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In this paper, one of analogies available in the literature between movement of a material particle and ray propagation of a sound in liquid is used. By means of the equations of Hamilton describing movement of a material particle, analytical expression of a tangent to a trajectory of a sound ray at non-uniform ocean on depth is received. The received expression for a tangent differs from traditional one, defined under law Snelius. Calculation of trajectories, and also other characteristics of a sound field is carried out by two methods: first—traditional, under law Snelius, and second—by the analogy to mechanics method. Calculations are made for canonical type of the sound channel. In the region near to horizontal rays, both methods yield close results, and in the region of steep slope, the small distinction is observed.

The ray description of a sound field is widely applied in problems on sound propagation at ocean. It is because of the big accuracy with which the ray theory defines the basic characteristics of a sound field. Among modern computing means, the ray method of calculation of a sound field does not cause any difficulties. Advantage of this method consists that it contains a minimum quantity of approximations and can serve as the standard in an evaluation of other methods. It proves to be true, for example, when calculation of a sound field of caustics is made [1,2]. Unique restriction of use of a ray method is performance of conditions of geometrical acoustics in non-uniform spaces [

Along with a traditional ray method of calculation of a sound field at the ocean, using Snelius’s law, other methods based on analogy between propagation of a sound wave at ocean and movement of a material particle are widely applied, see for example [

Other ways of an establishment of analogy between propagation of a sound and movement of a mechanical particle are offered in [

The purpose of the given work is to carry out one more variant of use of the equations of the classical mechanics, offered in [

It is known that in isotropic, homogeneous space not changing in time plane sound waves propagate in a direction of a wave vector k which does not vary neither on value, nor in a direction. At ocean, a sound speed depends on depth that makes the space not isotropic. It leads to the change of a wave vector k both on size, and in a direction. Therefore, the sound wave is not a plane one any more. However, representation about a plane wave can be used and in such space if on distances of an order of length of a wave, the value and a direction of a wave vector vary slightly. The method of replacement of a real sound wave with a set of plane waves is called as a method of geometrical acoustics [

Let’s consider briefly traditional way of definition of ray’s trajectories at ocean with speed of a sound depending on depth (a method 1). We will accept, that the ocean is a planelayered space in which speed of a sound varies with change depth z and does not depend on co-ordinates x, y. Thanks to uniform and isotropic space along axes x, y projections of a wave vector k on an axis also do not depend on co-ordinates x, y, [

Here a—amplitude of a wave,—a wave phase (eikonal), k—a wave vector, k—its module (length), R—a vector of a trajectory of the wave, parallel to a vector k, R—the module (length) of a vector, co-ordinates of the beginning of vector R—x_{0}, z_{0}, end co-ordinates (current co-ordinates of a trajectory)—x, z. We will accept, that absorption of a sound is a little, the amplitude of a wave does not vary.

According to Snelius law the component of a wave vector remains invariable for a concrete ray at non-uniform ocean on depth:

Here z_{0}—depth of a source of a sound, θ_{0}—an angle of an departure of a ray from a source,—speed of a sound on depth of a source. According to Snelius law, Equation (2), if a refraction angle on depth, where, “const” in Equation (2) is equal to. Apparently from Equation (2), the constant can be expressed through an angle of departure of a ray from a source θ_{0} and speed of a sound too, “const”. We will accept, that on distances of several lengths of waves the module of a wave vector varies slightly, considerably its direction varies only. Therefore we will enter a single vector n_{1}, coinciding on a direction with a wave vector,. Components of a single vector n_{1} are expressed through an angle. Using for a constant in Snelius law (2) expression we will receive:

Equations (3) defines components of an single vector n_{1} along axes of co-ordinates, and also a tangent to a trajectory of propagation of a wave tgθ, coinciding on a direction with a vector k. From Equation (3) it is visible, that components n_{1x}, n_{1z} depend from z, n_{1x} has the same dependence on depth, as. For a components of a wave vector taking into account Equation (3) we will receive:

It is visible, that k_{x}—a constant for the chosen ray with a turning point on depth z_{r}, k_{z}—depth function of z. Components of a wave vector, Equation (4), are received in the assumption, that periodicity of a sound field at change is not broken, conditions of geometrical acoustics are applicable. Trajectory co-ordinates x, z and length of a trajectory l, are connected by known parities:

which allow to calculate with the big accuracy a trajectory, and also time of propagation of a sound along a trajectory. Convincing acknowledgement of accuracy of calculation of a sound trajectory by a ray method can be found in many works, for example, in [

Let’s consider another method, method 2, calculation of trajectories of the sound rays, based on analogy between movement of a material particle and sound propagation in approach of the geometrical acoustics, offered in [

Let us conceder briefly principal features of the classical mechanics, describing movement of a material particle by the equations of Hamilton [

At known function of action S from first two equations it is defined Hamiltonian H of particle and its impulse p. From second two differential equations which are equations of Hamilton, co-ordinates of a trajectory of a particle q_{i} and components of its speed are calculated.

The equations of Hamilton define a particle trajectory, as well as Newton’s second equation. We will accept, that expression for eikonal in the definition of the sound wave’s field, Equation (1), is analogue of action S of a material particle. One of the functions entering in Equation (1), is a trajectory of a sound wave (a ray on which the sound wave propagates), vector R. It is natural to accept, that vector R is analogue of a vector of the generalized co-ordinate q of the particle, and projections to axes of co-ordinates of a vector of trajectory R_{i} - analogue qi. Continuing analogy between S and Ψ we will receive from Equations (6) the first equation, that analogue Hamiltonian H is frequency of radiation, , and from the second equation—analogue of an impulse p of the particle is the wave vector k, p_{i} - k_{i}. It is already formal analogy between particle’s Hamiltonian H and frequency of a sound wave and components of the generalized vector pi and a wave vector ki. The equations of Hamilton, 3, 4 in Equation (6), for a trajectory of a sound wave look like:

Let’s present a wave vector, as in a method 1, in the form of its module that don’t change on a distance of several wave’s lengths and single vector n_{2}, that varies its direction,. We will receive from the Equations (7) at stationary propagation of a sound in the motionless non-uniform space the equations for vectors n_{2} and R:

Authors [_{2}, containing an additional member, in our designations. This member has appeared at transition from to, through derivatives

and replacement on

, and on cn_{2}

from Equation (8), the second one. It is incorrect procedure since the generalized functions q and p for a mechanical particle, together with their analogues R and k in a sound wave, depend by their primary definition only on variables x, z, t, but not from each other. Therefore the correct equation for vector’s definition n_{2} is the first in Equations (8). It is easy to show, that the decision of the equation for n_{2}, resulted in [

Let’s write down system of the Equation (8) for projections of vectors R and n_{2} on an axis x, z:

The decision of the third of Equation (8’) can be or or. The first decision according to existing connection and leads to. This decision is fair only in the isotropic spaces. Remains. Since the component of a single vector n_{2x} can be expressed through becomes known after definition n_{2z}. Using a method of the decision of system of the ordinary stationary differential equations of 1st order, [

Solving the second of them, we will define a projection of a single vector n_{2z} as function of depth , where с_{1} – an integration constant. A constant с_{1} we will accept equal speed of a sound in a turning point of a ray, с_{1} = с_{r}. As a result for n_{2x}, n_{2z}, we will receive following expressions:

The first equation from system Equetion (9) defines a tangent inclination to a trajectory of a sound ray. For calculation of a trajectory of a ray it is necessary to substitute value tgθ, expressed through n_{2x}, n_{2z }in Equation (5). From Equation (10) also, as well as Equation (4) it is visible, that the component of a single vector along an axis z becomes zero in a ray’s turning points at.

Thus, at analogy use between the description of propagation the sound along a ray and movement of a material particle (method 2) analytical expressions for components of a single vector along a trajectory of the ray, Equation (10), distinct from the expressions in Equation (3) received by method (1) with use of Snelius law, Equation (2), are received.

Let’s spend comparison two ways of ray’s trajectory calculation for a profile of a sound’s speed of a canonical form. We will accept, that the axis of the sound channel lays on depth of 1 km, depth of a waveguide is 4 km, the sound source is on a channel axis. It is easy to show, that ratio is not a constant. However, calculation of this ratio for various rays shows, that it differs from a constant on small size. We will compare horizontal components of single vectors n_{1x}, n_{2x} trajectories of the rays, calculated under formulas (3) and (10). In _{1x}, n_{2x} show as functions of depth z for the rays having turning points (ТP) below an axis of a wave guide on depths z_{1} = 1.75 km and z_{2} = 3.6905 km. In _{1x}, n_{2x}, are postponed, on vertical axis—depth z. In ТP all 4 curves have the identical values equal 1, n_{1x} = n_{2x} = 1, see Equations (3) and (10), on corresponding depths. Numbers 3,

4 designate two curves coinciding on the scale of drawing having ТP on depth z_{1}. Appreciable difference n_{1x}, n_{2x} is observed near to a wave guide axis, for steep rays, curves 1, 2, , with ТP on depth z_{2}. Figures in drawing near to these curves specify, what method calculates curves. Difference of curves is caused, obviously, not only distinction of the functions describing a ray trajectory (a vector n) in methods 1, 2, but also distinction of angles of departure at which rays have the same depth.

On _{0}, calculated in two methods is shown. The divergence of curves on the scale of drawing also is observed only for steep rays (the least θ_{0}). The lower curve is calculated by a method 2, upper—on a method 1. It is visible, that length of cycle (a method 1) more than that calculated by a method 2,. So, at depth of a tuning point a ray, z = 1.0005 km, angles θ_{0 }coincide for two methods to within 5 th sign, a difference of cycle’s lengths, at

.

Let’s consider vertical structure of a sound field z(θ) in the canonical wave guide, calculated in two ways. On _{0}, ordinate—depth z arrival of a corresponding ray on distance x from a source. An initial (greatest) θ_{0} = 89.998˚. Extremes of curves is co-ordinates of the caustic’s centers on depth on the given distance x. The continuous curve on _{0 }the curves coinciding at the big angles θ_{0}, diverge a little, caustic’s co-ordinates do not coincide.

Similar curves with the caustic’s centers along an axis z only depending on time of arrival of rays, horizontal axis, are resulted in a number of works. On

arrival’s rays which have left the source towards a bottom, depending on time of their propagation, horizontal axis, is shown. By a shaped line the co-ordinates of rays calculated by a method 2 are shown. Distinction of the curves is visible only in that range of time and depth, in which the rays calculated by a method 1, are absent. In other points curves on the scale of drawing coincide. In co-ordinates the centers of caustics are tops of points of bend of curves. Comparison of numerical values of times of ray’s arrivals on the same depth shows, that times of propagation of the near horizontal rays, calculated by both methods, are very close to each other, considerable distinction is observed for steep rays. So, for corners θ ≥ 88˚ times of ray’s rrivals differ on ~ 1 - 3.10^{−4 c}, time of arrival of the most steep rays calculated on a method 2 on ~ 0.3с more than calculated on method 1.

Let’s consider structure of the module of an acoustic field along a vertical, calculated by two methods,

Distinction in caustic’s position and in value of amplitudes is visible.

As a result of comparison of two methods of obtaining of analytical expressions for a tangent to a ray trajectory, it is possible to draw the following conclusions. Practical coincidence of calculation of trajectories by two methods is the confirmation of legitimacy of the decision of a problem on sound propagation in the non-uniform space with the decision in ray approach. When the general structure of an acoustic field in a wave guide without quantitative comparison of separate characteristics is required, both methods of the account are equivalent. If it is necessary to compare calculated value with experiment, it is necessary to give preference to a traditional method 1. Under the physical concept, it is more proved than the 2nd method using formal analogy between action of a mechanical particle and eikonal of a sound wave.

The method of analogies is of interest as possibility of various comparisons by the nature of the physical phenomena, in which the general laws that promotes their deeper understanding can be found. In this case it is possibility to receive one more analytical expression for a tangent vector to a ray trajectory in approach of the geometrical acoustics, distinct from traditional, under Snelius’s law.