Wave equation is an equation expressing the dependence of the displacement of an oscillating particle participating in a wave process on the coordinate of its equilibrium position and time:

This function must be periodic both with respect to time and with respect to coordinates. In addition, points located at a distance l from each other, oscillate in the same way.

Let's find the type of function x in the case of a plane wave.

Let us consider a plane harmonic wave propagating along the positive direction of the axis in a medium that does not absorb energy. In this case, the wave surfaces will be perpendicular to the axis. All quantities characterizing the oscillatory motion of particles of the medium depend only on time and coordinates. The offset will depend only on and: . Let the oscillation of a point with a coordinate (the source of oscillation) be given by the function. Task: find the type of vibration of points in the plane corresponding to an arbitrary value. In order to travel from a plane to this plane, a wave requires time. Consequently, the oscillations of particles lying in the plane will lag in phase by a time from the oscillations of particles in the plane. Then the equation of particle oscillations in the plane will have the form:

As a result, we obtained the equation of a plane wave propagating in the increasing direction:

. (3)

In this equation, is the amplitude of the wave; – cyclic frequency; – initial phase, which is determined by the choice of the reference point and ; – plane wave phase.

Let the wave phase be a constant value (we fix the phase value in the wave equation):

Let's reduce this expression by and differentiate. As a result we get:

or .

Thus, the speed of propagation of a wave in the plane wave equation is nothing more than the speed of propagation of a fixed phase of the wave. This speed is called phase velocity .

For a sine wave, the speed of energy transfer is equal to the phase speed. But a sine wave does not carry any information, and any signal is a modulated wave, i.e. not sinusoidal (not harmonic). When solving some problems, it turns out that the phase speed is greater than the speed of light. There is no paradox here, because... the speed of phase movement is not the speed of transmission (propagation) of energy. Energy and mass cannot move at a speed greater than the speed of light c .

Usually the plane wave equation is given a relatively symmetrical form. To do this, enter the value , which is called wave number . Let us transform the expression for the wave number. Let's write it in the form (). Let's substitute this expression into the plane wave equation:

Finally we get

This is the equation of a plane wave propagating in the increasing direction. The opposite direction of wave propagation will be characterized by an equation in which the sign in front of the term will change.

It is convenient to write the plane wave equation in the following form.

Usually a sign Re are omitted, implying that only the real part of the corresponding expression is taken. In addition, a complex number is introduced.

This number is called the complex amplitude. The modulus of this number gives the amplitude, and the argument gives the initial phase of the wave.

Thus, the equation of a plane continuous wave can be represented in the following form.

Everything discussed above related to a medium where there was no wave attenuation. In the case of wave attenuation, in accordance with Bouguer's law (Pierre Bouguer, French scientist (1698 - 1758)), the amplitude of the wave will decrease as it propagates. Then the plane wave equation will have the following form.

a– wave attenuation coefficient. A 0 – amplitude of oscillations at a point with coordinates . This is the reciprocal of the distance at which the wave amplitude decreases by e once.

Let's find the equation of a spherical wave. We will consider the source of oscillations to be point-like. This is possible if we limit ourselves to considering the wave at a distance much greater than the size of the source. A wave from such a source in an isotropic and homogeneous medium will be spherical . Points lying on the wave surface of radius will oscillate with phase

The amplitude of oscillations in this case, even if the wave energy is not absorbed by the medium, will not remain constant. It decreases with distance from the source according to the law. Therefore, the spherical wave equation has the form:

or

Due to the assumptions made, the equation is valid only for , significantly exceeding the size of the wave source. Equation (6) is not applicable for small values, because the amplitude would tend to infinity, and this is absurd.

In the presence of attenuation in the medium, the equation of a spherical wave will be written as follows.

Group speed

A strictly monochromatic wave is an infinite sequence of “humps” and “valleys” in time and space.

The phase speed of this wave or (2)

It is impossible to transmit a signal using such a wave, because at any point in the wave all the “humps” are the same. The signal must be different. To be a sign (mark) on the wave. But then the wave will no longer be harmonic, and will not be described by equation (1). A signal (pulse) can be represented according to Fourier’s theorem as a superposition of harmonic waves with frequencies contained in a certain interval Dw . Superposition of waves that differ little from each other in frequency,

called wave packet or group of waves .

The expression for a group of waves can be written as follows.

(3)

Icon w emphasizes that these quantities depend on frequency.

This wave packet can be a sum of waves with slightly different frequencies. Where the phases of the waves coincide, an increase in amplitude is observed, and where the phases are opposite, a damping of the amplitude is observed (the result of interference). This picture is shown in the figure. In order for a superposition of waves to be considered a group of waves, the following condition must be met: Dw<< w 0 .

In a non-dispersive medium, all plane waves forming a wave packet propagate with the same phase velocity v . Dispersion is the dependence of the phase velocity of a sinusoidal wave in a medium on frequency. We will consider the phenomenon of dispersion later in the section “Wave Optics”. In the absence of dispersion, the speed of movement of the wave packet coincides with the phase speed v . In a dispersive medium, each wave disperses at its own speed. Therefore, the wave packet spreads out over time and its width increases.

If the dispersion is small, then the wave packet does not spread out too quickly. Therefore, a certain speed can be attributed to the movement of the entire package U .

The speed at which the center of the wave packet (the point with the maximum amplitude) moves is called group velocity.

In a dispersive environment v¹U . Along with the movement of the wave packet itself, the “humps” inside the packet itself move. "Humps" move in space at speed v , and the package as a whole with speed U .

Let us consider in more detail the movement of a wave packet using the example of a superposition of two waves with the same amplitude and different frequencies w (different wavelengths l ).

Let's write down the equations of two waves. For simplicity, let us assume the initial phases j 0 = 0.

Here

Let Dw<< w , respectively Dk<< k .

Let's add up the vibrations and carry out transformations using the trigonometric formula for the sum of cosines:

In the first cosine we will neglect Dwt And Dkx , which are much smaller than other quantities. Let's take into account that cos(–a) = cosa . We'll write it down finally.

(4)

The multiplier in square brackets changes with time and coordinates much more slowly than the second multiplier. Consequently, expression (4) can be considered as an equation of a plane wave with an amplitude described by the first factor. Graphically, the wave described by expression (4) is presented in the figure shown above.

The resulting amplitude is obtained as a result of the addition of waves, therefore, maxima and minima of the amplitude will be observed.

The maximum amplitude will be determined by the following condition.

(5)

m = 0, 1, 2…

xmax– coordinate of the maximum amplitude.

The cosine takes its maximum modulo value through p .

Each of these maxima can be considered as the center of the corresponding group of waves.

Resolving (5) relatively xmax we'll get it.

Since the phase speed is called group velocity. The maximum amplitude of the wave packet moves at this speed. In the limit, the expression for the group velocity will have the following form.

(6)

This expression is valid for the center of a group of an arbitrary number of waves.

It should be noted that when all terms of the expansion are accurately taken into account (for an arbitrary number of waves), the expression for the amplitude is obtained in such a way that it follows that the wave packet spreads out over time.
The expression for group velocity can be given a different form.

In the absence of variance

The maximum intensity occurs at the center of the wave group. Therefore, the speed of energy transfer is equal to the group speed.

The concept of group velocity is applicable only under the condition that wave absorption in the medium is low. With significant wave attenuation, the concept of group velocity loses its meaning. This case is observed in the region of anomalous dispersion. We will consider this in the “Wave Optics” section.

Before considering the wave process, let us give a definition of oscillatory motion. Hesitation - This is a periodically repeating process. Examples of oscillatory movements are very diverse: the change of seasons, heart vibrations, breathing, charge on the plates of a capacitor and others.

The oscillation equation in general form is written as

Where - amplitude of oscillations,
- cyclic frequency, - time, - initial phase. Often the initial phase can be taken to be zero.

From oscillatory motion we can move on to consider wave motion. Wave is the process of propagation of vibrations in space over time. Since oscillations propagate in space over time, the wave equation must take into account both spatial coordinates and time. 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, and acoustic waves are known.

Based on the type of vibration, 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 propagation of the wave (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 a transverse direction relative to the direction of propagation (Fig. 3.1b).

Electromagnetic waves are classified 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 with one frequency  propagates in space, 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
, called wave phase .

Schematically, wave propagation along one coordinate is shown in Fig. 3.2, in this case, propagation occurs along the z axis.

Period – time of one complete oscillation. The period is designated by the letter T and is measured in seconds (s). The reciprocal of the period is called linear frequency and is designated f, measured in Hertz (=Hz). Linear frequency is related to circular frequency. The relationship is expressed by the formula

(3.3)

If we fix time t, then from Fig. 3.2 it is clear that there are points, for example A and B, that vibrate equally, i.e. in phase (in phase). The distance between the nearest two points oscillating in phase is called wavelength . The wavelength is designated  and measured in meters (m).

Wave number  and wavelength  are related to each other by the formula

(3.4)

Wave number  is otherwise called the phase constant or propagation constant. From formula (3.4) it is clear that the propagation constant is measured in ( ). The physical meaning is that it shows how many radians the phase of the wave changes when passing one meter of path.

To describe the wave process, the concept of wave front is introduced. Wave front – this is the geometric location of the imaginary points of the surface to which the excitation has reached. A wave front is also called a wave front.

The equation describing the wave front of a plane wave can be obtained from equation (3.2) in the form

(3.5)

Formula (3.5) is the wavefront equation of a plane wave. Equation (3.4) shows that wave fronts are infinite planes moving in space perpendicular to the z axis.

The speed of movement of the phase front is called phase velocity . The phase velocity is denoted by V f and is determined by the formula

(3.6)

Initially, equation (3.2) contains a phase with two signs – negative and positive. Negative sign, i.e.
, indicates that the wave front propagates along the positive direction of propagation of the z-axis. Such a wave is called traveling or falling.

A positive sign of the wave phase indicates movement of the wave front in the opposite direction, i.e. opposite to the z-axis direction. Such a wave is called reflected.

In what follows we will consider traveling waves.

If a wave propagates in a real environment, then due to the heat losses occurring, a decrease in amplitude inevitably occurs. Let's look at a simple example. Let the wave propagate along the z axis and the initial value of the wave amplitude corresponds to 100%, i.e. A 0 =100. Let's say that when passing one meter of path, the amplitude of the wave decreases by 10%. Then we will have the following values ​​of wave amplitudes

The general pattern of amplitude changes has the form

The exponential function has these properties. Graphically the process can be shown in the form of Fig. 3.3.

In general, we write the proportionality relation as

, (3.7)

where  is the wave attenuation constant.

The phase constant  and the damping constant  can be combined by introducing a complex propagation constant , i.e.

, (3.8)

where  is the phase constant,  is the wave attenuation constant.

Depending on the type of wave front, plane, spherical, and cylindrical waves are distinguished.

Plane wave is a wave that has a plane wave front. A plane wave can also be given the following definition. A wave is called plane homogeneous if the vector field And at any point in the plane are perpendicular to the direction of propagation and do not change in phase and amplitude.

Plane wave equation

If the source generating the wave is a point source, then the wave front propagating in an unlimited homogeneous space is a sphere. Spherical wave is a wave that has a spherical wave front. The spherical wave equation has the form

, (3.10)

where r is the radius vector drawn from the origin, coinciding with the position of the point source, to a specific point in space located at a distance r.

The waves can be excited by an endless string of sources located along the z axis. In this case, such a thread will generate waves, the phase front of which is a cylindrical surface.

Cylindrical wave is a wave that has a phase front in the form of a cylindrical surface. The equation of a cylindrical wave is

, (3.11)

Formulas (3.2), (3.10, 3.11) indicate a different dependence of the amplitude on the distance between the wave source and the specific point in space to which the wave reached.

Helmholtz equations

Maxwell obtained one of the most important results in 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.

Let us write the first two Maxwell equations in complex form as

(3.12)

Let us take the second equation of system (3.12) and apply the rotor operation to it on the left and right sides. As a result we get

Let's denote
, which represents the propagation constant. Thus

(3.14)

On the other hand, based on the well-known identity in vector analysis, we can write

, (3.15)

Where
is the Laplace operator, which in the Cartesian coordinate system is expressed by the identity

(3.16)

Considering Gauss's law, i.e.
, equation (3.15) will be written in a simpler form

, or

(3.17)

Similarly, using the symmetry of Maxwell’s equations, we can obtain an equation for the vector , i.e.

(3.18)

Equations of the form (3.17, 3.18) are called Helmholtz equations. In mathematics it has been proven that if any process is described in the form of Helmholtz equations, 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 ,,- unit vectors along the corresponding coordinate axes

,

,

.(3.20)

Properties of plane waves when propagating in non-absorbing media

Let a plane electromagnetic wave propagate along the z axis, then the propagation of the wave is described by a system of differential equations

(3.21)

Where And - complex field amplitudes,

(3.22)

The solution to system (3.21) has the form

(3.23)

If the wave propagates in only one direction along the z axis, and the vector is directed along the x axis, then it is advisable to write the solution to the system of equations in the form

(3.24)

Where And - unit vectors along the x, y axes.

If there are no losses in the medium, i.e. environmental parameters  a and  a, and
are real quantities.

Let us list the properties of plane electromagnetic waves

For the medium, the concept of wave impedance of the medium is introduced

(3.25)

Where ,
- amplitude values ​​of field strengths. The characteristic impedance for a lossless medium is also a real value.

For air, the wave resistance is

(3.26)

From equation (3.24) it is clear that the magnetic and electric fields are in phase. The plane wave field is a traveling wave, which is written in the form

(3.27)

In Fig. 3.4 field vectors And change in phase, as follows from formula (3.27).

The Poynting vector at any time coincides with the direction of wave propagation

(3.28)

The Poynting vector modulus determines the power flux density and is measured in
.

The average power flux density is determined by

(3.29)

, (3.30)

Where
- effective values ​​of field strengths.

The field energy contained in a unit volume is called energy density. The electromagnetic field changes over time, i.e. is variable. The value of energy density at a given time is called instantaneous energy density. For the electric and magnetic components of the electromagnetic field, the instantaneous energy densities are respectively equal

Considering that
, from relations (3.31) and (3.32) it is clear that
.

The total electromagnetic energy density is given by

(3.33)

The phase speed of propagation of an electromagnetic wave is determined by the formula

(3.34)

The wavelength is determined

(3.35)

Where - wavelength in vacuum (air), s - speed of light in air,  - relative dielectric constant,  - relative magnetic permeability, f– linear frequency,  – cyclic frequency, V f – phase velocity,  – propagation constant.

The speed of energy movement (group speed) can be determined from the formula

(3.36)

Where - Poynting vector, - energy density.

If you paint and in accordance with formulas (3.28), (3.33), we obtain

(3.37)

Thus, we get

(3.38)

When an electromagnetic monochromatic wave propagates in a lossless medium, the phase and group velocities are equal.

There is a relationship between phase and group velocity expressed by the formula

(3.39)

Let's consider an example of the propagation of an electromagnetic wave in fluoroplastic having parameters  =2, =1. Let the electric field strength correspond to

(3.40)

The speed of wave propagation in such a medium will be equal to

The characteristic impedance of fluoroplastic corresponds to the value

Ohm (3.42)

The amplitude values ​​of the magnetic field strength take on the values

, (3.43)

The energy flux density is, accordingly, equal to

Wavelength at frequency
has the meaning

(3.45)

Umov–Poynting theorem

An electromagnetic field is characterized by its own field energy, 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

(3.46)

The energy of the electromagnetic field, in principle, cannot remain a constant value. The question arises: What factors influence the change in energy? It has been established that the change in energy inside a closed volume is influenced by the following factors:

part of the energy of the electromagnetic field can be converted into other types of energy, for example, mechanical;

inside a closed volume, external forces can act, which can increase or decrease the energy of the electromagnetic field contained in the volume under consideration;

the closed volume V under consideration can exchange energy with surrounding bodies through the process of energy radiation.

The radiation intensity is characterized by the Poynting vector . Volume V has a closed surface S. The change in the energy of the electromagnetic field can be considered as the flow of the Poynting vector through the closed surface S (Fig. 3.5), i.e.
, and options are possible
>0 ,
<0 ,
=0 . Note that the normal drawn to the surface
,is always external.

Let us remind you that
, Where
are instantaneous field strength values.

Transition from surface integral
to the integral over volume V is carried out on the basis of the Ostrogradsky-Gauss theorem.

Knowing that

Let's substitute these expressions into formula (3.47). After transformation, we obtain an expression in the form:

From formula (3.48) it is clear that the left side is expressed by a sum consisting of three terms, each of which we will consider separately.

Term
expresses instantaneous power loss , caused by conduction currents in the closed volume under consideration. In other words, the term expresses the thermal energy losses of the field enclosed in a closed volume.

Second term
expresses the work of external forces performed per unit of time, i.e. power of external forces. For such power the possible values ​​are
>0,
<0.

If
>0, those. energy is added to volume V, then external forces can be considered as a generator. If
<0 , i.e. in volume V there is a decrease in energy, then external forces play the role of load.

The last term for a linear medium can be represented as:

(3.49)

Formula (3.49) expresses the rate of change in the energy of the electromagnetic field contained inside the volume V.

After considering all terms, formula (3.48) can be written as:

Formula (3.50) expresses Poynting’s theorem. Poynting's theorem expresses the balance of energy within an arbitrary region in which an electromagnetic field exists.

Delayed potentials

Maxwell's equations in complex form, as is known, have the form:

(3.51)

Let there be external currents 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.

Let's take the equation
.Knowing that the characteristics And interconnected
, then we can write
Let us take into account that the magnetic field strength can be expressed using vector electrodynamic potential , which is introduced by the relation
, Then

(3.52)

Let's take the second equation of the Maxwell system (3.51) and perform the transformations:

(3.53)

Formula (3.53) expresses Maxwell’s second equation in terms of the vector potential . Formula (3.53) can be written as

(3.54)

In electrostatics, as is known, the following relation holds:

(3.55)

Where -field strength vector,
- scalar electrostatic potential. The minus sign indicates that the vector directed from a point of higher potential to a point of lower potential.

The expression in brackets (3.54), by analogy with formula (3.55), can be written in the form

(3.56)

Where
- scalar electrodynamic potential.

Let's take Maxwell's first equation and write it using electrodynamic potentials

In vector algebra the following identity has been proven:

Using identity (3.58), we can represent Maxwell’s first equation, written in the form (3.57), as

Let's give similar

Multiply the left and right sides by a factor (-1):

can be specified in an arbitrary way, so we can assume that

Expression (3.60) is called Lorentz gauge .

If w=0 , then we get Coulomb calibration
=0.

Taking into account the gauges, equation (3.59) can be written

(3.61)

Equation (3.61) expresses inhomogeneous wave equation for the vector electrodynamic potential.

In a similar way, based on Maxwell's third equation
, we can obtain a non-homogeneous equation for scalar electrodynamic potential as:

(3.62)

The resulting inhomogeneous equations for electrodynamic potentials have their own solutions

, (3.63)

Where M– arbitrary point M, - volumetric charge density, γ – propagation constant, r

(3.64)

Where V– volume occupied by external currents, r– 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
can be expressed taking into account
as

This factor corresponds to the finite speed of wave propagation from the source, and
Because the speed of wave propagation is a finite value, then the influence of the source generating the waves reaches an arbitrary point M with a time delay. The delay time value is determined by:
In Fig. 3.6 shows a point source U, which emits spherical waves propagating with speed v in the surrounding homogeneous space, as well as an arbitrary point M located at a distance r, which the wave reaches.

At a moment in time t vector potential
at point M is a function of the currents flowing in the source U at an earlier time
In other words,
depends on the source currents that flowed in it at an earlier moment

From formula (3.64) it is clear that the vector electrodynamic potential is parallel (co-directional) with the current density of external forces; its amplitude decreases according to the law; at large distances compared to the size of the emitter, the wave has a spherical wave front.

Considering
and Maxwell's first equation, the electric field strength can be determined:

The resulting relationships determine the electromagnetic field in the space created by a given distribution of external currents

Propagation of plane electromagnetic waves in highly conducting media

Let us consider the propagation of an electromagnetic wave in a conducting medium. Such media are also called metal-like media. A real medium is conductive if the density of conduction currents significantly exceeds the density of displacement currents, i.e.
And
, and
, or

(3.66)

Formula (3.66) expresses the condition under which a real medium can be considered conductive. In other words, the imaginary part of the complex dielectric constant must exceed the real part. Formula (3.66) also shows the dependence on frequency, and the lower the frequency, the more pronounced the properties of the conductor are in the medium. Let's look at this situation with an example.

Yes, at frequency f = 1 MHz = 10 6 Hz dry soil has parameters =4, =0.01 ,. Let's compare with each other And , i.e.
. From the obtained values ​​it is clear that 1.610 -19 >> 3.5610 -11, therefore dry soil should be considered conductive when a wave with a frequency of 1 MHz propagates.

For a real medium, we write down the complex dielectric constant

(3.67)

because in our case
, then for a conducting medium we can write

, (3.68)

where  is specific conductivity,  is cyclic frequency.

The propagation constant , as is known, is determined from the Helmholtz equations

Thus, we obtain a formula for the propagation constant

(3.69)

It is known that

(3.70)

Taking into account identity (3.49), formula (3.50) can be written in the form

(3.71)

The propagation constant is expressed as

(3.72)

Comparison of the real and imaginary parts in formulas (3.71), (3.72) leads to equality of the values ​​of the phase constant  and the damping constant , i.e.

(3.73)

From formula (3.73) we write out the wavelength that the field acquires when propagating in a well-conducting medium

(3.74)

Where - wavelength in metal.

From the resulting formula (3.74) it is clear that the length of the electromagnetic wave propagating in the metal is significantly reduced compared to the wavelength in space.

It was said above that the amplitude of a wave when propagating in a medium with losses decreases according to the law
. To characterize the process of wave propagation in a conducting medium, the concept is introduced surface layer depth or penetration depth .

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.

(3.75)

Where - wavelength in metal.

The depth of the surface layer can also be determined from the formula

, (3.76)

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 is clear that with increasing frequency and specific conductivity, the depth of the surface layer decreases.

Let's give an example. Conductivity copper
at frequency f = 10 GHz ( = 3cm) has a surface layer depth d =
. From this we can draw an important conclusion for practice: applying a layer of a highly conductive substance to a non-conducting coating will make it possible to produce device elements with low heat losses.

Reflection and refraction of a plane wave at the interface

When a plane electromagnetic wave propagates in space, which consists of regions with different parameter values
and the interface in the form of a plane, reflected and refracted waves arise. The intensities of these waves are determined through the coefficients of reflection and refraction.

Wave reflection coefficient is the ratio of the complex values ​​of the electric field strengths of the reflected to incident waves at the interface and is determined by the formula:

(3.77)

Pass rate waves into the second medium from the first is called the ratio of the complex values ​​of the electric field strengths of the refracted to the falling waves and is determined by the formula

(3.78)

If the Poynting vector of the incident wave is perpendicular to the interface, then

(3.79)

where Z 1 ,Z 2 – characteristic resistance for the corresponding media.

Characteristic resistance is determined by the formula:

Where
(3.80)

.

With oblique incidence, the direction of wave propagation relative to the interface is determined by the angle of incidence. Angle of incidence – the angle between the normal to the surface and the direction of beam propagation.

Incidence plane is the plane that contains the incident ray and the normal restored to the point of incidence.

From the boundary conditions it follows that the angles of incidence and refraction related by Snell's law:

(3.81)

where n 1, n 2 are the refractive indices of the corresponding 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 – polarization at which the vector oscillates in a plane perpendicular to the plane of incidence.

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 indicated by . Because the wave has horizontal polarization, i.e. the electric field strength vector oscillates in a plane perpendicular to the plane of incidence, then it is designated and in Fig. 3.7 is shown as a circle with a cross (directed away from us). Accordingly, the magnetic field strength vector lies in the plane of incidence of the wave and is designated . Vectors ,,form a right-hand triplet of vectors.

For a reflected wave, the corresponding field vectors are equipped with the index “neg”; for a refracted wave, the index is “pr”.

With horizontal (perpendicular) polarization, the reflection and transmission coefficients are determined as follows (Fig. 3.7).

At the interface between two media, boundary conditions are satisfied, i.e.

In our case, we must identify tangential projections of vectors, i.e. can be written down

The magnetic field strength lines for the incident, reflected and refracted waves are directed perpendicular to the plane of incidence. Therefore we should write

Based on this, we can create a system based on boundary conditions

It is also known that the electric and magnetic field strengths are interconnected through the characteristic impedance of the medium Z

Then the second equation of the system can be written as

So, the system of equations took the form

Let us divide both equations of this system by the amplitude of the incident wave
and, taking into account the definitions of the refractive index (3.77) and transmission (3.78), we can write the system in the form

The system has two solutions and two unknown quantities. Such a system is known to be solvable.

Vertical polarization – polarization at which the vector oscillates in the plane of incidence.

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 to 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
.

(3.84)

(3.85)

We emphasize that the Brewster angle when a plane electromagnetic wave is incident on a non-magnetic dielectric can only exist 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 must 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 boundary conditions of Shchukin-Leontovich

These boundary conditions are applicable 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

(3.86)

From the definition of the concept of a well-conducting medium it follows that
. Applying Snell's law, it can be noted that the angle of refraction  will be very small. From this we can assume that the refracted wave enters the well-conducting medium almost along the normal direction at any value of the angle of incidence.

Using Leontovich boundary conditions, you need to know the tangent component of the magnetic vector . It is usually assumed approximately that this value coincides with a similar component calculated for the surface of an ideal conductor. The error arising from such an approximation will be very small, since the reflection coefficient from the surface of metals is, as a rule, close to zero.

Emission of electromagnetic waves into free space

Let us find out what are the conditions for the radiation of electromagnetic energy into free space. To do this, consider a point monochromatic emitter of electromagnetic waves, which is placed at the origin of a spherical coordinate system. As is known, a 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 – meridional angle, measured from the Z axis (zenith) to the radius vector drawn to point M; φ – azimuthal angle, measured from the X axis to the projection of the radius vector drawn from the origin to point M′ (M′ is the projection of point M onto the XOY plane). (Fig.3.9).

A point emitter is located in a homogeneous medium with the parameters

A point emitter emits electromagnetic waves in all directions and any component of the electromagnetic field obeys the Helmholtz equation, except for the point r=0 . We can introduce a complex scalar function Ψ, which is understood as any arbitrary field component. Then the Helmholtz equation for the function Ψ has the form:

(3.87)

Where
- wave number (propagation constant).

(3.88)

Let us assume that the function Ψ has spherical symmetry, then the Helmholtz equation can be written as:

(3.89)

Equation (3.89) can also be written as:

(3.90)

Equations (3.89) and (3.90) are identical to each other. Equation (3.90) is known in physics as the oscillation equation. This equation has two solutions, which, if the amplitudes are equal, have the form:

(3.91)

(3.92)

As can be seen from (3.91), (3.92), the solution to the equation differs only in signs. Moreover, indicates an incoming wave from the source, i.e. the wave propagates from the source to infinity. Second wave indicates that the wave comes to the source from infinity. Physically, one and the same source cannot generate two waves at the same time: traveling and coming from infinity. Therefore, it is necessary to take into account that the wave does not physically exist.

The example in question is quite simple. But in the case of energy emission from a system of sources, choosing the right solution is very difficult. Therefore, an analytical expression is required, which is a criterion for choosing the correct solution. We need a general criterion in analytical form that allows us 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 ,the following relation holds:

(3.93)

This formula is called radiation condition or Sommerfeld condition .

Let's consider an elementary electric emitter in the form of a dipole. An electric dipole is a short piece of wire l compared to wavelength  ( 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 not difficult to show that the change in the electric field in the space surrounding the wire is of a wave nature. For clarity, let us consider an extremely simplified model of the process of formation and change in the electrical component of the electromagnetic field that the wire emits. In Fig. Figure 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 is known, electric current is caused by the movement of electric charges, namely

or

In the future, we will consider only the change in the position of positive and negative charges on the wire. The electric field strength line begins at a positive charge and ends at a negative charge. In Fig. 3.11 the power line is shown with a dotted line. It is worth remembering that the electric field is created in the entire space surrounding the conductor, although in Fig. Figure 3.11 shows one power line.

In order for alternating current to flow through a conductor, a source of alternating emf 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. Moment t=1 corresponds to the beginning of the process, i.e. EMF = 0. At the moment t=2, an alternating EMF appears, which causes the movement of charges, as shown in Fig. 3.11. with the appearance of moving charges in the wire, an electric field arises in space. over time (t = 3÷5) the charges move to the ends of the conductor and the power line covers an increasingly larger part of the space. the line of force expands at the speed of light in a direction perpendicular to the wire. At time t = 6 – 8, the emf, having passed through the maximum value, decreases. Charges move towards the middle of the wire.

At time t = 9, the half-period of EMF changes ends and it decreases to zero. In this case, the charges merge and they compensate each other. There is no electric field in this case. The strength line of the radiated electric field closes and continues to move away from the wire.

Next comes the second half-cycle of the EMF change, the processes are repeated taking into account the change in polarity. In Fig. Figure 3.11 at moments t = 10÷13 shows a picture of the process taking into account the electric field strength line.

We examined the process of formation of closed lines of force of a vortex electric field. But it is worth remembering that the emission of electromagnetic waves is a single process. The electric and magnetic fields are inextricably 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 communications technology. It must be remembered that the plane of oscillation of the electric field strength vector is mutually perpendicular to the plane of oscillation of the magnetic field strength vector .

The emission of electromagnetic waves is due to a variable process. Therefore, in the formula for the charge we can put the constant C = 0. For the complex value of the charge can be written.

(3.94)

By analogy with electrostatics, we can introduce the concept of the moment of an electric dipole with alternating current

(3.95)

From formula (3.95) it follows that the vectors of the moment of the electric dipole and the directed piece of wire are co-directional.

It should be noted that real antennas have wire lengths 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 emitted vector fields generated by the individual dipoles.

Mechanical waves- the process of propagation of mechanical vibrations in a medium (liquid, solid, gaseous). It should be remembered that mechanical waves transfer energy and shape, but do not transfer mass. The most important characteristic of a wave is the speed of its propagation. Waves of any nature do not propagate through space instantly; their speed is finite.

According to geometry they distinguish: spherical (spatial), one-dimensional (plane), spiral waves.

The wave is called plane, 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 lines.

Plane wave equation::

Options :

Oscillation period T is the period of time after which the state of the system takes on the same values: u(t + T) = u(t).

Oscillation frequency n is the number of oscillations per second, the reciprocal of the period: n = 1/T. It is measured in hertz (Hz), and has the unit s–1. A pendulum swinging once per second oscillates at a frequency of 1 Hz.

Oscillation phase j– a value showing how much 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 “span” of oscillation.

4.Doppler effect- a change in the frequency and length of waves perceived by the observer (wave receiver) due to the relative movement of the wave source and the observer. Let's imagine that the observer approaches a stationary source of waves at a certain speed. At the same time, he 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 propagation of the wave are related to each other 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 decrease in oscillations occurs.

Jung'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 passing through slits S1 and S2 overlapped. In the region where the light beams overlapped, 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 different types of sounds:

1. simple tone - a purely harmonic vibration emitted by a tuning fork (a metal instrument that produces a sound when struck):

2. complex tone - not sinusoidal, but periodic oscillation (emitted by various musical instruments).

According to Fourier's 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 an acoustic spectrum. The spectrum of a complex tone is linear.

3. noise - sound that is obtained from the addition of many inconsistent sources. Spectrum - continuous (solid):

4. sonic boom - short-term sound impact. Example: clap, explosion.

Wave impedance- the ratio of sound pressure in a plane wave to the speed of vibration of particles of the medium. 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=p/c, P-sound pressure, p-density, c-speed of sound, V-volume.

3 - characteristics independent of the properties of the receiver:

Intensity (sound strength) is the energy carried by a sound wave per unit time through a unit area installed perpendicular to the sound wave.

Fundamental frequency.

Sound spectrum - 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 environment, the speed of propagation of ultrasound is the same . Peculiarity- narrowness of the beam, which allows you to influence objects locally. In inhomogeneous media with small inclusions of particles, the phenomenon of diffraction (bending around obstacles) occurs. The penetration of ultrasound into another medium is characterized by the penetration coefficient() =L /L where the lengths of the ultrasound after and before penetration into the medium.

The effect of ultrasound on body tissue is mechanical, thermal, and chemical. Application in medicine is divided into 2 areas: the method of research and diagnosis, and the method of action. 1) echoencephalography- detection of tumors and cerebral edema ; cardiography- measurement of the heart in dynamics. 2) Ultrasound physiotherapy- mechanical and thermal effects on tissue; during operations like “ultrasonic scalpel”

6. Ideal liquid - an imaginary incompressible fluid devoid of viscosity and thermal conductivity. An ideal fluid has no internal friction, is continuous and has no structure.

Continuity equation -V 1 A 1 = V 2 A 2 The volumetric flow rate in any stream tube limited by adjacent stream lines must be the same at any time in all its cross sections

Bernoulli's equation - R v 2 / 2 + Rst + Rgh= const, in the case of steady flow, the total pressure is the same in all cross sections of the current tube. R v 2 / 2 + Rst= const – for horizontal plots.

7Stationary flow- a flow whose speed at any location in the fluid never changes.

Laminar flow- an ordered flow of liquid or gas, in which the liquid (gas) moves in layers parallel to the direction of flow.

Turbulent flow- a form of liquid or gas flow in which their elements perform disordered, unsteady movements along complex trajectories, which leads to intense mixing between layers of moving liquid or gas.

Lines– lines whose tangents coincide at all points with the direction of velocity at these points. In a steady 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 part relative to another

Newton's equation: F = (dv/dx)Sη.

Viscosity coefficient- Proportionality coefficient depending on the type of liquid or gas. A number used to quantitatively characterize the viscosity property. Coefficient of internal friction.

Non-Newtonian fluid called a fluid in 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 = rdv/m, where r is density, m is the dynamic coefficient of viscosity of a liquid or gas, v is the flow velocity. At R< Rekр возможно лишь ламинарное течение жидкости, а при Re >Rekр flow may become turbulent.

Kinematic viscosity coefficient- the ratio of the dynamic viscosity of a liquid or gas to its density.

9. Stokes method,Based on the method A Stokes contains the formula for the resistance force arising 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 of uniform motion: the vector sum of all forces acting on the ball is zero.

Mg + F A + F with =0 (everything is in vector form!!!)

Now we should express the force of gravity (mg) and the Archimedes force (Fa) in terms of known quantities. Equating the values ​​mg = Fa+Fc we obtain the expression for viscosity:

η = (2/9)*g*(ρ t - ρ l)* r 2 / v = (2/9) * g *(ρ t - ρ l)* r 2 * t / L. The radius is directly measured with a micrometer ball r (by diameter), L is the path of the ball in the liquid, t is the travel time of path L. To measure viscosity using the Stokes method, path L is taken not from the surface of the liquid, but between marks 1 and 2. This is caused by the following circumstance. When deriving the working formula for the viscosity coefficient using 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 you gain speed, the resistance force increases, and the resultant of the three forces decreases! Only after a certain mark can the movement be considered uniform (and then only approximately).

11.Poiseuille's formula: During steady laminar movement of a viscous incompressible fluid through a cylindrical pipe of circular cross-section, the second volumetric flow rate is directly proportional to the pressure drop per unit length of the pipe and the fourth power of the radius and inversely proportional to the viscosity coefficient of the liquid.

Wave processes

Basic concepts and definitions

Let's consider some elastic medium - solid, liquid or gaseous. If vibrations of its particles are excited in any place of this medium, then due to the interaction between the particles, the vibrations will, transmitted from one particle of the medium to another, propagate through the medium at a certain speed. Process propagation of vibrations in space is called wave .

If particles in a medium oscillate in the direction of propagation of the wave, then it is called longitudinal If particle oscillations occur in a plane perpendicular to the direction of propagation of the wave, then the wave is called transverse . Transverse mechanical waves can only arise in a medium with a non-zero shear modulus. Therefore, they can spread in liquid and gaseous media only longitudinal waves . The difference between longitudinal and transverse waves is most clearly seen in the example of the propagation of vibrations in a spring - see figure.

To characterize transverse vibrations, it is necessary to set the position in space plane passing through the direction of oscillation and the direction of wave propagation - plane of polarization .

The region of space in which all particles of the medium vibrate is called wave field . The boundary between the wave field and the rest of the medium is called wave front . In other words, wave front - the geometric location of the points to which the oscillations have reached at a given point in time. In a homogeneous and isotropic medium, the direction of wave propagation is perpendicular to the wave front.

While a wave exists in the medium, the particles of the medium oscillate around their equilibrium positions. Let these oscillations be harmonic, and the period of these oscillations is T. Particles separated by a distance

along the direction of wave propagation, oscillate in the same way, i.e. at any given moment in time their displacements are the same. The distance is called wavelength . In other words, wavelength is the distance a wave travels in one period of oscillation .

The geometric location of points that oscillate in the same phase is called wave surface . A wave front is a special case of a wave surface. Wavelength – minimum the distance between two wave surfaces in which the points vibrate in the same way, or we can say that the phases of their oscillations differ by .

If the wave surfaces are planes, then the wave is called flat , and if by spheres, then spherical. A plane wave is excited in a continuous homogeneous and isotropic medium when an infinite plane oscillates. The excitation of a spherical surface can be represented as a result of radial pulsations of a spherical surface, and also as a result of the action point source, the dimensions of which can be neglected compared to the distance to the observation point. Since any real source has finite dimensions, at a sufficiently large distance from it the wave will be close to spherical. At the same time, the section of the wave surface of a spherical wave, as its size decreases, becomes arbitrarily close to the section of the wave surface of a plane wave.

Equations of plane and spherical waves

Wave equation is an expression that determines the displacement of an oscillating point as a function of the coordinates of the equilibrium position of the point and time:

If the source commits periodic oscillations, then function (22.2) must be a periodic function of both coordinates and time. Periodicity in time follows from the fact that the function describes periodic oscillations of a point with coordinates; periodicity in coordinates - from the fact that points located at a distance along the direction of wave propagation oscillate in the same way

Let us limit ourselves to considering harmonic waves, when points in the medium perform harmonic oscillations. It should be noted that any non-harmonic function can be represented as the result of the superposition of harmonic waves. Therefore, considering only harmonic waves does not lead to a fundamental deterioration in the generality of the results obtained.

Let's consider a plane wave. Let us choose a coordinate system so that the axis Oh coincided with the direction of wave propagation. Then the wave surfaces will be perpendicular to the axis Oh and, since all points of the wave surface vibrate equally, the displacement of points of the medium from equilibrium positions will depend only on x and t:

Let the vibrations of points lying in the plane have the form:

(22.4)

Oscillations in a plane located at a distance X from the origin, lag in time from the oscillations in the period of time required for the wave to cover the distance X, and are described by the equation

which is equation of a plane wave propagating in the direction of the Ox axis.

When deriving equation (22.5), we assumed the amplitude of oscillations to be the same at all points. In the case of a plane wave, this is true if the wave energy is not absorbed by the medium.

Let's consider some value of the phase in equation (22.5):

(22.6)

Equation (22.6) gives the relationship between time t and place - X, in which the specified phase value is currently being implemented. Having determined from equation (22.6), we find the speed with which a given phase value moves. Differentiating (22.6), we obtain:

Where follows (22.7)

The wave equation is an expression that gives the displacement of an oscillating particle as a function of its coordinates x, y, z and time t:

(meaning the coordinates of the equilibrium position of the particle). This function must be periodic both with respect to time t and with respect to the coordinates x, y, z. Periodicity in time follows from the fact that it describes the oscillations of a particle with coordinates x, y, z. Periodicity in coordinates follows from the fact that points separated from each other by a distance K vibrate 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 axis coincides with the direction of wave propagation. Then the wave surfaces will be perpendicular to the axis and, since all points of the wave surface vibrate equally, the displacement will depend only on Let the oscillations of points lying in the plane (Fig. 94.1) have the form

Let us find the type of oscillation of points in the plane corresponding to an arbitrary value of x. In order to travel from the plane x = 0 to this plane, the wave requires time - the speed of propagation of the wave).

Consequently, the oscillations of particles lying in the x-plane will lag in time from the oscillations of particles in the plane, i.e., they will have the form

So, the equation of a plane wave (both longitudinal and transverse) propagating in the x-axis direction is as follows:

The quantity a represents the amplitude of the wave. The initial phase of the wave a is determined by the choice of origins. When considering a single wave, the origins of time and coordinates are usually chosen so that a is equal to zero. When considering several waves together, it is usually not possible to ensure that the initial phases for all of them are equal to zero.

Let us fix any value of the phase in equation (94.2) by putting

(94.3)

This expression defines the relationship between time t and the place x at which the phase has a fixed value. The resulting value gives the speed at which a given phase value moves. Differentiating expression (94.3), we obtain

Thus, the speed of wave propagation v in equation (94.2) is the speed of phase movement, and therefore it is called the phase speed.

According to (94.4). Consequently, equation (94.2) describes a wave propagating in the direction of increasing x. A wave propagating in the opposite direction is described by the equation

Indeed, by equating the phase of the wave (94.5) to a constant and differentiating the resulting equality, we arrive at the relation

from which it follows that wave (94.5) propagates in the direction of decreasing x.

The plane wave equation can be given a form that is symmetrical with respect to x and t. To do this, we introduce the quantity

which is called the wave number. Having reduced the numerator and denominator of expression (94.6) to frequency v, we can represent the wave number in the form

(see formula (93.2)). Opening the parentheses in (94.2) and taking into account (94.7), we arrive at the following equation for a plane wave propagating along the x axis:

The equation of a wave propagating in the direction of decreasing x differs from (94.8) only in the sign of the term

When deriving formula (94.8), we assumed that the amplitude of oscillations does not depend on x. For a plane wave, this is observed in the case when the wave energy is not absorbed by the medium. When propagating in an energy-absorbing medium, the intensity of the wave gradually decreases with distance from the source of oscillations - wave attenuation is observed. Experience shows that in a homogeneous medium such attenuation occurs according to an exponential law: with a decrease in time of the amplitude of the damped oscillations; see formula (58.7) of the 1st volume). Accordingly, the plane wave equation has the following form:

Amplitude at points of the plane

Now let's find the equation of a spherical wave. Every real source of waves has some extent. However, if we limit ourselves to considering waves at distances from the source that significantly exceed its dimensions, then the source can be considered a point source. 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’s oscillations is equal. Then the points lying on the wave surface of radius will oscillate with the phase