly inhomogeneous media.

To study some of the effects of ray propagation in randomly inhomogeneous media within these approximations, the index of refraction \( n\left(\overline{r}\right) \) in the representation (1) should be considered as 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 are random functions, then they are also described by the moments or probability distribution. Let us consider the momentary way of describing random functions within the correlation approximation, if *σ*_{n} ≪ 〈*ε*〉, where *σ*_{n} is the mean square deviation of the refractive index and 〈*ε*〉 is the mean square value of the refractive index. The average value and the correlation function of the refractive index \( {R}_n\left({\overline{r}}_1,{\overline{r}}_2\right)=\left\langle \overline{V}\left({\overline{r}}_1\right)\overline{V}\left({\overline{r}}_2\right)\right\rangle \) are considered to be given. Thus, it is required to find the coordinates of points of the average ray trajectory, as well as the average deviation of rays from the average trajectory.

*x*

_{3}-axis in the Cartesian coordinate system is a straight ray in a inhomogeneous medium, i.e., the trajectory of the middle ray in an inhomogeneous medium. Then the average value of the eikonal is \( {\tau}_0=\sqrt{\left\langle \varepsilon \right\rangle }{x}_3 \), and its first approximation is equal to

*R*

_{τ}of the eikonal

*τ*has the form

*x*

_{3min}is the minimum value from the magnitude \( {x}_3^1 \), \( {x}_3^2 \), and \( \overline{\rho}={\overline{\rho}}_1-{\overline{\rho}}_2 \).

*R*

_{τ}of the eikonal is expressed in terms of the spatial spectrum Φ

_{n}(æ) of fluctuations \( n\left(\overline{r}\right) \) according to the formula

*x*

_{3}

*R*

_{n}(

*ρ*) has a Gaussian shape

*ℓ*

_{n}is the correlation radius of

*n*and the dispersion of the eikonal depends linearly on

*x*

_{3}

The spatial correlation could be represented as an expansion in terms of the longitudinal and transverse correlation.

*x*

_{3max}from \( {x}_3^1 \) and \( {x}_3^2 \). The longitudinal correlation coefficient remains constant

*K*

_{II}∼

*x*

_{3}. Figure 1 shows the \( {x}_3^2 \)-dependence of \( {K}_{II}^{\left(\tau \right)} \) at the fixed magnitude of \( {x}_3^1 \).

As it follows from (38), the eikonal of ray bundle has a Gaussian correlation.

*L*

_{1}and

*L*

_{2}designate lengths of rays arriving at points \( {\overline{r}}_1 \) and \( {\overline{r}}_2 \), \( {\overline{{\ell}}}_{01} \) and \( {\overline{{\ell}}}_{02} \) are the unit vectors along these rays, and

*δ*(

*s*) is the distance between the rays \( \delta =\left|\overline{\rho}\right|=\left|{\overline{\rho}}_1-{\overline{\rho}}_2\right| \). The correlation function of the eikonal could be written in the form

*L*

_{1}=

*L*

_{2}=

*L*and

*δ*(

*s*) = 0 is calculated by the formula

From (39) it follows that the correlation of the eikonal does not depend on the type of the wave shape and thus will be the same for plane, spherical, and cylindrical waves.

*α*,

*β*is introduced (Fig. 3).

_{⊥}is the operator of cross-differentiation with respect to the undisturbed ray. Vector \( {\overline{{\ell}}}_1\,{\perp}\, {\overline{{\ell}}}_0 \) lies in the plane tangential to the undisturbed wave front

*τ*

_{0}= const.

*θ*

_{α}and

*θ*

_{β}will be equal

*θ*

_{α}and

*θ*

_{β}, considering that 〈

*τ*

_{1}〉 = 0, will be equal to

*θ*

_{α}and

*θ*

_{β}are equal to

From (46) it follows that the angles’ dispersion of a spherical wave arrival is less three times than that of a plane wave, what is caused by the fact that on the spherical wave the rays propagate on average nearer to each other than on the plane wave.

Reference to (47) shows that the ray propagates only in the transverse direction relative to the direction of the undisturbed ray. In (47) it is assumed that the initial surface is plane, and the front is flat on average, propagating in the *x*_{3}-direction, and then \( {\overline{r}}_1 \) depends on *x*_{1}, *x*_{2}.

*ρ*

_{1}=

*ρ*

_{2}are uncorrelated, i.e.,

*ε*〉 = const, what means that there is no regular refraction (bending) of the ray. Introducing

*L*as

*ε*〉 and

*V*are independent of

*ω*, and then from (56) it follows

*L*

_{1}coincides with the correction of the phase path (eikonal). This is a consequence of equality of the phase and group velocities in the 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’s processes and allows one to describe the diffusion of rays.

*x*

_{1},

*x*

_{2},

*x*

_{3}on the initial surface such that the

*x*

_{3}-axis has the unperturbed ray as a directional vector and

*x*

_{1},

*x*

_{2}-axes are located in the tangent plane to the normal \( {S}_t^0 \) to the ray \( {\overline{r}}^0 \) (Fig. 4).

*M*of a curved ray can be represented as \( \overline{r}=\left(\overline{\rho},{x}_3\right) \) with \( \overline{\rho}=\left({x}_1\left({x}_3\right),{x}_2\left({x}_3\right)\right) \). Then the equation describing the ray fluctuation can be written as

*τ*

_{⊥}= 1 corresponds to the point of the ray rotation on the angle \( \frac{\pi }{2} \). Considering that \( {\overline{\tau_{\perp}}}^2<<1 \), Eq. (59) takes the form of small-angle approximation

*μ*= ln

*n*.

*D*is the diffusion coefficient

*μ*.

The joint diffusion of two rays is described by a system of eight differential equations.

*k*= 1, 2 are the ray numbers.

In general case, Eq. (69) could not be solved analytically; however, for some models, approximate solutions could be found, allowing one to draw certain conclusions concerning the spread of rays.

If *d*_{n} << *ρ*, then *D*_{αβ} = 2*Dδ*_{αβ}. This corresponds to the model, when the distance *ρ* between the rays on the initial surface is much larger than the correlation radius *d*_{n} for a random field grad*n*. Since in this case their relative diffusion takes place with the double diffusion coefficient, each ray diffuses independently of the other. The joint probability distribution in this case could be regarded as the Gaussian.

*ρ*<<

*d*

_{n}, then \( {D}_{\alpha \beta}\left(\overline{\rho}\right)=\,\pi B\Big({\rho}^2{\delta}_{\alpha \beta}+ 2\overline{\rho_{\alpha}}\,\overline{\rho_{\beta }}\Big), B=\frac{\pi }{4}\underset{0}{\overset{\infty }{\int }}d{\mathrm{\ae \ae}}^3\Phi \left(\mathrm{\ae}\right) \), and EFE has the form of

In the case of diffusion of *N* rays forming a ray tube, it is assumed that the average cross-sectional area of the ray tube is retained in the plane *x*^{3} = const and 〈*S*(*x*_{3})〉 = *S*_{0}.

*t*or

*s*. Assume that the mobile point

*M*on the ray is connected with a fixed unit sphere, such that the Cartesian coordinate axes

*x*

_{1},

*x*

_{2},

*x*

_{3}with a center at the point

*M*are moving parallel to the fixed coordinate system on the initial surface. The Frenet trihedron moving along the ray makes the rotational movements, which could be described in a spherical coordinate system (Fig. 5).

*W*by the ratio \( P\left(s,\overline{r},\theta, \varphi \right)=W{\left(\sin \theta \right)}^{-1} \), then the normalization condition is performed for \( P\left(s,\overline{r},\theta, \varphi \right) \)

*P*satisfies the equation

From (79) it follows that \( \left\langle P\right\rangle \sim \left\langle {n}^2\left(\overline{r}\right)\right\rangle \), i.e., the average density of the probability of arrival ray trajectories is higher with high \( {n}^2\left(\overline{r}\right) \), what means the bending of rays to the direction of increasing 〈*n*^{2}〉. The same conclusion follows from Eq. (77).

## Deterministic Chaos of Ray Trajectories and Fronts in Inhomogeneous Media with Deterministic Structure

The equations describing the propagation of rays in inhomogeneous media are nonlinear due to the spatial coordinate dependence of the refractive index \( n\left(\overline{r}\right) \). The consequence is the possibility of deterministic chaos in the radiation pattern, when for a particular dependence \( n\left(\overline{r}\right) \) a set of possible trajectories is obtained instead of a concrete realization of the trajectory of the ray, in the same way as in the case for random functions. The optical-and-mechanical analogy allows one to consider this problem with the most common positions within the Hamiltonian mechanics. Issues of construction of ray trajectories for different types of the deterministic inhomogeneity are subject of numerous studies especially for 2D-layered models; however only a few papers are devoted to the chaotization of ray trajectories in deterministic inhomogeneous media (Abdullaev and Zaslavsky 1981; Borisov et al. 2015; Chigarev and Chigarev 1978, 2018), and Chigarev and Chigarev (1978) were among the first to consider the possibility of deterministic chaos in inhomogeneous media.

It is well known that 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 in inhomogeneous media greatly complicates the prediction of information and energy processes.

*s*is the distance along the ray trajectory.

*H*has the form of

*x*

_{1},

*x*

_{2}-coordinates are chosen on a locally plane wave and

*x*

_{3}-axis is directed along the straight ray orthogonal to the plane (0,

*x*

_{1},

*x*

_{2}) (Fig. 6).

Suppose that \( \overline{r}=\left({x}_1,{x}_2,{x}_3\right)=\left(\overline{\rho},{x}_3\right) \) and \( \overline{\rho}=\left({x}_1,{x}_2\right) \), where the function \( \overline{\rho} \) determines the coordinates of the ray point on the tangent plane of the wave front.

The waveguide in an inhomogeneous medium is a linearly extended volume, where the refractive index changes from its boundaries to the axis in such a way that rays falling or occurring therein propagate along its axis and in so doing experience fluctuations near the waveguide axis.

*x*

_{3}-axis of the waveguide is perpendicular to the plane 0

*x*

_{1}

*x*

_{2}, and it is an attractor for ray trajectories, if the change n along

*x*

_{1},

*x*

_{2}has the character of monotonic increase with a maximum on the

*x*

_{3}-axis (Fig. 7).

In this case, the rays emitted in the waveguide parallel to the *x*_{3}-axis and the rays entering into the waveguide at some angles oscillate near the *x*_{3}-axis. The ray acts like a ball received an initial impulse moving along an ideal (frictionless) chute along the *x*_{3}-axis (a potential well locates in the plane 0*x*_{1}*x*_{2}).

Taking account for the presence of a certain type of inhomogeneity along the *x*_{3}-axis may result in a deterministic chaos in radiation pattern.

*x*

_{3}-direction, \( V\left(\overline{\rho},{x}_3\right) \) takes into account the presence of inhomogeneity along the waveguide, and

*ε*≪ 1 is a small dimensionless parameter.

*H*

_{0}and

*H*

_{1}have the form

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 a quasi-cylindrical waveguide has an axially symmetrical nature of the change of the refractive index \( n\left(\overline{\rho}\right) \), and then at any section of the waveguide along the *x*_{3}-axis, the ray pattern will be the same, so for simplicity a 2D case will be considered below, when \( n\left(\overline{\rho}\right)=n\left({x}_1\right) \).

*x*

_{1},

*p*

_{1}, the phase trajectories of the ray have the form corresponding to the center (Fig. 8).

*I*,

*θ*)

*E*along the ray by the relation

*ω*(

*I*) is the frequency.

*I*and

*θ*yields

Equations (97) and (98) describe nonlinear oscillations of the ray, i.e., the frequency of the ray oscillation near the axis of the waveguide depends on its energy.

*α*defined by the formula

*H*

_{1}could be presented in the form of a double Fourier series

*T*/2

*π*is the frequency and

*T*is the period of disturbances of nonuniformity along the axis of the waveguide and

*σ*is the phase of vibrations’ disturbances.

The closeness of the relation (101) to zero depends on the width of the resonance.

*m*,

*n*= ± 1 (

*ω*≈ Ω), and the sum in (100) has two real summands, namely, with the argument \( \dot{\sigma}-\theta \) (resonance summand) and with the argument \( \dot{\sigma}+\theta \) (high-frequency disturbance). As it is known, if the localized resonances are far enough from each other, then leaving in (100), only the resonant summand yields

*I*.

In the case of the moderate nonlinearity characterized by the parameter *ε* ≪ *α* ≪ *ε*^{−1}, the difference Δ*I* = *I* − *I*_{P} is small (|Δ*I*| ≪ *I*_{P}), where *I*_{P} is the value of *I* which turns the approximate equality (101) into the exact equality, i.e., *m*Ω + *nω*(*I*_{P}) = 0.

_{mn}is the frequency of small phase oscillations.

*α*means little changes in the values of

*I*and

*ω*

Due to the nonlinearity, fluctuations are non-isochronous, and the frequency depends on the amplitude. As a result of the changes in the amplitude of oscillations at the resonance, the frequency changes and deviates from its resonance magnitude. Then the changes in amplitudes stabilize, resulting in the return of the oscillation frequency to its resonant value. This process is repeated, i.e., the nonlinearity stabilizes the isolated resonance, making the amount of change to be limited to Δ*I*.

For the moderate nonlinearity *α* ∼ 1, the decomposition is carried out by *ε*^{1/2}, indicating according to (106) that in the case of the resonance Δ*I* ∼ *ε*^{1/2}, i.e., more than in a non-resonance case.

*ω*) is defined by the formula (106) and represents the width of the resonance and Δ = |

*ω*

_{i + 1}−

*ω*

_{i}| is the distance from this resonance to the nearest neighboring resonance.

At *s* ≪ 1, an isolated resonance case occurs, while at *s* ≫ 1 resonances overlap, resulting in highly irregular movement. Value *s* ∼ 1 is the boundary of stochasticity (Chigarev and Chigarev 2018) or the criterion of instability.

For an arbitrary deterministic perturbation *H*_{1} in the phase space, within the neighborhood of the separatrix, it is formed a layer, within which the ray trajectories behave chaotically. The width of the layer is determined by the specific form of a periodic perturbation.

*f*(

*θ*) ∼ 1 is a dimensionless phase function and

*I*

_{0}is constantly acting value.

The transformation (110) describes the motion of the ray between the “points” of the inhomogeneity along the *x*_{3}-axis on the length interval *T*. At *T* → 0, \( \frac{\partial {H}_1}{\partial \theta }=0 \), resulting in the canonical Eq. (85).

If the perturbation possesses one harmonic *f*(*θ*) = sin *θ*, then the resonance condition is written as *Tω* = 2*πm*, where *m* is a whole number.

*δ*- function

*I*and

*σ*of the ray trajectory in the action-angle variables satisfy equations of the form

*f*and

*H*

_{1}in the Fourier series

*x*

_{3}-axis in the form of

*x*=

*NT*(

*N*≫ 1), and then Eq. (111) can be written as

At *K* ≫ 1,\( R\left({x}_3\right)\sim {e}^{\ln K-\Omega {x}_3+ i\omega {x}_3}. \)

*f*

_{0}is the density. Then at

*K*≫ 1, the function

*F*satisfies the diffusion equation

*x*

_{3}≫

*T*.

In this case Eq. (118) describes ray diffusion, i.e., deterministic chaos.

*K*≪ 1, averaging over

*σ*

_{0}gives an estimate

*K*(

*θ*) has the form

*K*(

*θ*) ∼

*s*

^{2}.

Stochastization of the rays at *K* ≫ 1, which is characterized by the correlation function (117), is similar to such processes as pitching of the ship, movements of various gyroscopic systems, in electrical circuits of the simplest form under the influence of periodic actions, etc.

## Cross-References

## References

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