Encyclopedia of Continuum Mechanics

Living Edition
| Editors: Holm Altenbach, Andreas Öchsner

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

  • Anatoly V. ChigarevEmail author
  • Yury V. Chigarev
Living reference work entry
DOI: https://doi.org/10.1007/978-3-662-53605-6_107-1



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.

Suppose the square of the refractive index in the form of
$$\begin{aligned} {n}^2\left(\overline{r}\right)&=\left\langle {n}^2\left(\overline{r}\right)\right\rangle +V\left(\overline{r}\right)\quad \mathrm{or}\\ \varepsilon (\overline{r})&=\left\langle \varepsilon (\overline{r})\right\rangle +V(\overline{r}), \end{aligned}$$
where \( \left\langle \varepsilon \left(\overline{r}\right)\right\rangle =\left\langle {n}^2\left(\overline{r}\right)\right\rangle \) is an unrecovered value of the refractive index, or the average and \( V\left(\overline{r}\right) \) is the fluctuation of the refractive index, which takes into account the medium’s inhomogeneity, with \( \max \left|V\left(\overline{r}\right)\right|\ll \left\langle \varepsilon \left(\overline{r}\right)\right\rangle \).
Accordingly, the expansions for the coordinates of the ray and the eikonal in any inhomogeneous medium could be represented in the Cartesian coordinate system in the form
$$ \overline{r}(t)={\sum}_{k=0}^{\infty }{\overline{r}}_k(t), $$
$$ \tau \left(\overline{r}\right)={\sum}_{k=0}^{\infty }{\tau}_k\left(\overline{r}\right), $$
where \( {\overline{r}}_0(t) \) and \( {\tau}_0\left(\overline{r}\right) \) are unperturbed (middle) trajectories of the ray and the eikonal, respectively, and \( {\overline{r}}_k(t) \) and \( {\tau}_k\left(\overline{r}\right) \) (k ≥ 1) give the corrections arising from the inhomogeneity.
Substituting (2) into the Newtonian equation for potential forces (see the entry “Rays Propagation in Inhomogeneous Media” for details), a recurrent system of ordinary differential equations could be obtained
$$ {\ddot{\overline{r}}}_0=\frac{1}{2}\nabla \varepsilon \left({r}_0\right), $$
$$ {\ddot{\overline{r}}}_n=\frac{1}{2}\left({r}_n\nabla \right)\nabla \varepsilon \left({r}_0\right)+{\overline{F}}_n\, \left(n=1,2\dots \right), $$
where Fn depend only on prior approximations, i.e., they are known functions.
Substituting (3) into the equation for eikonal, a recurrent system of the first-order partial differential equations could be found for the corrections to the eikonal
$$ {\left(\nabla \tau \right)}^2=\left\langle \varepsilon \right\rangle \left(\overline{r}\right), $$
$$ 2\left(\nabla {\tau}_0,\nabla {\tau}_1\right)=V\left(\overline{r}\right), $$
$$ 2\left(\nabla {\tau}_0,\nabla {\tau}_1\right)=-{\left(\nabla {\tau}_{n-1}\right)}^2{\displaystyle \begin{array}{cc}& \left(n=2,\dots \right)\end{array}} $$
If the medium is quasi-homogeneous, then 〈ε〉 = const, and the rays will be in average straight
$$ {\overline{r}}_0(t)={\overline{r}}^{(0)}+{\overline{p}}^{(0)}t,\qquad \tau ={\tau}^{(0)}+\left\langle \varepsilon \right\rangle t, $$
and the first-order corrections have the form of
$$ {\overline{p}}_1={\overline{r}}_1={{\overline{p}}^{(0)}}_1+\int {\overline{F}}_1\left({t}^1\right){dt}^1, $$
$$ \begin{aligned}{\overline{r}}_1&={{\overline{r}}_1}^{(0)}+t{{\overline{p}}^{(0)}}_1+{\int}_0^t{dt}^1{\int}_0^{t1}{\overline{F}}_1\left({t}^2\right){dt}^2\\&={{\overline{r}}_1}^{(0)}+t{{\overline{p}}^{(0)}}_1+{\int}_0^t\left(t-{t}^1\right){\overline{F}}_1\left({\tau}^1\right){dt}^1, \end{aligned}$$
where the initial values \( {\overline{r}}^{(0)} \) and \( {{\overline{p}}_1}^{(0)} \) correspond to the initial wave surface. Assuming that t0 = 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.
Thus, within the first approximation, the solution could be found as
$$ \overline{r}(t)={\overline{r}}_0\left(\xi, \eta, s\right)+{\overline{r}}_1\left(\xi, \eta, s\right). $$

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.

In the first case, considering that the solution of Eq. (7) for the unperturbed eikonal is unknown and that \( \nabla {\tau}_0=\overline{p}=d{\overline{r}}_0/ dt \), then Eq. (7) can be written as
$$ 2\left(\nabla {\tau}_0\nabla {\tau}_x\right)=2\frac{d{\tau}_1}{dt}=V. $$
Integrating (13) along the unperturbed ray yields
$$ {\tau}_1=\frac{1}{2}{\int}_{t_0}^tV{dt}^1=\frac{1}{2}{\int}_{s_0}^s\frac{V}{\sqrt{\left\langle \varepsilon \right\rangle }} ds. $$
Accordingly, Eq. (8) at n = 2 is reduced to
$$ {\tau}_2=-\frac{1}{2}{\int}_{t_0}^t{\left(\nabla {\tau}_1\right)}^2{dt}^1=-\frac{1}{2}{\int}_{s_0}^s\frac{{\left(\nabla {\tau}_1\right)}^2}{\sqrt{\left\langle \varepsilon \right\rangle }} ds. $$

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 .

Now consider the account of the transverse ray displacement, which is determined by the solution of Eq. (5). Then the solution of (6) in this case has the form
$$\begin{aligned} \tau &={\tau}^{(0)}+{\int}_{t_0}^t\left\{\left\langle \varepsilon \right\rangle \left[{r}_0\left({t}^1\right)+{r}_1\left({t}^1\right)\right]\right.\\&\quad\left.+V\left[{r}_0\left({t}^1\right)+{r}_1\left({t}^1\right)\right]\right\}{dt}_1. \end{aligned}$$

From (16) it follows that an approximation to (14) is obtained from (16) by neglecting the transverse displacements r1. This corresponds to the condition \( \left|{\overline{r}}_1\right|\ll {{\ell}}_{\perp } \).

In case when the medium is not quasi-homogeneous \( \left\langle n\left(\overline{r}\right)\right\rangle \ne \mathrm{const} \), the method of variation of arbitrary constants could be applied which is widely used in mechanics for solving equations with variable coefficients (Kujpers 1997). Thus, the solution of Eq. (5) at k = 1 is represented as
$$ {\overline{r}}_1(t)={\sum}_{k=1}^6{c}_k(t){\overline{\rho}}_k(t)={\sum}_{k=1}^6{c}_k(t)\frac{\partial {\overline{r}}_0(t)}{\partial {\alpha}_k}, $$
where ck(t) are coefficients to be determined and \( {\overline{\rho}}_k(t) \) are the fundamental solutions of the system of uniform linear equations
$$ \ddot{\overline{r}}=\frac{1}{2}\left({\overline{\rho}}_k\nabla \right)\nabla \left\langle \varepsilon \right\rangle \left({\overline{r}}_0\right)\quad \left(k=\overline{1,6}\right). $$
The equation of the trajectory for a particular ray \( {\overline{r}}_0(t) \) depends on the six initial conditions
$$ {\displaystyle \begin{array}{l}{\overline{r}}_0^{(0)}={\overline{r}}_0(0)={\overline{\alpha}}_k\quad \left(k=1,2,3\right)\\ {}{\overline{\rho}}_0^{(0)}={\overline{\alpha}}_k\qquad \quad \left(k=4,5,6\right),\end{array}} $$
and the vector components \( \overline{\rho} \) are determined by the formulas
$$ {\overline{\rho}}_k(t)=\partial {\overline{r}}_0(t)/\partial {\alpha}_k. $$
Substituting (17) in (5) and using conventional techniques of the method of constants’ variation, a system of equations for ck(t) could be obtained
$$ {\displaystyle \begin{array}{l}\sum \limits_{k=1}^6{\dot{c}}_k(t){\overline{\rho}}_k(t)=0,\\ {}{\sum}_{k=1}^6{\dot{c}}_k(t){\dot{\overline{\rho}}}_k(t)={\overline{F}}_1=\frac{1}{2}\nabla V\left({\overline{r}}_0\right).\end{array}} $$
The solution of the system (21) is written as
$$ {c}_k(t)={c}_k(0){+}{\sum}_{j=1}^3{\int}_0^t{Q}_{kj}\left({t}^1\right){F}_{1j}\left({t}^1\right){dt}^1, $$
where ck(0) should be found from the initial conditions.

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

Consider the vector \( \overline{p}=\dot{\overline{r}} \), which can be represented as a series
$$ \overline{p}(t)={\sum}_{k=0}^{\infty }{\overline{p}}_k(t)={\sum}_{k=0}^{\infty }{\dot{\overline{r}}}_k(t). $$
From (17) with regard to the first system in (21), it follows
$$ {\overline{p}}_1=\frac{d}{dt}{\sum}_{k=1}^6{c}_k{\overline{p}}_k={\sum}_{k=1}^6{c}_k{\dot{\overline{p}}}_k. $$
The correction \( {\overline{{\ell}}}_1 \) to the unperturbed unit vector \( {\overline{\mathrm{\ell}}}_0 \), which is tangent to the undisturbed ray \( {\overline{r}}_0 \), could be calculated in terms of \( {\overline{p}}_1 \) as
$$ {\overline{{\ell}}}_1=\frac{\left({\overline{p}}_1-{\overline{{\ell}}}_0\left({\overline{{\ell}}}_0{\overline{p}}_1\right)\right)}{\sqrt{\left\langle \varepsilon \right\rangle }}=\frac{p_{\perp }}{\sqrt{\left\langle \varepsilon \right\rangle }}, $$
i.e., the vector \( {\overline{{\ell}}}_1 \) characterizes the deflection angles of the perturbed ray from the unperturbed ray.

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 σn ≪ 〈ε〉, where σn 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.

In the case when the initial emitting surface is plane and the initial emitted wave is plane too, then the x3-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
$$ {\tau}_1=\frac{1}{2\sqrt{\left\langle \varepsilon \right\rangle }}{\int}_0^{x_3}V\left({x}_1,{x}_2,{x}_3^1\right){dx}_3^1. $$
Considering that
$$ {\sigma}_n\ll \left\langle \varepsilon \right\rangle, $$
the second-order approximation can be neglected.
The correlation function Rτ of the eikonal τ has the form
$$ {R}_{\tau}\left({\overline{r}}_1,{\overline{r}}_2\right)=\frac{x_{3\min }}{2{\left(\sqrt{\left\langle \varepsilon \right\rangle}\right)}^2}{\int}_0^{\infty }{R}_n\left(\overline{\rho},\xi \right) d\xi, $$
where \( \overline{r}=\left(\overline{\rho},{x}_3\right) \), \( \overline{\rho}=\left({x}_1,{x}_2\right) \), and x3min is the minimum value from the magnitude \( {x}_3^1 \), \( {x}_3^2 \), and \( \overline{\rho}={\overline{\rho}}_1-{\overline{\rho}}_2 \).
In the case of isotropic fluctuations of the refractive index, a linear dependence of the distance along the middle ray could be obtained
$$ {R}_{\tau}\left({\overline{r}}_1,{\overline{r}}_2\right)=\frac{x_{3\min }}{2{\left(\sqrt{\left\langle \varepsilon \right\rangle}\right)}^2}{\int}_0^{\infty }{R}_n\left(\sqrt{\rho^2+{\xi}^2}\right) d\xi . $$
The correlation function Rτ of the eikonal is expressed in terms of the spatial spectrum Φn(æ) of fluctuations \( n\left(\overline{r}\right) \) according to the formula
$$ {R}_{\tau}\left({\overline{r}}_1,{\overline{r}}_2\right)={\pi}^2{x}_{3\min }{\int}_0^{\infty }{\Phi}_n\left(\mathrm{\ae}\right){J}_0\left(\mathrm{\ae}\rho \right)\mathrm{\ae}d\mathrm{\ae}. $$
The dispersion of the eikonal, which is obtained from (30) at \( \overline{\rho}=0,{x}_3^1,{x}_3^2={x}_3 \), increases linearly with the increase in the distance along x3
$$ \begin{aligned}{D}_{\tau}\left({x}_3\right)&={\sigma}_{\tau}^2\left({x}_3\right)=\frac{x_{3\min }}{2{\left(\sqrt{\left\langle \varepsilon \right\rangle}\right)}^2}{\int}_0^{\infty }{R}_n\left(\xi \right) d\xi \\&=\frac{\pi^2{x}_3}{2{\left(\sqrt{\left\langle \varepsilon \right\rangle}\right)}^2}{\int}_0^{\infty }{\Phi}_n\left(\mathrm{\ae}\right)\mathrm{\ae}d\mathrm{\ae}. \end{aligned}$$
Designating the effective integral correlation radius \( {{\ell}}_{ef}={\int}_0^{\infty }{K}_n\left(0,\xi \right) d\xi \), relationship (31) is reduced to
$$ \begin{aligned}{\sigma}_{\tau}^2\left({x}_3\right)&=\frac{x_3}{2{\left(\sqrt{\left\langle \varepsilon \right\rangle}\right)}^2}{\int}_0^{\infty }{\sigma}_n^2{K}_n(0,\xi) d\xi\\ &=\frac{x_3}{2{\left(\sqrt{\left\langle \varepsilon \right\rangle}\right)}^2}{\sigma}_n^2{{\ell}}_{ef}.\end{aligned} $$
If Rn(ρ) has a Gaussian shape
$$ {R}_n\left(\rho \right)={\sigma}_n^2{e}^{-{\rho}^2/2{{\ell}}_{ef}^2}, $$
then \( {{\ell}}_{ef}=\sqrt{\pi /2}{{\ell}}_n \), where n is the correlation radius of n and the dispersion of the eikonal depends linearly on x3
$$\begin{aligned} {\sigma}_{\tau}^2\left({x}_3\right)&=\frac{x_3}{2{\left(\sqrt{\left\langle \varepsilon \right\rangle}\right)}^2}{\int}_0^{\infty }{\sigma}_n^2{K}_n\left(0,\xi \right) d\xi \\&=\frac{x_3}{2{\left(\sqrt{\left\langle \varepsilon \right\rangle}\right)}^2}{\sigma}_n^2{{\ell}}_{ef}. \end{aligned}$$

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

The longitudinal correlation \( {R}_{II}^{\left(\tau \right)} \)of the eikonal is calculated when the points \( {\overline{r}}_1 \) and \( {\overline{r}}_2 \) are located on one ray \( \left(\overline{\rho}={\overline{\rho}}_1-{\overline{\rho}}_2=0\right) \)
$$ \begin{aligned}{R}_{II}^{\left(\tau \right)}\left({x}_3^1,{x}_3^2\right)&=\frac{x_{3\min }}{2{\left(\sqrt{\left\langle \varepsilon \right\rangle}\right)}^2}{\int}_0^{\infty }{R}_n\left(0,\xi \right) d\xi\\ &=\frac{x_{3\min }}{2{\left(\sqrt{\left\langle \varepsilon \right\rangle}\right)}^2}{\sigma}_n^2{{\ell}}_{ef}.\\[-18pt] \end{aligned}$$
From (35) it follows that \( {R}_{II}^{\left(\tau \right)} \) is independent of x3max from \( {x}_3^1 \) and \( {x}_3^2 \). The longitudinal correlation coefficient remains constant
$$\begin{aligned} {K}_{II}^{\left(\tau \right)}\left({x}_3^1,{x}_3^2\right)&=\frac{R_{II}\left({x}_3^1,{x}_3^3\right)}{\sigma_{\tau}\left({x}_3^1\right){\sigma}_{\tau}\left({x}_3^2\right)}=\sqrt{\frac{x_{3\min }}{x_{3\max }}}\\&=\left\{\begin{array}{l}\sqrt{x_3^2/{x}_3^1}\, at\, {x}_3^2\vartriangleleft {x}_3^1\, \\ {}\sqrt{x_3^1/{x}_3^2}\, at\, {x}_3^2\vartriangleright {x}_3^1\end{array}\right. \end{aligned}$$
By fixing \( {x}_3^1 \) in formula (36), it could be found that the longitudinal correlation of the eikonal (phase) is extended over a distance as the path passed the ray KII ∼ x3. Figure 1 shows the \( {x}_3^2 \)-dependence of \( {K}_{II}^{\left(\tau \right)} \) at the fixed magnitude of \( {x}_3^1 \).
Fig. 1

The \( {x}_3^2 \)-dependence of the longitudinal correlation coefficient

Cross-correlation \( {R}_{\perp}\left(\overline{\rho},L\right) \) at \( {x}_3^1={x}_3^2=L \) has the form
$$ {R}_{\perp}^{\left(\tau \right)}\left(\overline{\rho},L\right)=\frac{L}{2{\left(\sqrt{\left\langle \varepsilon \right\rangle}\right)}^2}{\int}_0^{\infty }{R}_n\left(\overline{\rho},\xi \right) d\xi . $$
In the case of isotropic Gaussian fluctuations, Eq. (37) is reduced to
$$ {R}_{\perp}^{\left(\tau \right)}\left(\overline{\rho},L\right)={\sigma}_{\tau}^2(L){e}^{-{\rho}^2/2{{\ell}}_n^2} $$

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

The case of a non-planar initial wave is shown in Fig. 2, where L1 andL2 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
$$ {R}_{\tau}\left({\overline{r}}_1,{\overline{r}}_2\right)=\frac{L_{\mathrm{min}}}{2{\left(\sqrt{\left\langle \varepsilon \right\rangle}\right)}^2}{\int}_0^{\infty }{R}_n\left(\overline{\delta}+{{\ell}}_0\xi \right) d\xi . $$
Fig. 2

The case of a non-planar initial wave

The dispersion of the eikonal at L1 = L2 = L and δ(s) = 0 is calculated by the formula
$$ {\sigma}_{\tau}^2(L)=\frac{L}{2{\left(\sqrt{\left\langle \varepsilon \right\rangle}\right)}^2}{\int}_0^{\infty }{R}_n\left({\overline{{\ell}}}_0\xi \right) d\xi . $$

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.

In a randomly inhomogeneous medium, which is statistically homogeneous and isotropic, the rays propagating in them are in average direct and orthogonal to the initial surface, and the wave surfaces (phase fronts) retain the semblance of the initial surface on average, wherein the coordinate system α, β is introduced (Fig. 3).
Fig. 3

The unperturbed surface with the coordinate system α, β

In Fig. 3, \( {\overline{{\ell}}}_0 \)is the unperturbed vector tangential to the undisturbed ray and orthogonal to the unperturbed wave surface. Due to fluctuations on the front surface from the unperturbed surface normal to the front, \( \overline{{\ell}} \) will deviate from \( {\overline{{\ell}}}_0 \) by the value \( {\overline{{\ell}}}_1 \)
$$ {\overline{{\ell}}}_1=\overline{{\ell}}-{\overline{{\ell}}}_0=\frac{\nabla_{\perp}\tau }{\left\langle n\right\rangle }, $$
where ∇ 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 τ0 = const.
Let us introduce with the undisturbed ray the Frenet trihedron formed by the unit vectors \( {\overline{{\ell}}}_0,{\overline{\alpha}}_0,{\overline{\beta}}_0 \), where \( {\overline{{\ell}}}_0 \) is the tangent, \( {\overline{\alpha}}_0 \) is the principal normal, and \( {\overline{\beta}}_0 \) is binormal vector, such that
$$ {\overline{{\ell}}}_1={t}_{1\alpha }{\overline{\alpha}}_0+{t}_{1\beta }{\overline{\beta}}_0. $$
Then up to the smallness of the second order, the angles of arrival of the perturbed ray θα and θβ will be equal
$$ {\displaystyle \begin{array}{l}{\theta}_{\alpha}\approx {t}_{1\alpha }=\frac{1}{\left\langle n\right\rangle}\left({\overline{\alpha}}_0{\nabla}_{\perp }{\tau}_1\right)=\frac{1}{\left\langle n\right\rangle}\frac{\partial {\tau}_1}{\partial {\rho}_{\alpha }},\, \\ {}{\theta}_{\beta }=\frac{1}{\left\langle n\right\rangle}\frac{\partial {\tau}_1}{\partial {\rho}_{\beta }}.\end{array}} $$
From (43) it follows that the average values of the angles θα and θβ, considering that 〈τ1〉 = 0, will be equal to
$$ \left\langle {\theta}_{\alpha}\right\rangle =\left\langle {\theta}_{\beta}\right\rangle =0, $$
and the correlation matrixes are given by expressions
$$ {R}_{\alpha \beta}^{\theta}\left({\overline{\rho}}_1,{\overline{\rho}}_2\right){=}\left\langle {\theta}_{\alpha}\left({\overline{\rho}}_1\right){\theta}_{\beta}\left({\overline{\rho}}_2\right)\right\rangle {=}\frac{1}{\langle n\rangle}\frac{\partial^2{R}_{\perp}\left(,{\overline{p}}_2\right)}{\partial {\rho}_{1\alpha}\partial {\rho}_{2\alpha }}. $$
Dispersions of the angles θαand θβ are equal to
$$ {\left\langle {\theta}_{\alpha}^2\right\rangle}_{sph}=\frac{1}{3}{\left\langle {\theta}_{\alpha}^2\right\rangle}_{pl}. $$

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.

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 an average plane wave, the coordinates of points of the ray trajectory are written as (1), and then within the first approximation the following relationship is valid:
$$ {\overline{r}}_1={\int}_o^L{\overline{{\ell}}}_1 ds={\int}_0^L{\nabla}_{\perp }{\tau}_1{dx}_3^1. $$

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 x3-direction, and then \( {\overline{r}}_1 \) depends on x1, x2.

Calculating the elements of the correlation matrix, we obtain
$$ \begin{aligned}{R}_{\alpha \beta}^r&=(\rho, L)={R}_{\perp}^r(\rho, L)\left({\delta}_{\alpha \beta}-\frac{\rho_{\alpha }{\rho}_{\beta }}{\rho^2}\right)\\&\quad+{R}_{II}^r(\rho, L)\frac{\rho_{\alpha }{\rho}_{\beta }}{\rho^2}\end{aligned} $$
$$ \begin{aligned}{R}_{\perp}^r\left(\rho, L\right)&=-\frac{L^3}{6}{\int}_0^{\infty}\frac{\partial^2{R}_n\left(\sqrt{\rho^2+{\xi}^2}\right)}{\partial \rho } d\xi \\ {R}_{II}^r\left(\rho, L\right)&=-\frac{L^3}{6}{\int}_0^{\infty}\frac{\partial^2{R}_n\left(\sqrt{\rho^2+{\xi}^2}\right)}{\partial \rho } d\xi, \\ \rho &=\sqrt{\rho_{\alpha}^2+{\rho}_{\beta}^2}\end{aligned} $$
From formulas (48) and (49), it follows that \( {r}_{1\alpha}\left({\overline{\rho}}_1\right) \) and \( {r}_{1\beta}\left({\overline{\rho}}_2\right)\, \)at ρ1 = ρ2 are uncorrelated, i.e.,
$$ \left\langle {r}_{1\alpha }{r}_{1\beta}\right\rangle ={R}_{\alpha \beta}^r(0)=0, $$
and mean squares (dispersions) are equal to each other, which is a consequence of fluctuations of the isotropic medium inhomogeneity
$$ {\left\langle {r}_{1\alpha}^2\right\rangle}_{pl}=\left\langle {r}_{1\beta}^2\right\rangle ={R}_{\alpha \alpha}^r(0)=-\frac{L^2}{6}{\int}_0^{\infty}\frac{R_r^{\prime}\left(\xi \right)}{\xi } d\xi . $$
In the case of a spherical wave
$$ {\left\langle {r}_{1\alpha}^2\right\rangle}_{sph}=\frac{1}{10}{\left\langle {r}_{1\alpha}^2\right\rangle}_{pl}, $$
i.e., the mean square of the lateral displacement of the ray on a spherical wave is ten times less than on a plane wave.
In a randomly inhomogeneous medium, the time of signal propagation is determined by the group velocity
$$ {c}_{\Gamma P}=c/{\left(\frac{\partial \left(\omega n\right)}{\partial \omega}\right)}^{-1}, $$
$$ L=c{\int}_0^L\frac{ds}{C_{gr}}={\int}_0^L\frac{\partial \left(\omega n\right)}{\partial \omega } ds $$
In a statistically homogeneous isotropic medium 〈ε〉 = const, what means that there is no regular refraction (bending) of the ray. Introducing L as
$$ L={\sum}_{K=0}^{\infty }{L}_K, $$
the first approximation gives
$$ {L}_1=\frac{1}{2}{\int}_0^L\frac{\partial }{\partial \omega}\left(\omega \frac{V}{\sqrt{\left\langle \varepsilon \right\rangle }}\right) ds. $$
In the absence of a temporary (frequency) dispersion, 〈ε〉 and V are independent of ω, and then from (56) it follows
$$ {L}_1=\frac{1}{2}{\int}_0^L\left(\omega \frac{\nu }{\sqrt{\left\langle \varepsilon \right\rangle }}\right) ds={\tau}_1, $$
whence it is evident that within this approximation the group path correction L1 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
$$ {\sigma}_L^2={\sigma}_{\tau}^2=\frac{1}{2\left(\varepsilon \right)}{\int}_0^L dS{\int}_0^L{R}_n\left[\overline {l_0}\xi; \overline {r_0}(S)\right] dS. $$

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.

Let us introduce the Cartesian coordinate system x1, x2, x3 on the initial surface such that the x3-axis has the unperturbed ray as a directional vector and x1, x2-axes are located in the tangent plane to the normal \( {S}_t^0 \) to the ray \( {\overline{r}}^0 \) (Fig. 4).
Fig. 4

Scheme of ray fluctuation in an inhomogeneous medium

The coordinate \( \overline{r} \) of an arbitrary point 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
$$ \frac{d\overline{\rho}\left({x}_3\right)}{dx_3}=\frac{\overline{\tau_{\perp }}\left({x}_3\right)}{\sqrt{1-\overline{\tau_{\perp }}}};\frac{d\overline{\tau_{\perp }}}{dx_3}=\frac{\overline{a_{\perp }}\left(\overline{\rho},\overline{\tau_{\perp }},{x}_3\right)}{\sqrt{1-{\overline{\tau_{\perp}}}^2}}, $$
where \( \overline{\tau_{\perp }}=\left({\tau}_1,{\tau}_1\right) \), \( \overline{a_{\perp }}\left({a}_1,{a}_2\right) \), and \( \overline{\tau}\left({\tau}_1,{\tau}_2,{\tau}_3\right) \).
Equation (59) could be used at \( {\overline{\tau_{\perp}}}^2<1 \). The caseτ = 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
$$ \frac{d\overline{\rho}\left({x}_3\right)}{dx_3}={\tau}_{\perp}\left({x}_3\right),\frac{d\overline{\tau_{\perp }}\left({x}_3\right)}{dx_3}={\nabla}_{\perp}\mu \left(\overline{\rho},{x}_3\right), $$
where μ =  ln n.
Within an approximation of the diffusion random process, the Einstein-Fokker equation (EFE) for the density probability has the form (Chigarev 2000; Van Kampen 1988)
$$ \frac{\partial P\left(\overline{\rho},{x}_3\right)}{\partial {x}_3}+\overline{\tau_{\perp }}\frac{\partial P}{\partial \overline{\rho}}-D\frac{\partial^2P}{\partial {\overline{\tau_{\perp}}}^2}=0, $$
where D is the diffusion coefficient
$$ D={\pi}^2{\int}_0^{\infty }d{\mathrm{\ae \ae}}^2\Phi \left(\mathrm{\ae}\right), $$
and Φ(æ) is the three-dimensional spectral density of the correlation function for μ.
The initial condition for (61) has the form
$$ {P}_0\left(\overline{\rho},\overline{\tau_{\perp }}\right)=\delta \left(\overline{\rho}\right)\delta \left(\overline{\tau_{\perp }}\right). $$
The solution of (61) with the initial condition (63) is the Gaussian distribution with the moments
$$ {\displaystyle \begin{array}{l}\left\langle {\rho}_i\left({x}_3\right){\rho}_k\left({x}_3\right)\right\rangle =\left\langle {x}_i\left({x}_3\right){x}_k\left({x}_3\right)\right\rangle =\frac{2}{3}D{\delta}_{ik}{x}_3^3,\\ {}\left\langle {\rho}_i\left({x}_3\right){\tau}_{\perp k}\left({x}_3\right)\right\rangle =D{\delta}_{ik}{x}_3^2,\\ {}\left\langle {\tau}_{\perp i}\left({x}_3\right){\tau}_{\perp k}\left({x}_3\right)\right\rangle =2D{\delta}_{ik}{x}_3\begin{array}{cc}& \left(i,k=1,2\right)\end{array}.\end{array}} $$
The longitudinal correlation of the trajectories’ points located on one ray is calculated by the formula
$$ \left\langle \overline{\rho}\left({x}_3\right)\overline{\rho}\left({x}_3^{\prime}\right)\right\rangle =2D{\left({x}^{\prime}\right)}^2\left({x}_3-\frac{1}{3}{x}_3^1\right), $$
and the correlation coefficient according to the formula
$$ \frac{\left\langle \overline{\rho}\left({x}_3\right)\overline{\rho}\left({x}_3^{\prime}\right)\right\rangle }{\sqrt{\left\langle {\overline{\rho}}^2\left({x}_3\right){\overline{\rho}}^2\left({x}_3^{\prime}\right)\right\rangle }}=\left(1+\frac{3}{2}\xi \right){\left(1+\xi \right)}^{-3/2}, $$
where \( \xi =\frac{\left|{x}_3-{x}_3^1\right|}{x_{3\min }} \).

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

The set of equations for the coordinate points of the ray trajectory has the form
$$ \frac{d\overline{\rho_k}}{dx_3}=\overline{\tau_{\perp k}};\frac{d\overline{\tau_{\perp k}}}{dx_3}=\frac{\partial \mu \left({\rho}_k,{x}_3\right)}{\partial \overline{\rho_k}}, $$
where k = 1, 2 are the ray numbers.
The density of probability describing the relative diffusion of two rays
$$\begin{aligned} &P\left(\overline{\rho}^\ast, \overline{l}^\ast, {x}_3\right)=\left\langle \delta \left(\overline{\rho_1}-\overline{\rho_2}-\rho^\ast \right)\right.\\&\quad\left.\delta \left(\overline{\tau_{\perp 1}}-\overline{\tau_{\perp 2}}-\overline{l}^\ast \right)\right\rangle \overline{\rho}^\ast \end{aligned}$$
satisfies EFE, which has the form
$$ {\displaystyle \begin{array}{l}\frac{\partial P}{\partial {x}_3}+\overline{l}^\ast \frac{\partial P}{\partial \overline{\rho^\ast }}-{D}_{\alpha \beta}\left(\overline{\rho^\ast}\right)\frac{\partial^2P}{\partial {\overline{l}}_{\alpha}^{\ast}\partial {\overline{l}}_{\beta}^{\ast }}=0,\\ {}{D}_{\alpha \beta}=2\pi \int d\mathrm{\ae}\left[1-\cos \overline{\mathrm{\ae}}\overline{\rho^\ast}\right]{\mathrm{\ae}}_{\alpha }{\mathrm{\ae}}_{\beta}\Phi \left(\overline{\mathrm{\ae}}\right).\end{array}} $$

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 dn << ρ, then Dαβ = 2αβ. This corresponds to the model, when the distance ρ between the rays on the initial surface is much larger than the correlation radius dn for a random field gradn. 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.

If ρ<<dn, 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
$$ \begin{aligned}\frac{\partial P}{\partial {x}_3}+\overline{l_{\alpha }}\frac{\partial P}{\partial \overline{\rho_{\alpha }}}&=\pi B\left({\rho}^2{\delta}_{\alpha \beta}+2\overline{\rho_{\alpha }}\,\overline{\rho_{\beta }}\right)\frac{\partial^2P}{\partial {l}_{\alpha}\partial {l}_{\beta }}\\&=0 \end{aligned}$$

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 x3 = const and 〈S(x3)〉 = S0.

The end of the unit vector \( \overline{\tau} \), tangent to the ray, performs random walks along the unit sphere, depending on 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 x1, x2, x3 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).
Fig. 5

The Frenet trihedron in a spherical coordinate system

Vector \( \overline{\tau} \) has the components
$$ \overline{\tau}=\left({\tau}_1,{\tau}_2,{\tau}_3\right)=\left(\cos \varphi \sin \theta, \sin \varphi \sin \theta, \cos \theta \right), $$
and equations for the ray could be written as
$$ \begin{aligned}\frac{{d x}_1}{d s}&{=}\cos {\varphi} \sin {\theta}, \, \frac{{d x}_2}{d s}{=}\sin {\varphi} \sin {\theta}, \, \frac{{d x}_3}{d s}{=}\cos {\theta}, \\ \frac{d\theta}{d s}&{=}\cos \theta \left(\cos \varphi \frac{\partial \mu }{\partial {x}_1}{+}\sin \varphi \frac{\partial \mu }{\partial {x}_2}\right){-}\sin \theta \frac{\partial \mu }{\partial {x}_3},\\ \frac{d\varphi}{d s}&=-\frac{\sin \varphi }{\sin \theta}\frac{\partial \mu }{\partial {x}_1}+\frac{\cos \varphi }{\sin \theta}\frac{\partial \mu }{\partial {x}_2},\end{aligned} $$
where \( \mu =\ln n\left(\overline{r}\right) \) is a random function.
The equations for the probability distribution functions \( W\left(s,\overline{r},\theta, \varphi \right) \) satisfy the Liouville equation, which describes the preservation of the volume in the phase space (Chigarev 2000; Chigarev and Chigarev 2018). If the probability density \( P\left(s,\overline{r},\theta, \varphi \right) \) is connected with 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 \( P\left(s,\overline{r},\theta, \varphi \right) \)
$$ \underset{V}{\int }d\overline{r}{\int}_0^{\pi}\sin \theta d\theta \underset{-\pi }{\overset{\pi }{\int }} d\varphi P\left(s,\overline{r},\theta, \varphi \right)=1, $$
and the function \( P\left(s,\overline{r},\theta, \varphi \right) \) satisfies the equation
$$ \begin{aligned}&\frac{\partial P}{\partial s}{-}\frac{\partial P}{\partial \theta}\left\{\frac{\partial \mu }{\partial {x}_3}\sin \theta {-}\left[\frac{\partial \mu }{\partial {x}_1}\cos \varphi +\frac{\partial \mu }{\partial {x}_2}\sin \varphi \right]\right.\\&\quad\left.\cos \theta \vphantom{\frac{\partial P}{\partial s}}\right\}=\overline{\tau}\left(2P\nabla \mu -\nabla P\right). \end{aligned}$$
It is evident that the initial condition for (74) has the form
$$ \begin{aligned}&P\left(s,\overline{r},\theta, \varphi \right)=W\left({\overline{r}}_0,{\theta}_0,{\varphi}_0\right){\left(\sin {\theta}_0\right)}^{-1}\\&\quad=\delta \left(\overline{r}(0)-\overline{r}\right)\delta \left(\theta (0)-{\theta}_0\right){\left(\sin {\theta}_0\right)}^{-1}. \end{aligned}$$
In the process of ray diffusion, the probability density represents the average of the product of δ- functions in random variables
$$ \begin{aligned}P(s,\overline{r},\theta, \varphi)&=W(s,\overline{r},\theta, \varphi){(\sin {\theta}_0)}^{-1}\\&={\left(\sin {\theta}_0\right)}^{-1}\left\langle \delta \left(\overline{r}(s)-\overline{r}\right)\delta \right.1\\&\quad\left.\left(\theta (s)-{\theta}_0\right)\delta \left(\varphi (s)-\varphi \right)\right\rangle .\end{aligned} $$
In case if the medium is statistically homogeneous and isotropic, then the stationary ray mode of the fluctuation is possible, for which P satisfies the equation
$$ \nabla P-2P\nabla \mu =0, $$
the solution of which could be written in the form
$$ P\left(s,\overline{r},\theta, \varphi \right)=\frac{n^2\left(\overline{r}\right)}{4\pi \int {drn}^2(r)}, $$
and its average is
$$ \left\langle P\left(s,\overline{r},\theta, \varphi \right)\right\rangle =\left\langle \frac{n^2\left(\overline{r}\right)}{4\pi \int {n}^2\left(\overline{r}\right)d\overline{r}}\right\rangle . $$

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 〈n2〉. 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.

In accordance with the optical-and-mechanical analogy, differential equations for the ray trajectories in inhomogeneous media are derived from the Fermat principle
$$ \tau =\underset{M_0}{\overset{M}{\int }}n\left(\overline{r}\right) ds $$
and in the Cartesian coordinate system could be written as
$$ {\displaystyle \begin{array}{l}\frac{d\overline{r}}{ds}=\overline{l},\\ {}\frac{d}{ds}\left(n\left(\overline{r}\right)\frac{d\overline{r}}{ds}\right)=\overline{V}\tau (r)=\operatorname{grad}\tau .\end{array}} $$
The wave surface is determined by the eikonal equation
$$ \frac{d\tau}{d s}=n\left(\overline{r}\right), $$
where \( \overline{r}=\overline{r}\left({x}_1,{x}_2,{x}_3\right) \) and s is the distance along the ray trajectory.
Equation (81) could be rewritten in the Hamiltonian form, assuming that the Hamiltonian H has the form of
$$ H=-\sqrt{n^2-{p}^2}, $$
where \( \overline{p}=\left({p}_1,{p}_2,{p}_3\right) \) and \( p=\left|p\right|=\sqrt{{p_1}^2+{p_2}^2+{p_3}^2} \).
Then, dividing the spatial coordinates \( \left({x}_1,{x}_2,{x}_3\right)=\overline{r} \), for example, the x1, x2-coordinates are chosen on a locally plane wave and x3-axis is directed along the straight ray orthogonal to the plane (0, x1, x2) (Fig. 6).
Fig. 6

The spatial coordinates on a local plane wave at the time t = 0 and at the moment t

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.

Impulse \( \overline{\rho} \) is introduced by the relation
$$ \overline{p}=\frac{n\dot{\overline{p}}}{\sqrt{1-{p}^2}},\dot{\overline{p}}=\frac{d\overline{p}}{dx_3}. $$
Considering (83) and (84), the Hamiltonian form of the ray equations has the form
$$ \frac{d\overline{p}}{dx_3}=-\frac{\partial H}{\partial \overline{p}},\frac{d\overline{p}}{dx_3}=\frac{\partial H}{\partial \overline{p}}. $$

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.

The 0x3-axis of the waveguide is perpendicular to the plane 0x1x2, and it is an attractor for ray trajectories, if the change n along x1, x2 has the character of monotonic increase with a maximum on the x3-axis (Fig. 7).
Fig. 7

The dependence n(ρ) in a cross section of the waveguide

In this case, the rays emitted in the waveguide parallel to the x3-axis and the rays entering into the waveguide at some angles oscillate near the x3-axis. The ray acts like a ball received an initial impulse moving along an ideal (frictionless) chute along the x3-axis (a potential well locates in the plane 0x1x2).

Taking account for the presence of a certain type of inhomogeneity along the x3-axis may result in a deterministic chaos in radiation pattern.

Let us represent the refractive index \( {n}^2\left(\overline{\rho},{x}_3\right) \) as
$$ {n}^2\left(\overline{\rho},{x}_3\right)={n}^2\left(\overline{\rho}\right)+V\left(\overline{\rho},{x}_3\right), $$
where \( {n}^2\left(\overline{\rho}\right) \) is the refractive index in the waveguide excluding inhomogeneity in the x3-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.
In accordance with the representation (86), using the classical perturbation theory, it could be written (Chigarev and Chigarev 2018; Borisov et al. 2015)
$$ H={H}_0\left(\overline{p},\overline{p}\right)+\varepsilon {H}_1\left(\overline{p},\overline{p},{x}_3\right), $$
where the summands H0 and H1 have the form
$$ {H}_0\left(\overline{p},\overline{p}\right)=-\sqrt{n_{\left(\overline{p}\right)}^2-{p}^2}, $$
$$ {H}_1\left(\overline{p},\overline{p},{x}_3\right)=V\left(\overline{\rho},{x}_3\right){H}_0^{-1}. $$

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 x3-axis, the ray pattern will be the same, so for simplicity a 2D case will be considered below, when \( n\left(\overline{\rho}\right)=n\left({x}_1\right) \).

Equations (87, 88, and 89) and therefore (85) are reduced to
$$ H={H}_0+\varepsilon {H}_1\left({x}_1,p,{x}_3\right), $$
$$ {H}_0=-\sqrt{n^2\left({x}_1\right)-{p}^2}, $$
$$ {H}_1=V\left({x}_1,{x}_3\right){H}_0^{-1}, $$
$$ \frac{dp}{dx_3}=-\frac{\partial H}{\partial {x}_1},\quad \frac{dx_1}{dx_3}=\frac{\partial H}{\partial p},\quad p={p}_1. $$
In the phase plane x1, p1, the phase trajectories of the ray have the form corresponding to the center (Fig. 8).
Fig. 8

Phase trajectories of the rays in the waveguide

Let us introduce the action-angle variables (I, θ)
$$ I=\frac{1}{2\pi}\int {p}_1{dx}_1,\theta =\frac{\partial S\left({x}_1,I\right)}{\partial I}, $$
$$ S\left({x}_1,I\right)=\underset{0}{\overset{x_1}{\int }}{pdx}_1,p=\sqrt{n^2\left({x}_1\right)-{t}^2}, $$
which are associated with the wave front energy E along the ray by the relation
$$ \frac{dE(I)}{dI}=\omega (I),\theta =\omega t+\varphi, $$
where ω(I) is the frequency.
Writing the Hamiltonian and equations of the ray trajectories in the variables I and θ yields
$$ H\left(I,\theta, {x}_3\right)={H}_0(I)+\varepsilon {H}_1\left(I,\theta, {x}_3\right), $$
$$ I=-\varepsilon \frac{\partial {H}_1}{\partial \theta },\theta =\omega (I)+\varepsilon \frac{\partial {H}_1}{\partial I}. $$

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.

Parameter α defined by the formula
$$ \alpha =\left|\frac{I}{\omega}\ast \frac{d\omega}{d I}\right|=2\left|\frac{\partial^2{H}_0}{\partial^2I}/\frac{\partial {H}_0}{\partial {I}^2}\right|. $$
characterizes the degree of the system’s nonlinearity.
The disturbance H1 could be presented in the form of a double Fourier series
$$ {H}_1\left(I,\theta, {x}_3\right)=\sum \limits_{m,n}\frac{H_{mn}^{(1)}(I)}{2}{e}^{i\left(m\Omega + n\theta \right)}. $$
The condition for the occurrence of nonlinear resonance in the ray vibrations can be written as
$$ m\Omega + n\omega \approx 0,\quad \dot{\sigma}=\Omega, $$
where Ω = T/2π 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.

For a single 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
$$ \dot{I}=\frac{1}{2}\varepsilon {H}_{mn}^{(1)}\sin {\psi}_{mn}, $$
$$ {\dot{\psi}}_{mn}=m\Omega + n\omega (I)+\frac{1}{2}\varepsilon {nH}_{mn,I}^{(1)}\cos {\psi}_{mn}, $$
where \( {\psi}_{mn}=m\dot{\sigma}+ n\theta \) is the resonance phase and a comma denotes differentiation with respect to the action I.

In the case of the moderate nonlinearity characterized by the parameter ε ≪ α ≪ ε−1, the difference ΔI = I − IP is small (|ΔI| ≪ IP), where IP is the value of I which turns the approximate equality (101) into the exact equality, i.e., mΩ + (IP) = 0.

Equations (102) correspond to the Hamiltonian
$$ \begin{aligned}{H}_A&=m{\omega}_{,I}\frac{{\left(\Delta I\right)}^2}{2}+\varepsilon {nH}_{mn}^{(1)}\left({I}_P\right)\cos {\psi}_{mn},\\ {\omega}_{,I}&=\frac{\Delta \omega }{{\left.\Delta I\right|}_{I={I}_P}}.\end{aligned} $$
The maximum width of the nonlinear resonance could be obtained from (103)
$$ {\displaystyle \begin{array}{l}\left(\Delta I\right)=\sqrt[4]{\varepsilon {\left({\omega}_{,I}\right)}^{-1}{H}_{mn}^{(1)}},\\ {}\left(\Delta \omega \right)=\left({\omega}_{,I}\right)\left(\Delta I\right)=\sqrt[4]{{\varepsilon \omega}_{,I}{H}_{mn}^{(1)}}=4{n}^{-1}{\Omega}_{mn},\end{array}} $$
where Ωmn is the frequency of small phase oscillations.
The condition of moderate nonlinearity α means little changes in the values of I and ω
$$ \begin{aligned}\frac{\Delta I}{I}&=C{\left(\frac{\varepsilon }{\alpha}\right)}^{1/2},\quad \frac{\Delta \omega }{\omega }=C{\left(\varepsilon \alpha \right)}^{1/2},\\ C&=4{\left(\frac{H_{mn}^{(1)}}{I\omega}\right)}^{1/2}, \end{aligned}$$
or approximately
$$ \frac{\Delta I}{I}\sim {\left(\frac{\varepsilon }{\alpha}\right)}^{1/2},\frac{\Delta \omega }{\omega}\sim {\left(\varepsilon \alpha \right)}^{1/2}. $$

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 ΔI.

For the moderate nonlinearity α ∼ 1, the decomposition is carried out by ε1/2, indicating according to (106) that in the case of the resonance ΔI ∼ ε1/2, i.e., more than in a non-resonance case.

The condition of stabilization of nonlinear resonances has the form of
$$ \varepsilon {nH}_{mn,I}^{(1)}\le n\left(\Delta \omega \right), $$
$$ {\varepsilon \alpha}^{-1}\le 16\omega {H}_{mn,I}^{(1)}{I}^{-1}{\left({H}_{mn,I}\right)}^{-2}. $$
Consider now the case when the resonances strongly interact with each other, the consequence of which is a stochastic instability of the ray oscillations. Resonance coupling coefficient is introduced by the formula
$$ s=\left(\Delta \omega \right){\Delta}^{-1}, $$
where (Δω) is defined by the formula (106) and represents the width of the resonance and Δ = |ωi + 1 − ωi| 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 H1 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.

Let disturbance have the form of
$$ {H}_1\left(I,\theta, \sigma \right)={TH}_1\left(\theta \right)\sum \limits_k\delta \left({x}_3- kT\right). $$
Using the canonical transformations \( I\to \overset{\sim }{I} \) and \( \theta \to \overset{\sim }{\theta } \), the equations of motion could be written in the form
$$ {\displaystyle \begin{array}{l}\dot{I}=\dot{I}-\varepsilon T\frac{\partial {H}_1}{\partial \theta }=\dot{I}-\varepsilon {I}_0f\left(\theta \right),\\ {}\dot{\tilde{\theta}}=\dot{\theta}+ T\omega (I),\end{array}} $$
where f(θ) ∼ 1 is a dimensionless phase function and I0 is constantly acting value.

The transformation (110) describes the motion of the ray between the “points” of the inhomogeneity along the x3-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  = 2πm, where m is a whole number.

The system of resonances corresponds to the Fourier series expansion of the periodic δ- function
$$ \sum \limits_k\delta \left({x}_3- kT\right)=\frac{1}{T}\sum \limits_{m=-\infty}^{\infty }{e}^{im\Omega {x}_3}, $$
where the distance between the resonances is equal to \( \Delta =\frac{2\pi }{T}=\Omega \) and \( {H}_{m_1}^{(1)}=\frac{2{I}_0}{T} \).
Let the variables I and σ of the ray trajectory in the action-angle variables satisfy equations of the form
$$ {\displaystyle \begin{array}{l}\dot{I}=\varepsilon I\sum \limits_{k=-\infty}^{\infty}\delta \left({x}_3- kT\right)\sin \sigma =\varepsilon {H}_1\left(I,\sigma, {x}_3\right),\\ {}\dot{\sigma}=\omega (I).\end{array}} $$
The condition of conservation of the phase volume has the form of (Chigarev 2000)
$$ \frac{\partial f}{\partial {x}_3}+\omega \frac{\partial f}{\partial \sigma }+\varepsilon \frac{\partial }{\partial I}\left({H}_1f\right)=0. $$
Then expanding f and H1 in the Fourier series
$$ {\displaystyle \begin{array}{l}f\left(I,\sigma, {x}_3\right)=\sum \limits_{h=-\infty}^{\infty }{f}_n\left(I,{x}_3\right){e}^{in\sigma}\begin{array}{cc}& \left(n=\pm 1\right),\end{array}\\ {}{H}_1\left(I,\sigma, {x}_3\right)=\sum \limits_{h,k=-\infty}^{\infty }{H}_{nk}^{(1)}{e}^{i\left( n\sigma +k\Omega {x}_3\right)}\begin{array}{cc}& \Omega {=}\frac{2\pi }{T},\end{array}\end{array}} $$
and substituting (114) in (113) yield
$$ \frac{\partial {f}_n}{\partial {x}_3}+ in\omega {f}_n=-\varepsilon \frac{\partial }{\partial I}\sum \limits_{s,k}{H}_{sk}^{(1)}{}^{\ast} {f}_{n-s}{e}^{ik\Omega {x}_3}. $$
Consider the estimation of the probability of occurrence of deterministic chaos in a radiation pattern. Propose the discretization along the x3-axis in the form of x = NT (N ≫ 1), and then Eq. (111) can be written as
$$ {\displaystyle \begin{array}{l}{I}_{m+1}={I}_m+\varepsilon {I}_m\sin {\sigma}_m,\\ {}{\sigma}_{m+1}\approx {\sigma}_m+\omega T+{K}_m\sin {\sigma}_m,\end{array}} $$
where \( {K}_m=\frac{\varepsilon {I}_m}{\Omega}\frac{d\omega \left({I}_m\right)}{{d I}_m}. \)
The correlation function of the phase has the form
$$ R\left({x}_3\right)={R}_N=\frac{1}{2\pi }{\int}_0^{2\pi }d{\sigma}_0{e}^{i\left({\sigma}_N-{\sigma}_0\right)}. $$

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

Assume that the ergodicity conditions are met, and introduce the distribution function \( F={\left(2\pi \right)}^{-1}\underset{0}{\overset{2\pi }{\int }}{f}_0d{\sigma}_0 \), where f0 is the density. Then at K ≫ 1, the function F satisfies the diffusion equation
$$ \frac{\partial F}{\partial {x}_3}{=}2{\pi \varepsilon}^2\frac{\partial }{\partial I}\sum \limits_{n,k}\left|{H}_{nk}^{(1)}\right|\delta (n\omega {-}k\Omega)\frac{\partial }{\partial I}\left|{H}_{nk}^{(1)}\right|F, $$
where x3 ≫ T.

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

At K ≪ 1, averaging over σ0 gives an estimate
$$ R\left({x}_3\right)\sim \exp \left\{ i\omega {x}_3\left(1+O(k)\right)\right\}, $$
which shows that no randomization occurs, because correlation between the points of the ray does not decrease.
Coefficient K(θ) has the form
$$ K\left(\theta \right)\sim \frac{d\overset{\sim }{\theta }}{d\theta}-1=-{\varepsilon \omega}_{,I}{I}_0{Tf}_{,0}^{(0)}, $$
with K(θ) ∼ s2.

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.



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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Theoretical and Material MechanicsBelarusian National Technical UniversityMinskBelarus
  2. 2.Department of Theoretical Mechanics and Theory of Mechanisms and MashinesBelarusian State Agrarian Technical UniversityMinskBelarus

Section editors and affiliations

  • Marina V. Shitikova
    • 1
  1. 1.Research Center on Dynamics of Solids and Structures, Voronezh State Technical UniversityVoronezhRussia