Electromagnetic effects caused by sea surface waves. Propagation of radio waves Surface electromagnetic waves at the earth-air boundary

Surface electromagnetic waves

Surface waves are waves that propagate along the interface between two media and penetrate these media at a distance less than the wavelength. In surface waves, all the energy is concentrated in a narrow vicinity of the interface, and the state of the surface significantly affects their propagation. That is why surface waves are a source of information about the state of the surface. Moreover, the interaction of body and surface waves can lead to various surface effects, such as harmonic generation, rotation of the plane of polarization upon reflection, and so on. The properties of surface waves for ideal surfaces were theoretically studied quite a long time ago, back in the early twentieth century. But they learned to experimentally obtain clean surfaces only at the end of the twentieth century.

In 1901, Sommerfeld found special solutions to Maxwell's equations - exponentially decaying waves propagating along the interface between two media. At that time, no attention was paid to his work; it was believed that these were completely exotic objects. In 1902, Wood, while studying the properties of metal diffraction gratings, discovered deviations in the propagation of light from the laws of diffraction at some frequencies. These abnormalities were called Wood's anomalies. In 1941, Fano explained these anomalies - the energy turns into surface waves. In 1969, Otto proposed a scheme for exciting surface waves in a metal film using a prism. In 1971, Kretschmann proposed another geometry of the same thing. In 1988, German scientists Knohl and Rothenhäusler proposed and implemented a microscope design based on surface waves.

A little theory. Maxwell's equations in a medium

Material equations

We usually look for a solution in the form of propagating plane harmonic waves.

When substituting this type of solution into the material equations, we find that  and  depend on the frequency - time dispersion, and the wave vector - spatial dispersion. The relationship between frequency and wave vector through  and  is called the dispersion relation.

In this report we will assume that  does not depend on frequency and = 1. In the optical frequency range, this condition is satisfied quite well. Since  depends on frequency, it can take different values, including negative ones.

Let us consider the problem of the incidence of a plane monochromatic wave from a medium with  1 onto an ideal surface of some substance  2.

P
In this case, the following boundary conditions are met:

AND
From these boundary conditions, when substituting the usual form of solutions, the well-known Fresnel formulas, Snell's law, etc. are obtained. However, such solutions do not always exist. Let us consider the case when the dielectric constant of the medium is negative. This case is realized in a certain frequency range in metals. Then solutions in the form of propagating waves do not exist. We will look for solutions in the form of surface waves.

Substituting this representation into the equations and boundary conditions *, we find that there are waves of the TM (transverse-magnetic) type. These are partially longitudinal waves; the electric field vector may have a longitudinal component.

D
For these waves, dispersion relations can also be obtained from the boundary conditions.

Where
- wave vector in vacuum. The frequency dependence is also implicitly present in the functions  1 () and  2 ().

So what is negative dielectric constant in metals? The basic optical properties of metals are determined by the properties of electrons. Electrons in metals are free and can move under the influence of an electric field. Moreover, they move in such a way that the field they create is opposite in direction to the external electric field. This is where the negative sign comes from. Therefore, electrons in the metal partially screen the external field, and it penetrates into the metal to a depth significantly less than the wavelength. However, if the frequency of the external field is so high that the electrons do not have time to react, then the metal becomes transparent. The characteristic frequency at which this occurs is called plasma frequency .

Here is a simple formula - the Drude formula, which shows the dependence of the dielectric constant of a metal on frequency.

where  p is the plasma frequency,  is the collision frequency.

It is also possible to explain with ease why the polarization of surface waves is precisely TM, where the electric field is parallel to the surface. Electrons cannot simply leave the metal; to do this, work must be done (work function). Therefore, if the electric field is perpendicular to the surface, it will not lead to the excitation of surface waves - electrons will lose energy at the potential barrier - the surface. Moreover, the field is variable, and it either gives energy to the electrons or takes it away, so the electron does not leave the surface. If the field is parallel to the surface, then it excites electron oscillations in the same direction, where there is no potential barrier.

AND Thus, the dispersion curve for surface waves in a metal. In the figure it is a blue curve. The red line is the dispersion curve for vacuum.

The main condition for the excitation of any waves is the condition of phase matching. Phase matching is the equality of the phase velocities of the incident wave and the surface wave. From the dispersion curves it is clear that it is impossible to excite surface waves in a metal plate by a wave incident from a vacuum. There are two ways to excite surface waves - a) disrupted total internal reflection and b) the creation of resonant structures on the surface.

A) Disturbed total internal reflection is also known as the optical tunneling effect. At the dielectric boundary, at an angle of incidence greater than the angle of total internal reflection, surface waves arise, which are then converted into volumetric reflected waves. But when the conditions of phase matching at the boundary with the metal are met, these waves can be transformed into surface waves of the metal plate. This phenomenon is the basis of prismatic excitation of surface waves.

B
) By resonant structures we mean periodic structures with a period on the order of the wavelength of surface waves. In such periodic structures, the phase matching condition changes - , where is the reciprocal lattice vector. The excitation of surface waves leads to Wood's anomalies - a change in the intensity of light diffracting on a diffraction grating, contrary to the standard law of diffraction.

P surface plasmons are excited at certain angles of incidence of light, and the intensity of light reflected from the boundary depends very much on the angle of incidence. This is the so-called plasmon resonance. When the properties of the surface change, the angle of incidence at which this resonance is observed changes, therefore, by tuning to a certain angle of incidence, you can observe a change in the intensity of the light. The action of a microscope on surface plasmons is based on this effect.

1 - laser

2 - polarizer

3 - coordinate table

4 - prism with metal film

5 - telescope

6 - photodetector

The laser is focused on the surface of the silver film on which the object of observation is located. Using a coordinate stage, the angle of incidence is selected so that it corresponds to the plasmon resonance for a pure metal. When the properties of the film change, the light intensity at the photodetector changes, and from this change one can judge the change in film thickness.

-
detecting dielectric changes. permeability at a fixed film thickness

Detection of thickness changes at a fixed diel. permeability

The uncertainty relation here, however, is not violated: but in another coordinate, in the film plane, the resolution is quite low - the laser is focused into a spot with dimensions of about 2 microns.

AND
Another application of surface waves is the prospect of application in high-resolution optical lithography.

Photoresist onto which the image of the original is transferred. Image size is about 10 nm

Perforated metal film. Effective excitation of surface waves that convey information about the structure of the original

The original is a high-resolution image produced by electron beam lithography.

light

Electron beam lithography has high resolution, but requires sequential imaging (line by line, like on a TV), which is very time-consuming for industrial applications. If this ability to make copies is realized on an industrial scale, this will significantly reduce the cost of manufacturing integrated microstructures.

Bibliography:

1. S.I. Valyansky. Surface plasmon microscope, Soros Educational Journal, No. 8, 1999

2. M.N. Libenson Surface electromagnetic waves of the optical range, Soros Educational Journal, No. 10, 1996

3. Rothenhäusler B., Knoll W. Surface Plasmon Microscopy, Nature. 1988. No. 6165. p. 615-617.

4. Born, Wolf " Optics Basics", chapter "Optics of metals"

5. F. J. Garcia-Vidal, L. Martin-Moreno Transmission and focusing of light in one-dimensional periodically nanostructured metals, Phys. Rev.B 66, 155412 (2002)

6. N.A. Gippius, S. G. Tikhodeev, A. Christ, J. Kuhl, H. Giessen . Plasmon-waveguide polaritons in metal-dielectric photonic-crystal layers, Physics of Solid State, 2005, volume 47, issue. 1

The propagation of HF by an ionospheric wave occurs through sequential reflection from the F layer (sometimes the E layer) of the ionosphere and the Earth's surface. In this case, the waves pass through the lower region of the ionosphere - layers E and D, in which they undergo absorption (Fig. 5, a). To carry out radio communication on the HF, two conditions must be met: the waves must be reflected from the ionosphere and the electromagnetic field strength in a given location must be sufficient for reception, i.e., wave absorption in the layers of the ionosphere should not be too great. These two conditions limit the range of applicable operating frequencies.

To reflect a wave, it is necessary that the operating frequency is not too high, and the electron density of the ionospheric layer is sufficient to reflect this wave in accordance with (3-44). From this condition, the maximum applicable frequency (MUF) is selected, which is the upper limit of the operating range.

The second condition limits the operating range from below: the lower the operating frequency (within the short-wave range), the stronger the absorption of the wave in the ionosphere (see Fig. 5). The lowest applicable frequency (LOF) is determined from the condition that for a given transmitter power, the electromagnetic field strength must be sufficient for reception.

The electron density of the ionosphere changes throughout the day and throughout the year. This means that the boundaries of the operating range also change, which leads to the need to change the operating wavelength during the day:

During the day they work on waves of 10-25 m, and at night on waves of 35-100 m.

The need to select the correct wavelength for communication sessions at different times complicates the station design and the operator's work.

A KB silence zone is a ring-shaped area that exists at a certain distance from the transmitting station, within which it is impossible to receive radio waves. The appearance of a silence zone is explained by the fact that the ground wave attenuates and does not reach this area (point 6 in Fig. 3-39, a), and for ionospheric waves incident at small angles on the ionosphere, the reflection conditions are not met (3-44). The limits of the silent zone (SB) expand as the wavelength shortens and the electron density decreases.

Fading in the KB band is deeper than in the CB band. The main cause of fading is the interference of rays propagating through one or two reflections from the ionosphere (Fig. 3-39, o). In addition to this, fading is caused by the scattering of radio waves on irregularities in the ionosphere and the interference of scattered waves (Fig. 3-39,6), as well as the interference of the ordinary and extraordinary components of a magnetically split wave (Fig. 3-39, c). Processing of measurements over short time intervals (up to 5 minutes) showed that the amplitude distribution functions are close to the Rayleigh distribution (3-54). Over large observation time intervals, the distribution is closer to lognormal with a standard deviation of 6±1.25 dB. In both cases, the difference between the signal field strength levels exceeded 10% and 90% of the time is 16±3.2 dB.

The fading rate (§ 3-6) lies in the range of 6 - 16 fading per minute. On lines with a length of 3000 km, the fading rate is 2 - 6 times less than on a line with a length of 6000 km. The correlation time interval ranges from ?o = 4.5 - 1.5 s. The scale of spatial correlation depends on the length of the radio communication line, operating frequency, the nature of ionospheric inhomogeneities and lies within the range rо==210-560 m (10 - 25?). To combat fading, reception with spaced antennas is used. It is recommended to choose the separation direction perpendicular to the path direction; the separation distance is taken on the order of the correlation scale of 10?. Signals received at spaced antennas are added after detection. Polarization diversity is effective - reception by two antennas having mutually perpendicular polarization. Receiving antennas with
narrow radiation pattern, focused on receiving only one of the beams.

Under favorable propagation conditions, KB can circle the globe one or several times. Then, in addition to the main signal, a second signal can be received, delayed by about 0.1 s and called a radio echo. Radio echo has an interfering effect on meridional lines.

UDC 538.566.2: 621.372.8

Surface electromagnetic waves on flat boundaries of electrically conductive media with high conductivity, Zenneck wave

V. V. Shevchenko
Institute of Radio Engineering and Electronics named after. V.A. Kotelnikov RAS

annotation. The properties of a theoretical model of surface electromagnetic waves directed by flat boundaries of highly conductive media: metals, wet soil, sea and generally salt water are considered. The phase, “group” and energy velocities of such waves are calculated. It is shown that these waves belong to an unusual type of waves, in which the “group” speed differs from the energy speed, i.e. the speed of energy transfer by the wave. And although, depending on the parameters of the medium, the phase and “group” velocities of such waves can be greater than the speed of light With, their energy speed is always less than the speed of light. The type of waves considered is the so-called Zenneck wave.

Keywords: surface waves; phase, group, energy wave speeds; Zenneck wave.

Abstract.The properties of a theoretical model of surface electromagnetic waves, guided by the plane boundaries of high conductive media: metals, humid soils, sea and salty water in general are considered. The phase,”group” and energy flow velocities of these waves are calculated. These waves are related to the unusual type of waves, the “group” velocity of which is differed from the energy flow velocity, that is the wave energy transport velocity. Although depending on average parameters the phase and “group” velocities of these waves can be more than the light velocity c, their energy flow velocity is always less than the light velocity c. So named Zenneck’s wave is related to considered the type of waves.

Key words: surface waves; phase, group, energy flow velocities of waves; Zenneck's wave.

Introduction

The question of the surface waves indicated in the title of the article and, in particular, the so-called Zenneck wave has been raised for many years from time to time in scientific discussions in the field of applied electrodynamics, both by theorists and experimentalists. Since such discussions are reflected in many publications (see, for example, in and references in them), here we do not dwell on the details of published statements and doubts. Let us only note that the following questions are usually discussed. Is the Zenneck wave even possible from a physical point of view: does this not contradict physical laws, and if it is possible, then can it be excited by physically feasible sources and can it be used for signal transmission in communication systems and radar.

The theoretical analysis presented below gives, in the author’s opinion, a very definite answer to at least the first two of these questions, i.e. does not contradict and you can excite her. The remaining question relates to the technology of implementation and application of such surface waves.

1. Basic properties of a surface wave on a flat boundary of a highly conductive medium

Let the dependence of a stationary electromagnetic field on time have the form , where is the circular frequency of the field. Let us consider for simplicity, as is usually done [,], a two-dimensional model (the results are easily transferred to a three-dimensional model) of an electromagnetic surface wave on a flat boundary (Fig. 1) between free space with parameters , and an electrically conductive non-magnetic () medium with an effective dielectric constant, where is the complex dimensionless relative permeability

. (1)

Rice. 1. Flat boundary of an electrically conductive medium

, . (2)

For example, for wet soil, sea and simply salt water () in the radio wave range, and for metals () in the radio wave range, microwave, EHF and up to the infrared optical frequency range

, (3)

Where is the specific conductivity of the medium.

Complex magnetic and electric components of the field of a surface wave of corresponding polarization propagating along the flat boundary of the medium in the direction of the axis z(Fig.2), represent it in the form

, (4)

, (5)

(6)

Where A– amplitude constant, , With - speed of light and- wavelength in free space, ,

, (7)

Rice. 2. Localization of the wave field near the boundary of the medium

The original dispersion equation obtained by matching the field at the boundary of the medium at y =0 according to equalities

. (10)

Approximate equation and its solution for look like

, (11)

,, (12)

and the refined equation and its solution for , i.e. according to (12) –

, . (13)

Based on these relations and expressions (), (), the values are calculated

, (14)

. (15)

Thus, the wave is indeed a surface wave, since , , and it propagates along the boundary y =0 in the direction of the axis z.

It should be noted that result (15) can also be obtained from the relation

, (16)

(17)

which allows you to analyze the structure of the wave field corresponding to the expressions (), ().

Indeed, the quantity that describes the pressing of the wave field to the boundary of the medium, according to (16), increases the value of , which slows down the speed of movement of the phase front of the wave, and the quantity that describes the inclination of the phase front of the wave to the boundary of the medium (Fig. 3, the physical reason for the inclination is that that the medium partially absorbs the wave energy) reduces the value, that is, it accelerates the movement of the phase front of the wave along the boundary.

Fig.3. Inclination of the wave front to the boundary of the medium

Moreover, for the values of these quantities corresponding to the expressions (), the terms with the highest small value in () are compensated, so that

, (18)

and as a result, only terms proportional to the square of this small quantity remain in the real part in (). The above-mentioned inclination of the direction of propagation of the phase front of the wave to the boundary of the medium (Fig. 3), according to what has been said, is a small angle

. (19)

Expressions (),(),() allow us to estimate the extent of the surface wave field in transverse (L y) and longitudinal ( Lz)directions that are approximately equal

(20)

Here, the small transverse extent of the wave field inside the medium is not taken into account, equal, according to ()

. (21)

(32)

It should be noted here that the transitions of the values of the phase and group velocities of waves through the speed c occur under different environmental parameters. Considering the approximate nature of the introduced velocities, there is no reason to attach any physical meaning to the obtained specific values of the transient parameters of the medium.

4. Energy speed

Energy speed, i.e. the speed of energy wave transmission [ , , ] can be calculated using the following formula specified here:

, (33)

where time-averaged is the longitudinal (along the z axis) flow of power transferred by the wave and is the linear energy density per unit length moved along with the wave along the guide structure, i.e. flat boundary (also along the z axis). This kinematically determined energy velocity is based on the Umov-Poynting theorem. It is applicable both to waves propagating without loss of energy and to waves with loss. This definition does not include dissipative and absorbed energy by the medium, which does not propagate with the wave. In this case, a balance is achieved between the energy transported by the wave along the boundary of the medium.

For the wave under consideration we have

, (34)

where and are partial power flows above and below the plane y =0, which according to (), () are equal

(35)

and correspondingly , where at m we have

(36)

(37)

. (43)

Based on this expression and formula (), we obtain for the surface waves considered here

, (44)

Where - phase, and at small values it is also the energy velocity of a slow surface wave in the direction of movement of the phase front. As a result, based on () we get

. (45)

Essentially, the calculation made use of the property of waves with a plane phase front, applicable to plane and similar waves, which is that the inclination of the direction of motion of the phase front relative to the direction of wave propagation increases the phase velocity (), (), () and reduces the energy speed (45) of the wave.

As a result, we have that the energy speed of a surface wave is always less With, including the case corresponding to the Zenneck wave, for which the phase and group velocities are greater With.

5. Discussion of results

Let us discuss critically known versions, on the basis of which, it would seem, it can be argued that the theoretical model of surface waves discussed above does not describe physical surface waves directed by the boundary of an electrically conductive medium with high conductivity in the case when the phase and/or group velocities are greater than the speed of light With.

As follows from another, non-asymptotic, method of representing the total source field in the form of a spectral expansion in terms of natural waves (in transverse wave numbers with a discrete-continuous spectrum) of an open guide structure, here the boundaries of the medium [ , , ], such expansion in its original form contains, in addition to integral of a selected surface wave, regardless of whether it is slow or fast. This expansion can be obtained either directly on the basis of the theory of a singular (in an infinite interval) transverse boundary value problem on eigenvalues and eigenfunctions [, ], or by transforming the indicated integral Fourier expansion in longitudinal wave numbers into an expansion in transverse wave numbers. In the second case, when the integration contour is deformed in the complex plane of wave numbers, this contour equally sweeps out the poles of the integrand corresponding to both slow and fast surface waves [ , , ]. Thus, the surface wave, both slow and fast, is contained in the total field excited by the source, but it attenuates and disappears in asymptotics, where only the space wave field remains.

Conclusion

The waves considered are a special type of surface waves, the surface nature of which, i.e. The exponential decay of the field from the boundary of the highly conductive medium under consideration in the transverse direction occurs here not because of the slowness of its phase velocity relative to the speed of plane waves above the boundary of the medium, which turned out to be unnecessary here, but because of the partial absorption of energy in it during wave propagation. The presented results show that the considered model of such surface waves does not contradict physical laws. Therefore, there is no reason to doubt that it describes physical waves and when their phase velocity is less c, and when – more, and the generally accepted “group” speed for them, apparently, does not have a clear physical meaning.

However, such waves have significant disadvantages from the point of view of their use in technical applications. Firstly, they are weakly pressed against the boundary of the medium, i.e. their field has a sufficiently large extent in the transverse direction above the boundary, so to effectively excite them may require a source with a vertical aperture that is too large. Secondly, their phase speed is only slightly different from the speed of light With, therefore, any, even small, irregularities in the plane of the boundary of the medium can lead to scattering of the wave field and a significant increase in energy losses when propagating along the boundary. In particular, this can occur when the boundary deviates from the plane, i.e. in the presence of curvature of its surface. Analysis of the considered surface waves on an irregular boundary requires special research [,].

On the other hand, when trying to apply surface waves, for example, at the boundaries of metals in technical applications, it is necessary to take into account that the surfaces of real metals are usually covered with oxide films having a thickness of the order of fractions of a micron, micron or several microns (natural films) and of the order of tens of microns (artificially created films for mechanical protection of metal surfaces). In this case, it is necessary to use the results of a slightly different theoretical model of the guiding system: a layered structure such as metal substrate - dielectric film (necessarily taking into account energy losses in them) - free space. The presence of a film can significantly affect the pressure of the surface wave in the direction of its increase and, consequently, the possibility of simplifying the excitation of the wave and its greater stability with respect to structure irregularities.

As an afterword to the article, we note that in September 2012, this article was submitted to the journal UFN, which had previously published a series of articles devoted to the Zenneck wave, and, in essence, a discussion arose on this topic. However, the article was not accepted for publication due to the fact that the editorial board of UFN decided “not to accept new work on Zenneck waves for consideration.” As a result of this, the indicated publication of articles on this topic in UFN actually ended with the publication of an erroneous article.

Literature

1.Barlow H. M., Wait J. R. // Electron. Letters. 1967.T.3. No. 9.P.396.

2.Shevchenko V.V. // Radio engineering and electronics. 1969.T.14. No. 10.S.1768.

3., .: Golem Press, 1971).

17. Mandelstam L. I. Lectures on optics, relativity theory and quantum mechanics. M.: Nauka, 1972. P.420,431.

18. Zilbergleit A. S., Kopilevich Yu. I. // Letters to ZhTP. 1979.T.5.No.8. P. 454.

19. Brekhovskikh L. M. Waves in layered media. M.: Publishing house. USSR Academy of Sciences, 1957.

20.Barlow H. M., Brown J. Radio surface waves. Oxf.: Clarendon Press, 1962.

21. Shevchenko V.V.//Differential equations.1979.T.15. No. 11. WITH .2004 (ShevchenkoV.V.//Differential Equations.1980.V.15. No. 11.P.1431).

22.Shevchenko V.V. // Izv. Universities – Radiophysics. 1971.T.14. No. 5. P. 768.

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1 Syomkin Sergey Viktorovich, Smagin Viktor Pavlovich ELECTROMAGNETIC EFFECTS CAUSED BY SEA SURFACE WAVES Article address: The article was published in the author's edition and reflects the point of view of the author(s) on this issue. Source Almanac of modern science and education Tambov: Certificate, (59). C ISSN Journal address: Contents of this issue of the journal: Publishing house "Gramota" Information about the possibility of publishing articles in the journal is posted on the publishing house's website: The editors ask questions related to the publication of scientific materials to be sent to:

2 194 Publishing house "Gramota" Fig. 3. Filling out competencies To develop an information system for accounting for objects of an intelligent system. The PHP programming language was chosen, since this programming language allows you to create dynamic web pages and link them to a database implemented in MySQL. This approach allows you to place the system on the Internet and access it from anywhere without additional software. The developed information system for recording intellectual property contributes to: - reducing the time spent on participation in the development and implementation of a unified patent and licensing policy of the organization; - redistribution of the workload of the organization’s employees; - increasing the efficiency of accounting and control over the registration of intellectual property and timely registration of reports on them. The information system for recording intellectual property objects allows for convenient and reliable storage and management of department data, the ability to prepare documents for filing an application for official registration of a computer program or database. This will significantly improve the quality of services for the protection and protection of intellectual property and increase the efficiency of work with intellectual property objects. References 1. All-Russian Scientific and Technical Information Center [Electronic resource]. URL: (access date:). 2. Intellectual property: trademark, invention, patenting, patent attorney, patent bureau, Rospatent [Electronic resource]. URL: (access date:). 3. Sergeev A.P. Intellectual property rights in the Russian Federation: textbook. M., p. 4. Federal Institute of Industrial Property [Electronic resource]. URL: (access date:). UDC Physical and mathematical sciences Sergey Viktorovich Semkin, Viktor Pavlovich Smagin Vladivostok State University of Economics and Service ELECTROMAGNETIC EFFECTS CAUSED BY SEA SURFACE WAVES 1. Introduction Sea water, as is known, is a conductive liquid due to the presence of ions of different signs in it. Its electrical conductivity, depending on temperature and salinity, can Syomkin S.V., Smagin V.P., 2012

3 ISSN Almanac of modern science and education, 4 (59) change on the ocean surface within 3-6 Sym/m. Macroscopic movements of seawater in a geomagnetic field can be accompanied by the emergence of electric currents, which, in turn, generate an additional magnetic field. This induced field is influenced by a number of different factors. Firstly, the type of hydrodynamic source - sea surface waves, internal waves, currents and tides, long waves such as tsunamis, etc. An induced electromagnetic field can also be created by other types of macroscopic water movement - acoustic waves and artificial sources - underwater explosions and ship waves. Secondly, this field can be influenced by the electrical conductivity of seafloor rocks and seafloor topography. It can also be noted that a problem similar to calculating the induced field in the marine environment also arises in seismology - the movement of the lithosphere in the Earth's magnetic field leads to the emergence of induced currents. One of the directions for studying the spatiotemporal structure of the induced field is the case when it is generated by a two-dimensional surface wave. The calculation of the electromagnetic field induced by a surface wave can be carried out in various approximations and for various models of the marine environment. The field induced by sea surface waves in the approximation of an infinitely deep ocean was calculated in the works, and in the work the fields induced by wind waves in shallow zones taking into account a finite variable depth were theoretically studied. A more complex hydrodynamic model of sea waves - vortex waves with a finite crest - was considered in. That is, a significant number of different options for formulating the problem are possible, depending on the influence of which factors need to be taken into account. In this work, we study the influence of the electrical and magnetic properties of bottom rocks, namely their magnetic permeability and electrical conductivity, on the induced electromagnetic field. Typically, the study of the influence of the properties of bottom rocks on the magnetic field is limited to taking into account only their electrical conductivity, since bottom rocks, as a rule, do not have pronounced magnetic properties. However, in the coastal zone of the ocean it is quite possible that the bottom rocks also have magnetic properties. In addition, it turns out [Ibid] that for potential fluid movement, the emergence of currents in bottom rocks is possible only due to induction effects - a term in Maxwell’s equations. And discarding this term (quasi-static approximation) leads to the fact that the induced field does not depend at all on the conductivity of bottom rocks. Therefore, we will consider this formulation of the problem of determining the electromagnetic field induced by a surface wave, in which the bottom has not only electrical conductivity, but also magnetic properties, and we will also take into account the effect of self-induction. 2. Basic equations and boundary conditions To solve the problem of determining the electromagnetic field induced by the movement of sea water in the geomagnetic field, the Maxwell system of equations is used: (1) The relationship between pairs of vectors and (material equations) as well as the expression for the current density are different in different media . We will assume that in air (medium I) the connection between the vectors characterizing the electromagnetic field is the same as in vacuum, and there are no electric currents and space charges: (2) We will consider sea water (medium II) to be homogeneous in both hydrodynamic and and electromagnetic properties. The material equations in the coordinate system relative to which the fluid moves are described in. Assuming that the speed of water movement is low, and the induced magnetic field is significantly less than the geomagnetic field, we obtain: , (3) (4) where and are the electrical permeability and conductivity of sea water. Let's consider the question of volumetric electric charges inside water. From equations (1), relation (3), Ohm's law (4) and the conditions for conservation of electric charge, we obtain: (5) For the case of a stationary process, when and, solution (5) has the form: where is the characteristic time for establishing a stationary state. At,. This means that any established hydrodynamic and hydroacoustic processes can be

4 196 Publishing House "Gramota" be considered steady in the electrodynamic sense. Since the cyclic frequencies do not even exceed ultrasonic waves, we can assume with good accuracy that Thus, with the potential movement of sea water (), there are no space charges in sea water. We will assume that bottom rocks (medium III) are a semi-infinite homogeneous medium with conductivity, dielectric and magnetic permeability and, respectively. The material equations and Ohm's law in this medium are as follows: (6) The volume density of electric charges in medium III obeys an equation similar to (5), but with a zero right-hand side. Therefore, in a stationary periodic mode. The characteristic time for establishing equilibrium is of the same order as. As shown in , the boundary conditions at boundaries I-II and II-III have the same form for low velocities of water movement as for stationary media. That is, at the boundary I-II:, (7) At the boundary II-III:, (8) The surface charge densities are not known in advance and are found when solving the problem. 3. Two-dimensional surface wave Consider a two-dimensional surface wave propagating in the direction of the axis (the axis is directed vertically upward, and the plane coincides with the undisturbed surface of the water). The velocities of liquid particles will be as follows:, (9) - sea depth., and are related by the dispersion relation (10) Let us introduce the angles and that determine the orientation of the geomagnetic field vector (in the original coordinate system) as follows: That is, is the angle between the vertical and the vector , depending on the latitude of the place, and is the angle between the direction of wave propagation and the projection of the vector onto the horizontal plane. We will look for a solution to system (1) in the form Substituting these expressions into (1), we obtain: (11) (12) (13) (14) (15) ( () (16) ( (17) ( () (18) Equations (11)-(18) can be divided into two groups: equations (11), (13), (16) and (18) for the components, and equations (12), (14), (15) and (17 ) for components, and. We solve the equations of the second group as follows. and express them through: and the equations for have the form Here,. Finding the general solution (20) and using (19), we obtain in environment I: (19) (20)

5 ISSN Almanac of modern science and education, 4 (59) in environment II:, (21) (22) in environment III:, (23) To determine the coefficients, and we use the boundary conditions (7) and (8) Excluding and, we reduce system to two equations for and which we write in matrix form: () () () Solving this system, we find the coefficients and through which the components of the electromagnetic field are expressed, and. In a similar way, we solve the system of equations (11), (13), (16) and (18) for the components, and the equations for have the form The component is expressed from (19). Solving (25) and using (23) and (19) we find the components in medium I: in medium II: (24) (25) (26) (27) in medium III: Using boundary conditions (7) and (8), we get: (28) Hence and. Thus, in all three media and ( (29) ( (30) The component has discontinuities at the boundaries between the media. This means that there are surface charges at the boundaries, the densities of which are determined from conditions (7) and (8): (boundary I -II) (31) (boundary II-III) (32) From the obtained solution it follows that the current density components and are equal to zero in all three media, which is consistent with the condition of conservation of electric charge. The component is not zero and

6 198 Publishing House "Gramota" in order of magnitude is. The existence of periodically changing surface charges at first glance contradicts the condition: since the medium is not superconducting, there are no surface currents, and a change in the surface charge can only be associated with the existence of a volume current component normal to the boundary. We will find the value of this component from the condition of charge conservation. Thus, the ratio will be of the order that for sea water and typical frequencies of wind waves is approximately. That is, when discarding, we do not go beyond the accuracy with which the material equations (2), (4) and (6) and boundary conditions (7) and (8) are considered. 4. Calculation results and conclusions Thus, for a two-dimensional surface wave having an arbitrary direction relative to the magnetic meridian, we calculated the components of the magnetic and electric fields in all media, as well as surface electric charges on the bottom and free surface. The influence of the electrical and magnetic properties of bottom rocks on the wave-induced magnetic field is manifested as follows. Rice. 1 In Fig. Figure 1 shows the dependences of the amplitudes of the components equal above the surface and (in units) on the wave period for waves of the same amplitude. Curve 2 corresponds to the case of a non-magnetic and non-conducting bottom (,), curve 1 to the case of a non-magnetic conducting bottom (,), curve 4 to the case of a magnetic non-conducting bottom (,), and curve 3 to the case of a magnetic conducting bottom (,). All curves are calculated for the case. It turns out that for any value of the wave period, the induced field monotonically increases with increasing magnetic permeability of the bottom and decreases with increasing its conductivity. The dependence of the magnetic field on the wave period can be either monotonically increasing or having a maximum, depending on the orientation of the wave relative to the geomagnetic field. Rice. 2

7 ISSN Almanac of modern science and education, 4 (59) In Fig. Figure 2 shows the dependences of the induced magnetic field (in the same units as in Fig. 1) on the sea depth (in kilometers) for waves with a period of,. Curves 1, 2, 3 and 4 correspond to values equal to 1, 2, 10 and 100. From the results obtained, the following general conclusions can be drawn: 1. Volumetric electric charges do not arise either in sea water or in conductive bottom rocks in the case of potential motion sea water. 2. Surface electric charges (30), (31) are determined only by the component of the geomagnetic field, the amplitude and frequency of the wave and the depth of the ocean and do not depend on the magnetic permeability and electrical conductivity of bottom rocks and sea water. 3. The along-ridge component of the induced magnetic field is zero in all media. 4. The along-ridge component of the induced electric field is zero in the quasi-static approximation, and the and components, like surface electric charges, do not depend on the electrical and magnetic properties of water and bottom rocks. 5. For all values of ocean depth and wave period, the magnitude of the induced magnetic field monotonically increases to a final limit value with increasing magnetic permeability of bottom rocks and monotonically decreases with increasing their conductivity. References 1. Gorskaya E. M., Skrynnikov R. T., Sokolov G. V. Magnetic field variations induced by the movement of sea waves in shallow water // Geomagnetism and Aeronomy S. Guglielmi A. V. Ultra-low-frequency electromagnetic waves in the Earth’s crust and magnetosphere // UFN T S Sommerfeld A. Electrodynamics. M., Savchenko V.N., Smagin V.P., Fonarev G.A. Issues of marine electrodynamics. Vladivostok: VGUES, p. 5. Semkin S.V., Smagin V.P., Savchenko V.N. Magnetic field of an infrasonic wave in an oceanic waveguide // Geomagnetism and Aeronomy T S Semkin S.V., Smagin V.P., Savchenko V.N. Generation of magnetic field disturbances during an underwater explosion // Izvestia RAS. Physics of the atmosphere and ocean T S Smagin V. P., Semkin S. V., Savchenko V. N. Electromagnetic fields induced by ship waves // Geomagnetism and Aeronomy T S Sretensky L. N. Theory of wave motions of fluid. M.: Science, p. 9. Fonarev G. A., Semenov V. Yu. Electromagnetic field of sea surface waves // Study of the geomagnetic field in the waters of the seas and oceans. M.: IZMIRAN, S Fraser D. C. The Magnetic Fields of Ocean Waves // Geophys. Journal Royal Astron. Soc Vol P Larsen J. C. Electric and Magnetic Fields Induced by Deep Sea Tides // Geophys. Journal Royal Astron. Soc Vol. 16. P Pukhtyar L. D., Kukushkin A. S. Investigation of the Electromagnetic Fields Induced by Sea Motion // Physical Oceanography Vol P Sanford T. B. Motionally Induced Electric and Magnetic Fields in the Sea // J. Geophys. Res Vol P Warburton F., Caminiti R. The Induced Magnetic Field of Sea Waves // J. Geophys. Res Vol P Weaver J. T. Magnetic Variation Associated with Ocean Waves and Swell // J. Geophys. Res Vol P UDC 34 Legal sciences Victoria Vitalievna Sidorenko, Aigul Sharifovna Galimova Bashkir State University THE PROBLEM OF EFFECTIVENESS OF USE OF WORKING TIME Working time is an important category in the organization of labor in an enterprise. It represents the time during which the employee, in accordance with the internal labor regulations and the terms of the employment contract, must perform labor duties, as well as other periods of time that, in accordance with laws and other legal acts, relate to working time. Working time is a natural measure of labor, existing at the same time as a multifaceted category, because The general health and vital activity of a person depends on the length of working hours. The duration and intensity of working time directly affects the length of rest time a person needs to recuperate, expend energy, fulfill family responsibilities for education, etc. Therefore, the strictest compliance with working time legislation is at the same time ensuring the most important constitutional human right - the right to rest. Regulation of working hours solves such important problems as: establishing the possible participation of citizens in public labor, ensuring labor protection, and ensuring the right to rest. Sidorenko V.V., Galimova A.Sh., 2012

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2005.

A little theory. Maxwell's equations in a medium

Material equations

When substituting this type of solution into the material equations, we find that e and m depend on the frequency - time dispersion, and the wave vector - spatial dispersion. The relationship between frequency and wave vector through e and m is called the dispersion relation.

In this report we will assume that m does not depend on frequency and = 1. In the optical frequency range, this condition is satisfied quite well. Since e depends on frequency, it can take different values, including negative ones.

Let us consider the problem of the incidence of a plane monochromatic wave from a medium with e1 onto an ideal surface of some substance e2.

From these boundary conditions, when substituting the usual form of solutions, the well-known Fresnel formulas, Snell's law, etc. are obtained. However, such solutions do not always exist. Let us consider the case when the dielectric constant of the medium is negative. This case is realized in a certain frequency range in metals. Then solutions in the form of propagating waves do not exist. We will look for solutions in the form of surface waves.

where Vacuum" href="/text/category/vakuum/" rel="bookmark">vacuum. The dependence on frequency is also implicitly present in the functions e1(w) and e2(w).

Here is a simple formula - the Drude formula, which shows the dependence of the dielectric constant of a metal on frequency.

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So, the dispersion curve for surface waves in a metal. In the figure it is a blue curve. The red line is the dispersion curve for vacuum.

a) Disturbed total internal reflection is also known as the optical tunneling effect. At the dielectric boundary, at an angle of incidence greater than the angle of total internal reflection, surface waves arise, which are then converted into volumetric reflected waves. But when the conditions of phase matching at the boundary with the metal are met, these waves can be transformed into surface waves of the metal plate. This phenomenon is the basis of prismatic excitation of surface waves.

b) Resonant structures here are understood as periodic structures with a period on the order of the wavelength of surface waves. In such periodic structures, the phase matching condition changes - , where is the reciprocal lattice vector. The excitation of surface waves leads to Wood's anomalies - a change in the intensity of light diffracting on a diffraction grating, contrary to the standard law of diffraction.

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Detection of thickness changes at a fixed diel. permeability

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Bibliography:

1. . Surface plasmon microscope, Soros Educational Journal, No. 8, 1999

2. Surface electromagnetic waves of the optical range, Soros Educational Journal, No. 10, 1996

3. Rothenhäusler B., Knoll W. Surface Plasmon Microscopy, Nature. 1988. No. 000. p. 615-617.

4. Born, Wolf " Optics Basics", chapter "Optics of metals"

5. F. J. Garcia-Vidal, L. Martin-Moreno Transmission and focusing of light in one-dimensional periodically nanostructured metals, Phys. Rev. B 66, 155

6. , S. G. Tikhodeev, A. Christ, J. Kuhl, H. Giessen . Plasmon-waveguide polaritons in metal-dielectric photonic-crystal layers, Physics of Solid State, 2005, volume 47, issue. 1