Plane traveling wave equation. Plane wave equation. Phase velocity Plane wave equation in complex form
mechanical waves- distribution process mechanical vibrations in a medium (liquid, solid, gaseous). It should be remembered that mechanical waves transfer energy, form, but do not transfer mass. The most important characteristic wave is the speed of its propagation. Waves of any nature do not propagate in space instantly, their speed is finite.
Geometry distinguishes: spherical (spatial), one-dimensional (plane), spiral waves.
The wave is called flat, if its wave surfaces are planes parallel to each other, perpendicular to the phase velocity of the wave (Fig. 1.3). Consequently, the rays of a plane wave are parallel straight lines.
Plane wave equation::
Options :
Oscillation period T is the period of time after which the state of the system takes the same values: u(t + T) = u(t).
Oscillation frequency n is the number of oscillations in 1 second, the reciprocal of the period: n = 1/T. It is measured in hertz (Hz), has the dimension s–1. A pendulum swinging once per second oscillates at a frequency of 1 Hz
Oscillation phase j- a value showing what part of the oscillation has passed since the beginning of the process. It is measured in angular units - degrees or radians.
Oscillation amplitude A- the maximum value that the oscillatory system takes, the “range” of the oscillation.
4.Doppler effect- change in the frequency and length of the waves perceived by the observer (wave receiver), due to the relative motion of the wave source and the observer. Imagine that the observer is approaching at a certain speed to a stationary source of waves. At the same time, it encounters more waves in the same time interval than in the absence of movement. This means that the perceived frequency is greater than the frequency of the wave emitted by the source. So the wavelength, frequency and speed of wave propagation are interconnected by the relation V= / , - wavelength.
Diffraction- the phenomenon of bending around obstacles, which are comparable in size to the wavelength.
Interference- a phenomenon in which, as a result of the superposition of coherent waves, either an increase or a decrease in oscillations occurs.
Young's experience The first interference experiment to be explained on the basis of the wave theory of light was Young's experiment (1802). In Young's experiment, light from a source, which served as a narrow slit S, fell on a screen with two closely spaced slits S1 and S2. Passing through each of the slits, the light beam broadened due to diffraction, therefore, on the white screen E, the light beams that passed through the slits S1 and S2 overlapped. In the region of overlapping light beams, an interference pattern was observed in the form of alternating light and dark stripes.
2.Sound - mechanical longitudinal wave, which propagates in elastic media, has a frequency from 16 Hz to 20 kHz. There are types of sounds:
1. simple tone - purely harmonic vibration emitted by a tuning fork (a metal instrument that makes a sound when struck):
2. complex tone - not sinusoidal, but periodic oscillation (radiated by various musical instruments).
According to the Fourier theorem, such a complex oscillation can be represented by a set of harmonic components with different frequencies. The lowest frequency is called the fundamental tone, and multiple frequencies are called overtones. A set of frequencies indicating their relative intensity (wave energy flux density) is called the acoustic spectrum. The spectrum of complex tone is linear.
3. noise - sound, which is obtained from the addition of many inconsistent sources. Spectrum - continuous (continuous):
4. sonic impact - short-term sound impact. For example: cotton, explosion.
Wave resistance- the ratio of sound pressure in a plane wave to the velocity of oscillation of the particles of the medium. It characterizes the degree of rigidity of the medium (i.e., the ability of the medium to resist the formation of deformations) in a traveling wave. Expressed by the formula:
P / V \u003d p / c, P- sound pressure, p- density, c- speed of sound, V- volume.
3 - characteristics that do not depend on the properties of the receiver:
Intensity (strength of sound) - the energy carried by sound wave per unit time through a unit area, set perpendicular to the sound wave.
pitch frequency.
The spectrum of sound is the number of overtones.
At frequencies below 17 and above 20,000 Hz, pressure fluctuations are no longer perceived by the human ear. Longitudinal mechanical waves with a frequency of less than 17 Hz are called infrasound. Longitudinal mechanical waves with a frequency exceeding 20,000 Hz are called ultrasound.
5. UZ- mechanical wave with a frequency of more than 20 kHz. Ultrasound is an alternation of condensation and rarefaction of the medium. In each medium, the speed of propagation of ultrasound is the same . Peculiarity- the narrowness of the beam, which allows you to act on objects locally. In inhomogeneous media with small inclusions of particles, diffraction phenomena (enveloping obstacles) take place. The penetration of ultrasound into another medium is characterized by the penetration coefficient () =L /L where the length of the ultrasound after and before penetration into the medium.
The effect of ultrasound on body tissues is mechanical, thermal, chemical. Application in medicine is divided into 2 areas: the method of research and diagnosis, and the method of action. one) echoencephalography- detection of tumors and cerebral edema ; cardiography- measurement of the heart in dynamics. 2) Ultrasound physiotherapy- mechanical and thermal effects on the fabric; during operations as an "ultrasound scalpel"
6. The ideal liquid imaginary incompressible fluid, devoid of viscosity and thermal conductivity. An ideal fluid has no internal friction, it is continuous and has no structure.
Continuity equation -V 1 A 1 = V 2 A 2 The volume flow in any stream tube limited by adjacent stream lines must be the same at any time in all its cross sections.
Bernoulli equation - R v 2 / 2 + Rst + Rgh= const, in the case of a steady flow, the total head is the same in all cross sections of the current tube. R v 2 / 2 + Rst= const – for horiz. plots.
7Stationary flow A flow whose velocity never changes anywhere in the fluid.
laminar flow- an ordered flow of a liquid or gas, in which the liquid (gas) moves, as it were, in layers parallel to the direction of flow.
turbulent flow- the form of the flow of a liquid or gas, in which their elements make disorderly, unsteady movements along complex trajectories, which leads to intense mixing between the layers of a moving liquid or gas.
lines- lines, the tangents to which coincide at all points with the direction of the velocity at these points. In a stationary flow, the streamlines do not change with time.
Viscosity - internal friction, the property of fluid bodies (liquids and gases) to resist the movement of one of their parts relative to another
Newton's equation: F = (dv/dx)Sη.
Viscosity factor- Proportionality factor depending on the type of liquid or gas. A number used to quantify the property of viscosity. Coefficient of internal friction.
non-newtonian fluid is called a liquid, during which its viscosity depends on the velocity gradient, the flow of which obeys Newton's equation. (Polymers, starch, liquid soap blood)
Newtonian - If in a moving fluid its viscosity depends only on its nature and temperature and does not depend on the velocity gradient. (Water and diesel fuel)
.Reynolds number- characterizing the relationship between inertial forces and viscous forces: Re \u003d rdv / m, where r is the density, m is the dynamic coefficient of viscosity of the liquid or gas, v is the flow velocity. At R< Rekр возможно лишь ламинарное течение жидкости, а при Re >Rekp the flow can become turbulent.
Kinematic viscosity coefficient- the ratio of the dynamic viscosity of a liquid or gas to their density.
9. Stokes method, based method a Stokes's formula for the resistance force that occurs when a ball moves in a viscous fluid, obtained by Stokes: Fc = 6 π η V r. To indirectly measure the viscosity coefficient η, one should consider the uniform motion of a ball in a viscous fluid and apply the condition uniform motion: the vector sum of all forces acting on the ball is zero.
Mg + F A + F c \u003d 0 (everything in vector form !!!)
Now it is necessary to express the force of gravity (mg) and the force of Archimedes (Fa) through known quantities. By equating the values mg = Fa + Fс, we obtain the expression for viscosity:
η \u003d (2/9) * g * (ρ t - ρ w) * r 2 / v \u003d (2/9) * g * (ρ t - ρ w) * r 2 * t / L. Radius is directly measured with a micrometer ball r (in diameter), L is the path of the ball in the liquid, t is the travel time of the path L. To measure the viscosity according to the Stokes method, the path L is taken not from the surface of the liquid, but between marks 1 and 2. This is due to the following circumstance. When deriving the working formula for the viscosity coefficient by the Stokes method, the condition of uniform motion was used. At the very beginning of the movement (the initial speed of the ball is zero), the resistance force is also zero and the ball has some acceleration. As the speed increases, the drag force increases, the resultant of the three forces decreases! Only after a certain mark, the movement can be considered uniform (and then, approximately).
11.Poiseuille formula: With steady laminar motion of a viscous incompressible fluid through a cylindrical tube of circular cross section, the volume flow per second is directly proportional to the pressure drop per unit length of the tube and the fourth power of the radius, and inversely proportional to the coefficient of viscosity of the fluid.
PLANE WAVE
PLANE WAVE
A wave in which the direction of propagation is the same at all points in space. The simplest example is a homogeneous monochromatic undamped P. v.:
u(z, t)=Aeiwt±ikz, (1)
where A - amplitude, j= wt±kz - , w=2p/Т - circular frequency, Т - oscillation period, k - . Surfaces of constant phase (phase fronts) j=const P.v. are planes.
In the absence of dispersion, when vph and vgr are the same and constant (vgr = vph = v), there exist stationary (i.e., moving as a whole) traveling P.V., which admit a general representation of the form:
u(z, t)=f(z±vt), (2)
where f is an arbitrary function. In nonlinear media with dispersion, stationary propagating waveforms are also possible. type (2), but their shape is no longer arbitrary, but depends both on the parameters of the system and on the nature of the motion. In absorbing (dissipative) media P. century. decrease their amplitude as they propagate; with linear damping, this can be taken into account by replacing k in (1) by the complex wave number kd ± ikm, where km is the coefficient. attenuation P. in.
A uniform waveform that occupies the whole of the infinite is an idealization, but any waveform concentrated in a finite region (for example, guided by transmission lines or waveguides) can be represented as a superposition of the waveform. with one space or another. spectrum k. In this case, the wave may still have a flat phase front, but an inhomogeneous amplitude. Such P. in. called plane inhomogeneous waves. Separate sections of spherical and cylindrical. waves that are small compared to the radius of curvature of the phase front behave approximately like P.V.
Physical Encyclopedic Dictionary. - M.: Soviet Encyclopedia. . 1983 .
PLANE WAVE
- wave, uk-swarm direction of propagation is the same at all points in space.
where BUT - amplitude, - phase, - circular frequency, T - oscillation period, k- wave number. = const P. c. are planes.
In the absence of dispersion, when the phase velocity v f and group v gr are the same and constant ( v gr = v f = v) there are stationary (i.e., moving as a whole) traveling P. c., which can be represented in a general form
where f- arbitrary function. In nonlinear media with dispersion, stationary traveling parametric waves are also possible. type (2), but their shape is no longer arbitrary, but depends both on the parameters of the system and on the nature of the wave motion. In absorbing (dissipative) media P. k on the complex wavenumber k d ik m, where k m - coefficient. attenuation P. in. A homogeneous wave field occupying everything infinite is an idealization, but any wave field concentrated in a finite region (for example, directed transmission lines or waveguides), can be represented as a superposition. in. with one or another spatial spectrum k. In this case, the wave may still have a flat phase front, in a non-uniform amplitude distribution. Such P. in. called plane inhomogeneous waves. Dep. spherical plots or cylindrical. waves that are small compared to the radius of curvature of the phase front behave approximately like P.V.
Lit. see at Art. Waves.
M. A. Miller, L. A. Ostrovsky.
Physical encyclopedia. In 5 volumes. - M.: Soviet Encyclopedia. Editor-in-Chief A. M. Prokhorov. 1988 .
When describing the wave process, it is required to find the amplitudes and phases of the oscillatory motion at various points in the medium and the change in these quantities over time. This problem can be solved if it is known according to which law it oscillates and how the body that caused the wave process interacts with the medium. However, in many cases it does not matter what body the given wave is excited by, but a simpler problem is solved. Given the state of oscillatory motion at some points in the medium at a certain point in time and needs to be determined the state of oscillatory motion at other points in the medium.
As an example, consider the solution of such a problem in a simple, but at the same time important case of propagation of a plane or spherical harmonic wave in a medium. Let us denote the fluctuating value by u. This value can be: the displacement of the particles of the medium relative to their equilibrium position, the deviation of the pressure in a given place of the medium from the equilibrium value, etc. Then the task will be to find the so-called wave equations - an expression that specifies a fluctuating value u as a function of the coordinates of the points of the medium x, y, z and time t:
u = u(x, y, z, t). (2.1)
Let, for simplicity, u be the displacement of points in an elastic medium when a plane wave propagates in it, and the oscillations of the points have a harmonic character. In addition, we direct the coordinate axes so that the axis 0x coincides with the direction of wave propagation. Then the wave surfaces (family of planes) will be perpendicular to the axis 0x(Fig. 7), and since all points of the wave surface oscillate in the same way, the displacement u will only depend on X and t: u = u(x, t). For harmonic oscillations of points lying in the plane X= 0 (Fig. 9), the equation is valid:
u(0, t) = A cos ( ωt + α ) (2.2)
Let us find the type of oscillations of the points of the plane corresponding to an arbitrary value X. To go the way from the plane X= 0 to this plane, the wave needs time τ = x/s (With is the speed of wave propagation). Consequently, oscillations of particles lying in the plane X, will look like:
So, the equation of a plane wave (both longitudinal and transverse) propagating in the direction of the 0x axis looks like this:
Value BUT is the amplitude of the wave. Initial phase of the wave α determined by the choice of reference points X and t.
Let us fix some value of the phase in square brackets of Eq. (2.3) by setting
Let us differentiate this equality with respect to time, taking into account that the cyclic frequency ω and initial phase α are permanent:
Thus, the wave propagation speed With in equation (2.3) is the speed of the phase movement, in connection with which it is called phase speed . According to (2.5) dx/dt> 0. Therefore, equation (2.3) describes a wave propagating in the direction of increasing X, the so-called traveling progressive wave . A wave propagating in the opposite direction is described by the equation
and called traveling regressive wave . Indeed, by equating the phase of the wave (2.6) to a constant and differentiating the resulting equality, we arrive at the relation:
from which it follows that wave (2.6) propagates in the direction of decreasing X.
We introduce the quantity
which is called wave number and is equal to the number of wavelengths that fit into the interval of 2π meters. Using formulas λ = c/v and ω = 2π ν wavenumber can be represented as
Opening the brackets in formulas (2.3) and (2.6) and taking into account (2.8), we arrive at the following equation for plane waves propagating along (the "-" sign) and against (the "+" sign) axis 0 X:
When deriving formulas (2.3) and (2.6), it was assumed that the oscillation amplitude does not depend on X. For a plane wave, this is observed when the wave energy is not absorbed by the medium. Experience shows that in an absorbing medium, the intensity of the wave gradually decreases with distance from the source of oscillations - the wave attenuation is observed according to the exponential law:
Accordingly, the equation of a plane damped wave has the form:
where A 0 - amplitude at the points of the plane X= 0, and γ is the attenuation coefficient.
Now let's find the equation spherical wave . Any real source of waves has some extent. However, if we confine ourselves to considering the wave at distances from the source, much larger than its size, then the source can be considered pinpoint . In an isotropic and homogeneous medium, the wave generated by a point source will be spherical. Let us assume that the phase of the source oscillations ωt+α. Then the points lying on the wave surface of radius r, will oscillate with the phase
The oscillation amplitude in this case, even if the wave energy is not absorbed by the medium, will not remain constant - it decreases depending on the distance from the source according to the law 1/ r. Therefore, the spherical wave equation has the form:
where BUT is a constant value numerically equal to the oscillation amplitude at a distance from the source equal to unity.
For an absorbing medium, in (2.11) we must add the factor e-γr. Recall that, by virtue of the assumptions made, Eq. (2.11) is valid only for r, significantly exceeding the dimensions of the vibration source. When striving r to zero, the amplitude goes to infinity. This absurd result is explained by the inapplicability of equation (2.11) for small r.
Before considering the wave process, let's give a definition of oscillatory motion. hesitation is a recurring process. Examples of oscillatory movements are very diverse: the change of seasons, the fluctuation of the heart, breathing, the charge on the capacitor plates, and others.
The oscillation equation in general form is written as
where 
From oscillatory motion, we can proceed to the consideration of wave motion. Wave is the process of propagation of vibrations in space over time. Since oscillations propagate in space over time, both spatial coordinates and time must be taken into account in the wave equation. The wave equation has the form
where A 0 - amplitude, - frequency, t - time, - wave number, z - coordinate.
The physical nature of waves is very diverse. Sound, electromagnetic, gravitational, acoustic waves are known.
According to the type of oscillations, all waves can be classified into longitudinal and transverse. Longitudinal waves - these are waves in which the particles of the medium oscillate along the direction of wave propagation (Fig. 3.1a). An example of a longitudinal wave is a sound wave.
transverse waves - these are waves in which the particles of the medium oscillate in the transverse direction relative to the direction of propagation (Fig. 3.1b).
Electromagnetic waves are referred to as transverse waves. It should be taken into account that in electromagnetic waves the field oscillates, and no oscillation of the particles of the medium occurs. If a wave propagates in space with one frequency , then such wave called monochromatic .
To describe the propagation of wave processes, the following characteristics are introduced. The cosine argument (see formula (3.2)), i.e. expression 
Schematically, wave propagation along one coordinate is shown in fig. 3.2, in this case, the propagation occurs along the z-axis.
Period is the time of one complete oscillation. The period is denoted by the letter T and is measured in seconds (s). The reciprocal of a period is called line frequency and denoted f, measured in hertz (= Hz). The line frequency is related to the circular frequency. The connection is expressed by the formula
If we fix the time t, then from Fig. 3.2 it can be seen that there are points, for example, A and B, which oscillate in the same way, i.e. in phase (in-phase). The distance between the nearest two points that oscillate in phase is called wavelength . Wavelength is denoted and is measured in meters (m).
Wave number and wavelength are related by the formula
The wave number is otherwise called the phase constant or propagation constant. It can be seen from formula (3.4) that the propagation constant is measured in ( 
To describe the wave process, the concept of a wave front is introduced. wave front is the locus of imaginary points on the surface to which the excitation has reached. The wave front is also called the wave front.
The equation describing the wavefront of a plane wave can be obtained from equation (3.2), in the form
Formula (3.5) is the wavefront equation for a plane wave. Equation (3.4) shows that wavefronts are infinite planes moving in space perpendicular to the z axis.
The speed of the phase front is called phase speed . The phase velocity is denoted by V f and is determined by the formula
Initially, equation (3.2) contains a phase with two signs - negative and positive. Negative sign, i.e. 
The positive sign of the wave phase indicates the movement of the wave front in the opposite direction, i.e. opposite direction of the z-axis. Such a wave is called reflected.
In what follows, we will consider traveling waves.
If the wave propagates in a real medium, then due to the occurring heat losses, the amplitude will inevitably decrease. Let's consider a simple example. Let the wave propagate along the z axis and the initial value of the wave amplitude corresponds to 100%, i.e. A0=100. Assume that when passing one meter of the path, the amplitude of the wave decreases by 10%. Then we will have the following wave amplitudes
The general pattern of amplitude change has the form
An exponential function has these properties. Graphically, the process can be shown in the form of Fig. 3.3.
In general, the proportionality relation can be written as
where is the damping constant of the wave.
The phase constant and the damping constant can be combined by introducing the complex propagation constant , i.e.
where is the phase constant, is the damping constant of the wave.
Depending on the type of wave front, waves are plane, spherical, and cylindrical.
plane wave
is a wave with a flat wave front. A plane wave can also be given the following definition. A wave is said to be plane homogeneous if the vector field 
Plane wave equation
If the source that generates the wave is a point, then the wave front propagating in an unlimited homogeneous space is a sphere. spherical wave is a wave with a spherical wave front. The spherical wave equation has the form
where r is the radius vector drawn from the origin, which coincides with the position of the point source, to a specific point in space located at a distance r.
Waves can be excited using an infinite string of sources located along the z axis. In this case, such a thread will generate waves whose phase front is a cylindrical surface.
cylindrical wave is a wave with a phase front in the form of a cylindrical surface. The cylindrical wave equation has the form
Formulas (3.2), (3.10, 3.11) indicate a different dependence of the amplitude on the distance between the source of the wave and a specific point in space to which the wave has reached.
Helmholtz equations
Maxwell obtained one of the most important results of electrodynamics, proving that the propagation of electromagnetic processes in space over time occurs in the form of a wave. Let us consider the proof of this proposition, i.e. Let us prove the wave nature of the electromagnetic field.
We write the first two Maxwell equations in complex form as
Let us take the second equation of the system (3.12) and apply the rotor operation to it to the left and right parts. As a result, we get
Denote 
On the other hand, based on the well-known identity in vector analysis, one can write
where 
Considering the Gauss law, i.e. 
Similarly, using the symmetry of Maxwell's equations, one can obtain an equation with respect to the vector 
Equations of the form (3.17, 3.18) are called Helmholtz equations. It has been proven in mathematics that if any process is described in the form of Helmholtz equations, then this means that the process is a wave process. In our case, we conclude: time-varying electric and magnetic fields inevitably lead to the propagation of electromagnetic waves in space.
In coordinate form, the Helmholtz equation (3.17) is written as
where 
Properties of plane waves during propagation in non-absorbing media
Let a plane electromagnetic wave propagate along the z axis, then the wave propagation is described by a system of differential equations
where 
The solution to system (3.21) has the form
If the wave propagates in only one direction along the z-axis, and the vector 
where 
If there are no losses in the medium, i.e. environment parameters a and a, and 
We list the properties of plane electromagnetic waves
For the medium, the concept of the wave resistance of the medium is introduced
where 
For air, the wave resistance is
Equation (3.24) shows that the magnetic and electric fields are in phase. The field of a plane wave is a traveling wave, which is written in the form
On fig. 3.4 field vectors 
The Poynting vector at any time coincides with the direction of wave propagation
The Poynting vector modulus defines the power flux-density and is measured in 
The average power flux-density is determined
where 
The field energy contained in a unit volume is called the energy density. The electromagnetic field changes over time, i.e. is variable. The value of the energy density at a given time is called the instantaneous energy density. For the electric and magnetic components of the electromagnetic field, the instantaneous energy densities are respectively equal to
Given that 
The total electromagnetic energy density is given by
The phase velocity of propagation of an electromagnetic wave is determined by the formula
The wavelength is determined
where 
The energy transfer rate (group velocity) can be determined from the formula
where 
If you paint 
Thus, we get
When an electromagnetic monochromatic wave propagates in a lossless medium, the phase and group velocities are equal.
There is a relationship between the phase and group velocity, expressed by the formula
Consider an example of the propagation of an electromagnetic wave in a fluoroplast having parameters =2, =1. Let the electric field strength correspond to
The speed of wave propagation in such a medium will be equal to
Wave impedance of fluoroplast corresponds to the value
The amplitude values of the magnetic field strength take the values
The energy flux density, respectively, is equal to
Wavelength at frequency 
Umov–Poynting theorem
The electromagnetic field is characterized by its own energy of the field, and the total energy is determined by the sum of the energies of the electric and magnetic fields. Let the electromagnetic field occupy a closed volume V, then we can write
The energy of the electromagnetic field, in principle, cannot remain constant. The question arises: What factors influence the change in energy? It has been established that the following factors influence the change in energy inside a closed volume:
part of the energy of the electromagnetic field can turn into other types of energy, for example, mechanical;
outside forces can act inside a closed volume, which can increase or decrease the energy of the electromagnetic field contained in the volume under consideration;
the considered closed volume V can exchange energy with the surrounding bodies due to the process of energy radiation.
The radiation intensity is characterized by the Poynting vector 
Recall that 
Passing from an integral over a surface 
Knowing that 
It can be seen from formula (3.48) that the left side is expressed as a sum consisting of three terms, each of which we will consider separately.
term 
Second term 
If a 
The last term for a linear medium can be represented as:
Formula (3.49) expresses the rate of change of the energy of the electromagnetic field contained within the volume V.
After considering all the terms, formula (3.48) can be written as:
Formula (3.50) expresses the Poynting theorem. Pointing's theorem expresses the balance of energy within an arbitrary region in which an electromagnetic field exists.
Retarded potentials
Maxwell's equations in complex form, as is known, have the form:
Let external currents exist in a homogeneous medium. Let's try to transform Maxwell's equations for such a medium and obtain a simpler equation that describes the electromagnetic field in such a medium.
Take the equation 
Let's take the second equation of the Maxwell system (3.51) and perform transformations:
Formula (3.53) expresses the second Maxwell equation in terms of the vector potential
In electrostatics, as is known, the relation is fulfilled:
where 
The expression in brackets (3.54), by analogy with formula (3.55), can be written as
where 
Let's take Maxwell's first equation and write it down using electrodynamic potentials
In vector algebra, the identity is proved:
Using identity (3.58), the first Maxwell equation written in the form (3.57) can be represented as
Here are similar
Multiply the left and right parts by the factor (-1):
Expression (3.60) is called Lorentz gauge .
If a w=0
, then we get Coulomb gauge
Taking into account gauges, equation (3.59) can be written
Equation (3.61) expresses itself inhomogeneous wave equation for the vector electrodynamic potential.
In a similar way, based on the third Maxwell equation 
The resulting inhomogeneous equations for electrodynamic potentials have their own solutions
where M- arbitrary point M, 
where V is the volume occupied by external currents, r is the current distance from each element of the source volume to point M.
The solution for the vector electrodynamic potential (3.63), (3.64) is called Kirchhoff integral for retarded potentials .
Factor 
This factor corresponds to the final speed of wave propagation from the source, and 
At the point in time t vector potential 
From formula (3.64) it can be seen that the vector electrodynamic potential is parallel (codirectional) with the current density of external forces; its amplitude decreases according to the law; at large distances compared to the dimensions of the emitter, the wave has a spherical wave front.
Considering 
The obtained relations determine the electromagnetic field in the space created by a given distribution of external currents
Propagation of plane electromagnetic waves in highly conductive media
Consider the propagation of an electromagnetic wave in a conducting medium. Such media are also called metal-like. A real medium is conductive if the density of conduction currents significantly exceeds the density of displacement currents, i.e. 
Formula (3.66) expresses the condition under which a real medium can be considered conductive. In other words, the imaginary part of the complex permittivity must exceed the real part. Formula (3.66) also shows the dependence 
Yes, at the frequency f
= 1 MHz = 10 6 Hz dry soil has parameters =4, =0.01 
For a real medium, we write the complex permittivity
because in our case 
where - specific conductivity, - cyclic frequency.
The propagation constant is known to be determined from the Helmholtz equations
Thus, we obtain the formula for the propagation constant
It is known that
Taking into account identity (3.49), formula (3.50) can be written as
The propagation constant is expressed as
Comparison of the real and imaginary parts in formulas (3.71), (3.72) leads to the equality of the values of the phase constant and damping constant , i.e.
From formula (3.73) we write the wavelength that the field acquires when propagating in a well-conducting medium
where 
From the obtained formula (3.74) it can be seen that the length of an electromagnetic wave propagating in a metal is significantly reduced in comparison with the wavelength in space.
It was said above that the amplitude of the wave during propagation in a medium with losses decreases according to the law 
Surface layer depth - this is the distance d at which the amplitude of the surface wave decreases by a factor of e compared to its initial level.
where 
The depth of the surface layer can also be determined from the formula
where is the cyclic frequency, a is the absolute magnetic permeability of the medium, is the specific conductivity of the medium.
From formula (3.76) it can be seen that with an increase in frequency and conductivity, the depth of the surface layer decreases.
Let's take an example. Copper Conductivity 
Reflection and refraction of a plane wave at the interface between media
When a plane electromagnetic wave propagates in space, which is a region with different values of parameters 
wave reflection coefficient
passing ratio
waves
If the Poynting vector of the incident wave is perpendicular to the interface, then
where Z 1 ,Z 2 - characteristic resistance for the respective media.
The characteristic resistance is determined by the formula:
where 
With oblique incidence, the direction of wave propagation with respect to the interface is given by the angle of incidence. Angle of incidence is the angle between the normal to the surface and the direction of beam propagation.
plane of incidence is the plane that contains the incident ray and the normal restored to the point of incidence.
It follows from the boundary conditions that the angles of incidence 
where n 1 , n 2 are the refractive indices of the respective media.
Electromagnetic waves are characterized by polarization. There are elliptical, circular and linear polarizations. In linear polarization, horizontal and vertical polarization are distinguished.
Horizontal polarization
is the polarization at which the vector 
Let a plane electromagnetic wave with horizontal polarization fall on the interface between two media, as shown in Fig. 3.7. The Poynting vector of the incident wave is denoted 
For the reflected wave, the corresponding field vectors are provided with the index "neg", for the refracted one - with the index "pr".
With horizontal (perpendicular) polarization, the reflection and transmission coefficients are found as follows (Fig. 3.7).
At the interface between two media, the boundary conditions are satisfied, i.e.
In our case, we must identify the tangential projections of the vectors, i.e. can be written
The lines of the magnetic field strength are directed for the incident, reflected and refracted waves perpendicular to the plane of incidence. Therefore, one should write
Based on this, we can compose a system based on the boundary conditions
It is also known that the strengths of the electric and magnetic fields are interconnected through the wave resistance of the medium Z
Then the second equation of the system can be written as
So, the system of equations has taken the form
Let us divide both equations of this system by the amplitude of the incident wave 
The system has two solutions and two unknowns. Such a system is known to be decidable.
Vertical polarization
is the polarization at which the vector 
With vertical (parallel) polarization, the reflection and transmission coefficients are expressed as follows (Fig. 3.8).
For vertical polarization, a similar system of equations is written as for horizontal polarization, but taking into account the direction of the electromagnetic field vectors
Such a system of equations can be similarly reduced to the form
The solution of the system is the expressions for the reflection and transmission coefficients
When plane electromagnetic waves with parallel polarization are incident on the interface between two media, the reflection coefficient can become zero. The angle of incidence at which the incident wave completely, without reflection, penetrates from one medium to another is called the Brewster angle and is denoted as 
We emphasize that the Brewster angle when a plane electromagnetic wave is incident on a nonmagnetic dielectric can exist only with parallel polarization.
If a plane electromagnetic wave is incident at an arbitrary angle on the interface between two media with losses, then the reflected and refracted waves should be considered inhomogeneous, since the plane of equal amplitudes should coincide with the interface. For real metals, the angle between the phase front and the plane of equal amplitudes is small, so we can assume that the angle of refraction is 0.
Approximate Schukin-Leontovich Boundary Conditions
These boundary conditions apply when one of the media is a good conductor. Let us assume that a plane electromagnetic wave is incident from air at an angle onto a plane interface with a well-conducting medium, which is described by the complex refractive index
It follows from the definition of the concept of a well-conducting medium that 
Using the Leontovich boundary conditions, it is necessary to know the tangent component of the magnetic vector 
Emission of electromagnetic waves into free space
Let us find out what are the conditions for the emission of electromagnetic energy into free space. To do this, consider a point monochromatic emitter of electromagnetic waves, which is placed at the origin of the spherical coordinate system. As is known, the spherical coordinate system is given by (r, Θ, φ), where r is the radius vector drawn from the origin of the system to the observation point; Θ is the meridional angle measured from the Z axis (zenith) to the radius vector drawn to the point M; φ is the azimuthal angle measured from the X axis to the projection of the radius vector drawn from the origin to the point M′ (M′ is the projection of the point M onto the XOY plane). (Fig.3.9).
The point emitter is located in a homogeneous medium with parameters
A point emitter radiates electromagnetic waves in all directions and any component of the electromagnetic field obeys the Helmholtz equation, except for the point r=0 . One can introduce a complex scalar function Ψ, which is understood as any arbitrarily taken component of the field. Then the Helmholtz equation for the function Ψ has the form:
where 
Let us assume that the function Ψ has spherical symmetry, then the Helmholtz equation can be written as:
Equation (3.89) can also be written as:
Equations (3.89) and (3.90) are identical to each other. Equation (3.90) is known in physics as the oscillation equation. Such an equation has two solutions, which, if the amplitudes are equal, have the form:
As can be seen from (3.91), (3.92), the solution of the equation differs only in signs. Moreover, 
The example under consideration is quite simple. But in the case of radiation of energy by a system of sources, it is very difficult to choose the right solution. Therefore, an analytical expression is required, which is a criterion for choosing the right solution. We need a general criterion in an analytical form, which makes it possible to choose an unambiguous physically determined solution.
In other words, we need a criterion that distinguishes a function that expresses a traveling wave from a source to infinity, from a function that describes a wave coming from infinity to a radiation source.
This problem was solved by A. Sommerfeld. He showed that for a traveling wave described by the function 
This formula is called radiation condition or Sommerfeld condition .
Consider an elementary electric emitter in the form of a dipole. An electric dipole is a short piece of wire l compared to the long wave ( l<< ), по которому протекает переменный ток (рис. 3.9). Т.к. соблюдается выполнение условия l<< , то можно считать, что во всех сечениях провода в данный момент времени протекает одинаковый ток
It is easy to show that the change in the electric field in the space surrounding the wire has a wave character. For clarity, let's consider an extremely simplified model of the process of formation and change of the electric component of the electromagnetic field emitted by the wire. On fig. 3.11 shows a model of the process of radiation of the electric field of an electromagnetic wave over a period of time equal to one period
As you know, electric current is due to the movement of electric charges, namely
In the future, we will consider only the change in the position of the positive and negative charges on the wire. The electric field strength line starts at a positive charge and ends at a negative one. On fig. 3.11 the line of force is shown by a dotted line. It is worth remembering that the electric field is created in the entire space surrounding the conductor, although in Fig. 3.11 shows one line of force.
In order for an alternating current to flow through a conductor, an alternating EMF source is required. Such a source is included in the middle of the wire. The state of the electric field emission process is shown by numbers from 1 to 13. Each number corresponds to a certain point in time associated with the state of the process. The moment t=1 corresponds to the beginning of the process, i.e. EMF = 0. At the moment t=2, a variable EMF appears, which causes the movement of charges, as shown in fig. 3.11. With the advent of moving charges in the wire, an electric field arises in space. over time (t = 3÷5) the charges move towards the ends of the conductor and the line of force covers an increasing part of the space. the line of force expands at the speed of light in a direction perpendicular to the wire. At the time t = 6 - 8, the EMF, having passed through the maximum value, decreases. Charges move towards the middle of the wire.
At the time t = 9, the half-cycle of the EMF change ends, it decreases to zero. In this case, the charges merge, they compensate each other. there is no electric field in this case. The line of force of the radiated electric field closes and continues to move away from the wire.
Then comes the second half-cycle of the change in EMF, the processes are repeated taking into account the change in polarity. On fig. 3.11 at the moments t = 10÷13 shows the picture of the process taking into account the force line of the electric field.
We have considered the process of formation of closed lines of force of a vortex electric field. But it is worth remembering that the radiation of electromagnetic waves is a single process. The electric and magnetic fields are inseparable interdependent components of the electromagnetic field.
The radiation process shown in fig. 3.11 is similar to the radiation of an electromagnetic field by a symmetrical electric vibrator and is widely used in radio communication technology. It must be remembered that the plane of oscillations of the electric field strength vector 
The emission of electromagnetic waves is due to a variable process. Therefore, in the formula for the charge, you can put the constant C \u003d 0. For the complex value of the charge can be written.
By analogy with electrostatics, we can introduce the concept of the moment of an electric dipole with alternating current
From formula (3.95) it follows that the moment vectors of the electric dipole and the directed wire segment 
It should be noted that real antennas have wire lengths that are usually comparable to the wavelength. To determine the radiative characteristics of such antennas, the wire is usually mentally divided into separate small sections, each of which is considered as an elementary electric dipole. the resulting antenna field is found by summing the radiated vector fields generated by the individual dipoles.
Function (78.1) must be periodic both with respect to time t and with respect to x, y and z coordinates. Periodicity in t follows from the fact that it describes the fluctuations of a point with coordinates x, y, z. Periodicity in coordinates follows from the fact that points spaced at a distance from each other oscillate in the same way.
Let us find the form of the function in the case of a plane wave, assuming that the oscillations are harmonic in nature. To simplify, let us direct the coordinate axes so that the x axis coincides with the direction of wave propagation. Then the wave surfaces will be perpendicular to the x-axis and, since all points of the wave surface oscillate in the same way, the displacement will depend only on x and t:
Let the fluctuations of points lying in the x=0 plane (Fig. 195) have the form
Let us find the type of oscillation of particles in the plane corresponding to an arbitrary value of x. In order to go from the x=0 plane to this plane, the wave needs time
Where is the speed of wave propagation. Consequently, oscillations of particles lying in the x plane will lag behind the oscillations of particles in the x=0 plane in time, i.e. will look like
So, the plane wave equation will be written as follows;
Expression (78.3) gives the relationship between time (t) and the place (x) in which the fixed value of the phase is carried out at the moment. Having determined the value of dx /dt resulting from it, we will find the speed with which this phase value moves. Differentiating expression (78.3), we get:
Indeed, by equating the wave phase (78.5) to a constant and differentiating, we obtain:
whence it follows that wave (78.5) propagates in the direction of decreasing x.
The plane wave equation can be given a form that is symmetric with respect to t and x. To do this, we introduce the so-called wave number k;
Replacing in equation (78.2) its value (78.7) and putting in brackets , we obtain the equation of a plane wave in the form
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(78 .8) |
The equation of a wave propagating in the direction of decreasing x will differ from (78.8) only in sign at the term kx.
Now let's find the equation of a spherical wave. Any real source of waves has some extent. However, if we confine ourselves to considering the wave at distances from the source that are much larger than its dimensions, then the source can be considered as a point source.
In the case when the speed of wave propagation in all directions is the same, the wave generated by a point source will be spherical. Let us assume that the phase of the source oscillation is . Then the points lying on the wave surface of radius r will oscillate with phase (it takes time for the wave to travel the path r). The oscillation amplitude in this case, even if the wave energy is not absorbed by the medium, does not remain constant - it decreases with distance from the source according to the law 1/r (see §82). Therefore, the spherical wave equation has the form
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(78 .9) |
where a is a constant value numerically equal to the amplitude at a distance from the source equal to unity. The dimension a is equal to the dimension of the amplitude multiplied by the dimension of length (the dimension r).
Recall that, by virtue of the assumptions made at the outset, Eq. (78.9) is valid only when the source dimensions are much larger. As r tends to zero, the expression for the amplitude goes to infinity. This absurd result is explained by the inapplicability of the equation for small r.
We mean the coordinates of the equilibrium position of the point.