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There is a new method of calculating the trajectory of sound waves (rays) in layered stratified speed of sound in ocean without dispersion. A sound wave in the fluid is considered as a vector. The amplitudes occurring at the boundary layers of the reflected and refracted waves are calculated according to the law of addition of vectors and using the law of conservation of energy, as well as the laws that determine the angles of reflection and refraction. It is shown that in calculating the trajectories, the reflected wave must be taken into account. The reflecting wave’s value may be about 1 at certain angles of the initial wave output from the sours. Reflecting wave forms the so-called water rays, which do not touch the bottom and the surface of the ocean. The conditions of occurrence of the water rays are following. The sum of the angles of the incident and refracted waves (rays) should be a right angle, and the tangent of the angle of inclination of the incident wave is equal to the refractive index. Under these conditions, the refracted wave amplitude vanishes. All sound energy is converted into the reflected beam, and total internal reflection occurs. In this paper, the calculation of the amplitudes and beam trajectories is conducted for the canonical type of waveguide, in which the speed of sound is asymmetric parabola. The sound source is placed at the depth of the center of the parabola. Total internal reflection occurs in a narrow range of angles of exit beams from the source 43 ° - 45 °. Within this range of angles, the water rays form and not touch the bottom and surface of ocean. Outside this range, the bulk of the beam spreads, touching the bottom and the surface of the ocean. When exit corners, equal and greater than 77 °, at some distance the beam becomes horizontal and extends along the layer, without leaving it. Calculation of the wave amplitudes excludes absorption factor. Note that the formula for amplitudes of the sound waves applies to light waves.

The work is devoted to the study of the propagation of sound waves in inhomogeneous media, the speed of sound in which smoothly changes with depth. This environment may be modeled with a plurality of horizontal layers of constant speed of sound in each layer. This type of consideration depending on the depth of the sound speed is widely used in acoustics. A sound wave is described by a traveling wave phase of which consists of two components, depending on the spatial and temporal coordinates.

The direction of wave propagation in the majority of the work is determined by the wave phase. This is wrong. The wave phase does not contain its direction, as shown in [

It is known that at the boundary between two media of the incident wave arises two waves, reflection and refraction. The newly arisen wave also put pressure on the liquid. In [

In the future, for the wave passing through the boundary, instead of the term “refracted”, it will be used the term “passing”, leaving the term “refracted” to refer to beam angle, passing through the boundary of the space.

The purpose of this work is to calculate the trajectories of sound rays, using the formula for amplitudes of reflected and passing waves, received in [

We assume that the sound wave propagation occurs in an not uniform space where the speed of sound varies continuously in depth, while remaining constant in the layer thickness δz equal to the speed of sound at a depth of layer. The amplitudes of the reflected and passed waves at the boundaries between the layers of the liquid with constant speed of sound in each layer determined by the following formulas [

Here V, W―modules amplitude of the reflected wave and the wave passing through the boundary between the layers to the next layer. These formulas are applied at the boundaries of layers in depth, at which the waveguide divide. The value n = c/c_{1}―refractive index at the interface of adjacent layers, c―sound velocity in the layer where the initial wave comes, c_{1}―followed next layer. The angles θ and θ_{1} of incident and passed waves took place in the neighboring layer waves, measured from the vertical axis. Formula (1) applies for any value of the coefficient n, greater than or less than 1, it is easy to show by replacing in the formulas (1) n for n' = 1/n.

It can be seen that the amplitude of the reflected wave V as a function of n is less than the amplitude of the transmitted wave W and vanishes only in a homogeneous medium with n = 1. The amplitude of the transmitted wave W depends on the sum of the angles θ and θ_{1}, by which the waves propagate in the adjacent layers. If θ + θ_{1} = π/2, then V = 1 and W = 0. This means that the transmitted wave W is absent, its amplitude is zero. As a result the incident on the boundary layer wave is reflected totally with the opposite sign of the vertical component of the vector V at an angle θ falling. This process is called total internal reflection (TIR). When TIR arise sound wave with the changed direction enters the layer, from which it came, and already is the incident wave with respect to the adjacent layer. At the boundary of the next layer there form two waves: transmitted wave W and the reflected wave of the V, directed towards the layer, where the transmitted wave disappeared. There has been a change in direction of propagation of the wave without contact with the bottom or surface. The water wave (ray) appears. From (1) it follows that the formation of water rays in inhomogeneous media can only occur when the corners of the output rays from a source less than 45˚. As a result of refraction the sum of angles the sum of θ + θ_{1} increases, when it reaches 90˚ TIR comes. It follows that in calculating the trajectories of waves propagating in not uniform environment, you must take into account both waves, the reflected and transmitted. At certain output angles, and the value of the refractive index, they can pass one another without touching the bottom and surface forming a water rays in the waveguide.

Thus, the TIR process, the change of direction of propagation of the beam about depth, both above and below the axis of the waveguide, leads to the formation of water ray cycles without touching the surface of the waveguide and a bottom.

It is known that cosφ, while φ ≈ π/2 is changing quickly enough in comparison with the region where the phase φ is close to zero. Therefore, the change in direction of the sound wave amplitude at TIR takes place in the space of δx, δz small extent.

At present, the trajectories calculation is carried out without regard to the reflected wave, [

The difference between the angles θ, θ_{1} is small, θ_{1} = θ + δθ. Let us assume that θ_{1}-angle of refraction is equal to 90˚. According to Snell’s law, sinθ = nsin (θ_{1}) = sin (θ + δθ) = n, the refracted beam is horizontal. For horizontal beam trajectory is an infinite horizontal line at the depth where the angle of refraction become equal to 90˚. There is no way to stop this line and turn in the desired direction. Where the beam becomes horizontal, the trajectory becomes infinite. In formula (1) W consists of angles incident on the layer, and transmitted waves: cos (θ + θ_{1}) = cos (θ + θ + δθ) = cos (180 − δθ) = −cos (δθ)―finite value. According to formulas (1) it is possible to calculate the amplitude of the transmitted and reflected beams for the different propagation angles.

To calculate the trajectories adopted canonical type waveguide [

Assume the following boundary conditions for V and W: when the sound wave touches the ocean surface and the bottom there is a full reflection of the waves. Upon reflection, according to [

We assume that the sound source emits a spherical wave. The refractive indices n_{0} =c_{1}/c_{0} = 0.9762, n_{4} = c_{1}/c_{4} = 0.9942, c_{1}-speed at the depth of source z_{1} = 1 km, c_{0}, c_{4}-at the surface and bottom. During the propagation the beam from source to the bottom or to surface refraction occurs, and its trajectory is deflected towards the source location of the horizon, z_{1} = 1 km. Using the law of refraction there is can be calculated, at which the output angle the beam becomes horizontal: sin θ = n sinθ_{1}. When θ_{1} = 90˚ sin θ = n, θ ≈ 77˚ ÷ 84˚.

The trajectory of the sound beam is based on the coordinates of the beam changes from layer to layer. We introduce the following notation: i―layer number, sign (+) means that the wave propagates in the direction of the bottom, (−) in the direction of the surface, z_{i} = z_{0} ± stz * i, x_{i} = x_{o} + stz * tg (θ_{i}), x_{0} = 0, z_{0} = 1 km―initial coordinates of the beam at the exit of the source, θ_{i}―angle of beam spread in a layer i, stz = 0.25 m-thickness of each layer along the z axis.

_{1} increase at 2˚ and reach 90˚. There was TIR and, according to

the formulas (1), the amplitudes W = 0, V = 1. The direction of propagation of sound turn in depth to the contrary, the beam turns towards the surface. In the second break point x ≈ 5 km, z ≈ 0.111 km near the surface occurs again TIR: the sum of the angles θ + θ_{1} also is 90˚, W = 0, V = 1. Energy transmitted wave transmit reflected wave directed toward the bottom. After the second TIR beam returns to the source at the horizon of source, z = 1 km, completed the first full cycle of its propagation in the waveguide. Cycle length D ≈ 6 km, thus, a water ray (wave) appear and not touching the bottom and the surface of the waveguide.

_{1} along the trajectory shown in _{1} (x, z) in _{1}. Angles are both the functions of depth.

_{1} along the trajectory shown in _{1} (x, z) in _{1}. Angles are both the functions of depth.

When the waves propagate the angle changes θ, θ_{1} accumulate. The angle of inclination of the trajectory when ray exit from the source to the bottom increases from 44˚ at the horizon of the source (z_{0} = 1 km), to the 45˚ at break point of trajectory with z ≈ 3 km, x ≈ 2 km, forming first point TIR. Reflected at this point the wave propagates in the direction to the surface of the waveguide, its angle decreases (

reflected wave V between the points of TIR is so small that in

Pay attention to the following. The value of the amplitude of the transmitted wave (red line) is close to the maximum value, unit, all along the path with the exception of TIR region. As shown in _{v} = stz *10 = 2.5 m, for W-order 5 points, δz_{w} = 1.3 m. In TIR area transmitted wave W can be smaller than 1 and equal zero. The amplitude of the reflected wave V outside of TIR is several orders of magnitude less 1, but never vanishes. At TIR points amplitude V is close to 1. The above calculation shows that take account of the reflected wave in the construction of trajectories is necessary. Without its accounting reversed beam direction and the formation of water rays cannot be recorded.

The lowest water exit angle of rays in the waveguide is θ ≈ 43˚. For smaller angles of exit rays reflect from the surface and bottom of the waveguide.

The first break point of the curve is near the bottom, at a depth of z = 3.888 km and does not touch the bottom. The second break point touches the surface.

As it follows from _{1} = 1 km. The cycle of this beam is finished.

Obviously, at lower exit angles rays touch the bottom and the surface.

It is shown that at an angle θ = 45˚ output rays touch the bottom and the surface, the point of TIR absent. With further increase in the angle of exit rays are retained by the bottom and surface up to angles θ ≈ 75˚.

At this beam output angle the curvature of the trajectory can be seen as a result of refraction and on the horizon the source z = 1 km noticeable inflection curve.

_{1} = 90˚: sin θ = n = 0.9762, θ = 77˚30', sin θ = n = 0.9942, θ = 83˚50'.

The formulas for the reflection coefficient (CR) and passing (CP) obtained in [

In optics, the light wave is created atoms of matter that emit light waves of different polarization and with different initial phases. It is known to observe the interference in natural light, a special device is used, for example, Fresnel mirrors [

It is clear that the CR and the CP for any wave must not depend on the phase and polarization at the boundary of spaces. To eliminate wave’s phases of the processes at the boundary in [

Effect of polarization of light can also be excluded as the influence of phase waves at the point of reflection and refraction at the interface. We assume that during the formation of the incident wave reflected and transmitted all the characteristics of the original wave, including the polarization of the incident wave and its phase is passed unchanged the newly formed waves. If we take the boundary conditions are the same as in [

These findings can be confirmed by calculation. The paper [

Trajectories of sound waves make using a new approach to the description of sound propagation in inhomogeneous media in depth.

It is assumed that the amplitude of the sound wave is a vector whose direction is given by the sound source. The pressure produced by the sound wave in the liquid is a vector too parallel to the direction of wave propagation.

To calculate the amplitude of the reflected and transmitted waves, vector summation of amplitudes of the waves is used, which is similar to the law of conservation of momentum of material particles. The second condition for calculating the amplitudes of the waves is the law of conservation of energy.

It was found that there exist water rays that not touch the bottom and the surface of the medium under certain conditions. They are the results of total internal reflection of the wave, when all the sound energy is converted to the reflected beam and the amplitude of the transmitted wave vanishes.

When total internal reflection axis, a sign of the vertical component of the amplitude of the reflected wave, is opposite to the wave incident on the interface, the direction of wave propagation along the vertical is reversed, it remains the same along the horizontally.

Water rays occur when the sum of the angles of the incident and transmitted waves is 90˚, and the tangent of the angle of inclination of the incident wave is equal to the refractive index.

The range of output angles of rays from a source in which there are water rays does not exceed 45˚. The lower boundary of occurrence of water rays in the waveguide is considered the angle of ≈ 43˚.

When exit angles equal to 45˚ and later 77˚ in this waveguide, rays reflect from the bottom and the surface. At large angles of rays of output as a result of refraction, rays not close to the surface or the bottom, and become horizontal and distributed in a liquid made on the horizon, without changing its direction of propagation.

After the elimination of errors in the understanding of the scale production of vectors k, R, calculation of wave amplitudes can be made at any depth of the waveguide.

It is assumed that the phase of the incident at the boundary of a sound wave is the initial phase of the reflected and transmitted waves. This leads to the continuous of phases of the arising waves and their coherence.

V. P. Ivanov,G. K. Ivanova, (2016) New Method in Calculating the Trajectory of Sound Waves at Stratified Ocean. Open Journal of Acoustics,06,13-21. doi: 10.4236/oja.2016.62002