A wave is the process of propagation of an oscillation (or some other signal) in space.

Imagine, for example, that at all points of the plane YOZ some physical parameter changes in time according to the harmonic law

Let the oscillations of this abstract parameter propagate along the axis OX with speed v(Fig. 13.1.). Then in the plane with coordinate x the original oscillations will be repeated again, but with a delay of seconds:

Rice. 13.1.

Function (13.1) is called the plane wave equation. This important function is often written in this form

Here: E 0 and w - amplitude and frequency of oscillations in the wave,

(w t – kx+ - wave phase,

a - initial phase,

wave number,

v- wave propagation speed.

The set of all points in space at which oscillations occur in the same phase determines phase surface. In our example, this is a plane.

(w t – kx+ = F = const - the equation of motion of the phase surface in the process of wave propagation. Let's take the derivative of this equation with respect to time:

w - k= 0.

Here = vφ - velocity of the phase surface - phase velocity.

= v f = .

Thus, the phase velocity is equal to the wave propagation velocity.

The phase surface separating the space covered by the wave process from the part where the wave has not yet reached is called the wave front. The wave front, as one of the phase surfaces, also moves with the phase velocity. This speed, for example, of an acoustic wave in air is 330 m/s, and of a light (electromagnetic) wave in vacuum - 3×10 8 m/s.

wave equation E = E 0 ×cos(w t – kx+ j) is a solution differential wave equation. To find this differential equation, we differentiate the wave equation (13.2) twice in time, and then twice in coordinate:

,

Comparing these two expressions, we find that

.

But the wave number k= , so

. (13.3)

This is the differential equation of the wave process - wave equation.

Once again, we note that wave equation(13.2) there is a solution wave equation (13.3).

The wave equation can, of course, be written as

It is now obvious that in the wave equation the coefficient of the second derivative with respect to the coordinate is equal to the square of the phase velocity of the wave.

If, solving the problem of motion, we obtain a differential equation of the type

this means that the movement under study is own damped oscillations…

If, when solving a regular problem, a differential equation arose

then this means that the study wave process, and the propagation speed of this wave.

For most of the problems associated with waves, it is important to know the state of oscillations of various points of the medium at one time or another. The states of the points of the medium will be determined if the amplitudes and phases of their oscillations are known. For transverse waves, it is also necessary to know the nature of the polarization. For a plane linearly polarized wave, it is sufficient to have an expression that allows one to determine the displacement c(x, t) from the equilibrium position of any point of the medium with the coordinate X, at any point in time t. Such an expression is called wave equation.

Rice. 2.21.

Consider the so-called running wave, those. a wave with a flat wavefront propagating in any one specific direction (for example, along the x-axis). Let the particles of the medium directly adjacent to the source of plane waves oscillate according to the harmonic law; %(0, /) = = rsobcoG (Fig. 2.21). In figure 2.21, a through ^(0, t) the displacement of the particles of the medium lying in the plane perpendicular to the figure and having the coordinate in the chosen coordinate system is indicated X= 0 at time t. The origin of the time reference is chosen so that the initial phase of the oscillations, defined through the cosine function, is equal to zero. Axis X compatible with the beam, i.e. with the direction of vibration propagation. In this case, the wave front is perpendicular to the axis X, so that particles lying in this plane will oscillate in the same phase. The wave front itself in this medium moves along the axis X with speed and wave propagation in a given medium.

Let's find the expression? (x, t) displacement of particles of the medium, remote from the source at a distance x. This is the distance the wave front travels

over time Therefore, oscillations of particles lying in a plane remote from the source at a distance X, will lag behind in time by the value m from the oscillations of particles directly adjacent to the source. These particles (with coordinate x) will also make harmonic vibrations. In the absence of damping, the amplitude BUT oscillations (in the case of a plane wave) will not depend on the x coordinate, i.e.

This is the required equation longing running wave(not to be confused with the wave equation discussed below!). The equation, as already noted, allows us to determine the displacement % particles of the medium with coordinate x at the moment of time t. The oscillation phase depends

on two variables: on the x-coordinate of the particle and time t. At a given fixed moment in time, the phases of oscillations of various particles will, generally speaking, be different, but it is possible to single out such particles whose oscillations will occur in the same phase (in-phase). It can also be assumed that the difference between the phases of oscillations of these particles is equal to 2pt(where t = 1, 2, 3,...). shortest distance between two traveling wave particles oscillating in the same phase is called wavelength x.

Let's find the connection of the wavelength X with other quantities characterizing the propagation of oscillations in the medium. In accordance with the introduced definition of wavelength, we can write

or after abbreviations Since , then

This expression allows us to give a different definition of the wavelength: the wavelength is the distance over which the oscillations of the particles of the medium have time to propagate in a time equal to the period of the oscillations.

The wave equation reveals a double periodicity: in coordinate and in time: ^(x, t) = Z,(x + nk, t) = l,(x, t + mT) = ​​Tx + pX, ml), where pit - any whole numbers. One can, for example, fix the coordinates of the particles (put x = const) and consider their offset as a function of time. Or, conversely, to fix a moment in time (take t = const) and consider particle displacement as a function of coordinates (the instantaneous state of displacements is an instantaneous photograph of a wave). So, being on the pier you can use the camera at the time t photograph the sea surface, but you can throw a chip into the sea (i.e., fixing the coordinate X), keep track of its fluctuations over time. Both of these cases are shown in the form of graphs in Fig. 2.21, a-c.

The wave equation (2.125) can be rewritten differently

The ratio is denoted to and called wave number

Because , then

The wave number thus shows how many wavelengths fit into a segment of 2n units of length. By introducing the wavenumber into the wave equation, we obtain the equation for a wave traveling in the positive direction Oh waves in the most commonly used form

Let us find an expression relating the phase difference Dp of oscillations of two particles belonging to different wave surfaces X and x 2. Using the wave equation (2.131), we write:

If we denote or according to (2.130)

A plane traveling wave propagating in an arbitrary direction is described in general case equation

where G-radius vector drawn from the origin to the particle lying on the wave surface; to - a wave vector equal in absolute value to the wave number (2.130) and coinciding in direction with the normal to the wave surface in the direction of wave propagation.

Also possible complex form writing the equation of the wave. So, for example, in the case of a plane wave propagating along the axis X

and in the general case of a plane wave of an arbitrary direction

The wave equation in any of the listed forms of writing can be obtained as a solution to a differential equation called wave equation. If we know the solution of this equation in the form (2.128) or (2.135) - the traveling wave equation, then it is not difficult to find the wave equation itself. Differentiate 4(x, t) = % from (2.135) twice in coordinate and twice in time and obtain

expressing ?, through the obtained derivatives and comparing the results, we get

Keeping relation (2.129) in mind, we write

This is the wave equation for the one-dimensional case.

AT general view for?, = c(x, y, z/) the wave equation in Cartesian coordinates looks like that

or in a more compact form:

where D is the Laplace differential operator

phase speed called the speed of propagation of the points of the wave, oscillating in the same phase. In other words, this is the speed of movement of the "crest", "trough", or any other point of the wave, the phase of which is fixed. As noted earlier, the wave front (and, consequently, any wave surface) moves along the axis Oh with speed and. Consequently, the speed of propagation of vibrations in the medium coincides with the speed of movement of a given phase of vibrations. Therefore, the speed and, defined by relation (2.129), i.e.

called phase speed.

The same result can be obtained by finding the speed of the points of the medium that satisfy the condition of the constancy of the phase co/ - fee = const. From here, the dependence of the coordinate on time (co / - const) and the speed of movement of this phase are found

which coincides with (2.142).

Plane traveling wave propagating in the negative direction of the axis Oh, is described by the equation

Indeed, in this case the phase velocity is negative

The phase velocity in a given medium may depend on the oscillation frequency of the source. The dependence of the phase velocity on frequency is called dispersion, and the environments in which this dependence takes place are called dispersing media. It should not be thought, however, that the expression (2.142) is the indicated dependence. The point is that in the absence of dispersion, the wavenumber to in direct ratio

with and therefore . Dispersion occurs only when w depends on to non-linear).

A traveling plane wave is called monochromatic (having one frequency), if the oscillations in the source are harmonic. Monochromatic waves correspond to an equation of the form (2.131).

For a monochromatic wave, the angular frequency ω and the amplitude BUT do not depend on time. This means that a monochromatic wave is infinite in space and infinite in time, i.e. is an idealized model. Any real wave, no matter how carefully the constancy of frequency and amplitude is maintained, is not monochromatic. A real wave does not last indefinitely, but begins and ends at certain times in a certain place, and, therefore, the amplitude of such a wave is a function of time and the coordinates of this place. However, the longer the time interval during which the amplitude and frequency of oscillations are maintained constant, the closer this wave is to monochromatic. Often in practice, a sufficiently large segment of the wave is called a monochromatic wave, within which the frequency and amplitude do not change, just as a segment of a sinusoid is shown in the figure, and it is called a sinusoid.

As a manuscript

Physics

Lecture notes

(Part 5. Waves, wave optics)

For students of direction 230400

« Information Systems and technologies"

Electronic educational resource

Compiled by: Candidate of Physical and Mathematical Sciences, Associate Professor V.V. Konovalenko

Minutes No. 1 dated 04.09.2013

Wave processes

Basic concepts and definitions

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 particles, the vibrations will, being transmitted from one particle of the medium to another, propagate in the medium with a certain speed. Process propagation of vibrations in space is called wave .

If the particles in the medium oscillate in the direction of wave propagation, then it is called longitudinal. If the oscillations of particles occur in a plane perpendicular to the direction of wave propagation, then the wave is called transverse . transverse mechanical waves can arise only in a medium with a nonzero shear modulus. Therefore, 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 oscillations in a spring - see figure.

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

The region of space in which all particles of the medium oscillate 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 locus of points to which oscillations have reached a given point in time. In a homogeneous and isotropic medium, the direction of wave propagation perpendicular to the front of the wave.

As long as there is a wave 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 equal to T. Particles separated from each other 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 traveled by a wave in one period of oscillation .

The locus of points that oscillate in one phase is called wave surface . Wave front - special case wave surface. Wavelength – minimum the distance between two wave surfaces in which the points oscillate 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 during oscillations of an infinite plane. 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, whose dimensions compared with the distance to the observation point can be neglected. 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.

The equation of a plane wave propagating

In any direction

We will receive. Let oscillations in a plane parallel to the wave surfaces and passing through the origin of coordinates have the form:

In a plane separated from the origin by a distance l, the oscillations will lag behind in time by . Therefore, the equation of oscillations in this plane has the form:

From analytical geometry It is known that the distance from the origin of coordinates to a certain plane is equal to the scalar product of the radius vector of a certain point of the plane and the unit vector of the normal to the plane: . The figure illustrates this situation for the two-dimensional case. Substitute the value l into equation (22.13):

(22.14)

The vector equal in absolute value to the wave number and directed along the normal to the wave surface is called wave vector . The plane wave equation can now be written as:

Function (22.15) gives the deviation from the equilibrium position of a point with a radius vector at the moment of time t. In order to represent the dependence on coordinates and time in an explicit form, it is necessary to take into account that

. (22.16)

Now the plane wave equation takes the form:

Often useful represent the wave equation in exponential form . To do this, we use the Euler formula:

where , we write the equation (22.15) in the form:

. (22.19)

wave equation

The equation of any wave is a solution to a second-order differential equation called wave . In order to establish the form of this equation, we find the second derivatives with respect to each of the arguments of the plane wave equation (22.17):

, (22.20)

, (22.21)

, (22.22)

We add the first three equations with derivatives with respect to coordinates:

. (22.24)

We express from equation (22.23): , and take into account that:

(22.25)

We represent the sum of the second derivatives on the left side of (22.25) as the result of the action of the Laplace operator on , and in the final form we represent wave equation as:

(22.26)

It is noteworthy that in the wave equation Square root from the reciprocal coefficient of the time derivative gives the wave propagation velocity.

It can be shown that the wave equation (22.26) is satisfied by any function of the form:

and each of them is wave equation and describes some wave.

Elastic wave energy

Consider in a medium in which an elastic wave (22.10) propagates, an elementary volume is small enough so that the deformation and the speed of particles in it can be considered constant and equal:

Due to the wave propagation in the medium, the volume has an elastic deformation energy

(22.38)

In accordance with (22.35), Young's modulus can be represented as . That's why:

. (22.39)

The volume under consideration also has kinetic energy:

. (22.40)

Total volume energy:

And the energy density:

, a (22.43)

Substitute these expressions in (22.42) and take into account that:

In this way, the energy density is different at different points in space and varies with time according to the square sine law.

The average value of the square of the sine is 1/2, which means average in time, the value of the energy density at each point of the medium , in which the wave propagates:

. (22.45)

Expression (22.45) is valid for all types of waves.

So, the medium in which the wave propagates has an additional supply of energy. Consequently, wave carries energy .

X.6 Dipole radiation

Oscillating electric dipole, i.e. a dipole whose electric moment changes periodically, for example, according to a harmonic law, is the simplest system that emits electromagnetic waves. One of important examples An oscillating dipole is a system consisting of a negative charge that oscillates near a positive charge. It is this situation that is realized when an electromagnetic wave acts on an atom of a substance, when, under the action of the wave field, the electrons oscillate in the vicinity of the atomic nucleus.

Let us assume that the dipole moment changes according to the harmonic law:

where is the radius vector of the negative charge, l- oscillation amplitude, - unit vector directed along the dipole axis.

We confine ourselves to considering elementary dipole , whose dimensions are small compared to the emitted wavelength and consider wave zone dipole, i.e. a region of space for which the modulus of the radius vector of the point is . In the wave zone of a homogeneous and isotropic medium, the wave front will be spherical - Figure 22.4.

An electrodynamic calculation shows that the wave vector lies in a plane passing through the dipole axis and the radius vector of the considered point. Amplitudes and depend on distance r and the angle between and the dipole axis. in a vacuum

Since the Poynting vector , then

, (22.33)

and it can be argued that the dipole radiates most of all in the directions corresponding to , and radiation pattern dipole has the form shown in figure 22.5. radiation pattern called graphic image distribution of the radiation intensity in different directions in the form of a curve constructed so that the length of the beam segment drawn from the dipole in a certain direction to the point of the curve is proportional to the radiation intensity.

The calculations also show that power R dipole radiation is proportional to the square of the second time derivative of the dipole moment :

Because the

, (22.35)

then average power

turns out proportional to the square of the amplitude of the dipole moment and fourth power frequency.

On the other hand, considering that and , we get that radiation power is proportional to the square of the acceleration:

This statement is true not only for charge oscillations, but also for an arbitrary charge movement.

wave optics

In this section, we will consider such light phenomena in which the wave nature of light is manifested. Recall that light is characterized by corpuscular-wave dualism and there are phenomena that can be explained only on the basis of the concept of light as a stream of particles. But we will consider these phenomena in quantum optics.

General information about light

So we think of light as an electromagnetic wave. AT electromagnetic wave fluctuates and. It has been experimentally established that the physiological, photochemical, photoelectric and other effects of light are determined by the vector of the light wave, therefore it is called light. Accordingly, we will assume that the light wave is described by the equation:

where is the amplitude,

- wave number (wave vector),

Distance along the propagation direction.

The plane in which it oscillates is called vibration plane. A light wave travels at a speed

, (2)

called refractive index and characterizes the difference between the speed of light in a given medium and the speed of light in vacuum (emptiness).

In most cases, transparent substances have magnetic permeability, and it can almost always be assumed that the refractive index is determined by the dielectric constant of the medium:

Meaning n used to characterize optical density of the medium: the larger n, the more optically dense the medium is called .

Visible light has wavelengths in vacuum in the range and frequency

Hz

Real light receivers are not able to keep track of such fleeting processes and register time-averaged energy flux . By definition , light intensity is called the modulus of the time-averaged value of the energy flux density carried by the light wave :

(4)

Since in an electromagnetic wave

, (6)

Ι ~ ~ ~ (7)

I~A2(8)

Rays we will call the lines along which light energy propagates.

The vector of the average energy flux is always directed tangentially to the beam. In isotropic media coincides in direction with the normal to the wave surfaces.

In natural light, there are waves with very different orientations of the oscillation plane. Therefore, despite the transverseness of light waves, the radiation of ordinary light sources does not reveal asymmetry with respect to the direction of propagation. This feature of light (natural) is explained by the following: the resulting light wave of the source is composed of waves emitted by various atoms. Each atom emits a wave within seconds. During this time, space is formed wave train (sequence of "humps and valleys") about 3 meters long.

The plane of oscillation of each train is quite definite. But at the same time, a huge number of atoms radiate their trains, and the plane of oscillations of each train is oriented independently of the others, in a random way. That's why in the resulting wave from the body oscillations of different directions are represented with equal probability. It means that, if some device is used to investigate the intensity of light with different orientations of the vector , then in natural light the intensity does not depend on the orientation .

Measuring the intensity is a long process compared to the wave period, and the considered ideas about the nature of natural light are convenient for describing sufficiently long processes.

However, at a given point in time, at a specific point in space, as a result of the addition of the vectors of individual trains, some specific one is formed. Due to random "on" and "off" individual atoms a light wave excites at a given point an oscillation close to harmonic, but the amplitude, frequency, and phase of the oscillations depend on time and change randomly. The orientation of the vibrational plane also changes randomly. uy. Thus, oscillations of the light vector at a given point in the medium can be described by the equation:

(9)

Moreover, and there are chaotically time-varying functions ii. Such a conception of natural light is convenient if time intervals comparable to the period of a light wave are considered.

Light in which the directions of the vector oscillations are ordered in some way is called polarized.

If the oscillations of the light vector occur only in one plane passing through the beam, then the light is called flat - or linearly polarized. In other words, in plane polarized light, the oscillation plane has a strictly fixed position. Other types of ordering are also possible, that is, types of light polarization.

Huygens principle

In the geometrical optics approximation, light should not penetrate into the region of the geometrical shadow. In reality, light penetrates into this area, and this phenomenon becomes the more significant, the smaller the size of the obstacles. If the dimensions of the holes or slots are comparable to a long wavelength, then geometric optics is not applicable.

Qualitatively, the behavior of light behind the barrier is explained by the Huygens principle, which makes it possible to construct the wave front at the moment from the known position at the moment .

According to Huygens' principle, each point reached by the wave motion becomes a point source of secondary waves. The envelope along the fronts of the secondary waves gives the position of the wave front.

Light interference

Let at some point of the medium two waves (plane polarized) excite two oscillations same frequency and same direction:

and . (24.14)

The amplitude of the resulting oscillation is determined by the expression:

For incoherent waves, it changes randomly and all values ​​are equally probable. Therefore, from (24.15) it follows:

6 If the waves are coherent and , then

But depends on , are the lengths of the paths from the wave sources to the given point and different for different points of the environment. Consequently, when coherent waves are superimposed, the light flux is redistributed in space, as a result of which the light intensity increases at some points of the medium, and decreases at others -. This phenomenon is called interference.

The absence of interference in everyday life when using several light sources is explained by their incoherence. Individual atoms emit pulses for c and the length of the train is ≈ 3 meters. For the new train, not only is the orientation of the polarization plane random, but the phase is also unpredictable.

Really coherent waves are obtained by dividing the radiation of one source into two parts. When overlapping parts, interference can be observed. But at the same time, the separation of optical lengths should not be of the order of the train length. Otherwise, there will be no interference, because various trains are superimposed.

Let the separation occur at the point O, and the overlap at the point P. Oscillations are excited in P.

and (24.17)

Wave propagation velocities in the corresponding media.

Spread phases at a point R:

where is the wavelength of light in vacuum.

Value , i.e. equal to the difference in optical path lengths between the considered points is called optical path difference.

then , in (24.16) equal to one, and the light intensity in will be maximum.

(24.20)

then , oscillations at a point occur in antiphase, which means that the light intensity is minimal.

COHERENCE

Coherence - coordinated flow of two or more wave processes. Absolute consistency never happens, so we can talk about varying degrees of coherence.

Distinguish between temporal and spatial coherence.

Temporal coherence

Real wave equation

We have considered the interference of waves described by equations of the form:

(1)

However, such waves are a mathematical abstraction, since the wave described by (1) must be infinite in time and space. Only then can the quantities be definite constants.

A real wave, formed as a result of the superposition of trains from different atoms, contains components whose frequencies lie in a finite frequency range (respectively, wave vectors in ), and A and a experience continuous chaotic changes. Oscillations excited at some point by superimposed real waves, can be described by the expression:

and (2)

Moreover, the chaotic changes of functions with time in (2) are independent.

For ease of analysis, we assume that the wave amplitudes are constant and identical (experimentally, this condition is implemented quite simply):

Frequency and phase changes can be reduced to only frequency or only phase changes. Indeed, let us assume that the inharmonicity of functions (2) is due to phase jumps. But, according to what is proved in mathematics Fourier theorem, any non-harmonic function can be represented as a sum of harmonic components whose frequencies are enclosed in some . In the limiting case, the sum goes into an integral: any finite and integrable function can be represented by the Fourier integral:

, (3)

where is the amplitude of the harmonic frequency component, analytically determined by the relation:

(4)

So, a non-harmonic function due to a phase change can be represented as a superposition of harmonic components with frequencies in some .

On the other hand, a function with variable frequency and phase can be reduced to a function with only variable phase:

Therefore, to tame further analysis, we assume:

i.e. we implement phase approach to the concept of "Temporal coherence".

Stripes of equal slope

Let a thin plane-parallel plate be illuminated by diffuse monochromatic light. Let us place a converging lens parallel to the plate, in its focal plane - the screen. Scattered light contains rays of various directions. Rays incident at an angle give 2 reflected rays, which will converge at a point. This is true for all rays incident on the surface of the plate at a given angle, at all points on the plate. The lens ensures that all such rays converge to one point, since parallel rays incident on the lens at a certain angle are collected by it at one point of the focal plane, i.e. on the screen. At point O, the optical axis of the lens intersects the screen. At this point, rays are collected that run parallel to the optical axis.

Rays incident at an angle, but not in the plane of the figure, but in other planes, will gather at points located at the same distance from the point as the point. As a result of the interference of these rays, at a certain distance from the point, a circle is formed with a certain intensity of the incident light. Rays incident at a different angle form a circle on the screen with a different illumination, which depends on their optical path difference. As a result, alternating dark and light bands in the form of circles are formed on the screen. Each of the circles is formed by rays falling at a certain angle, and they are called stripes of equal slope. These bands are localized at infinity.

The role of the lens can be played by the lens, and the screen - the retina. In this case, the eye must be accommodated to infinity. In white light, multi-colored stripes are obtained.

Stripes of equal thickness

Let's take a plate in the form of a wedge. Let her fall parallel beam of light. Consider the rays reflected from the upper and lower faces of the plate. If these rays are brought together by a lens at a point, then they will interfere. With a small angle between the faces of the plate, the difference in the path of the rays can be calculated from the form
le for a plane-parallel plate. The rays formed from the fall of the beam to some other point of the plate will be collected by the lens at the point . The difference in their course will be determined by the thickness of the plate in the corresponding place. It can be proved that all points of type P lie in the same plane passing through the vertex of the wedge.

If the screen is positioned so that it is conjugated with the surface in which the points P, P 1 P 2 lie, then a system of light and dark stripes will appear on it, each of which is formed due to reflections from the plate in places of a certain thickness. Therefore, in this case, the stripes are called stripes of equal thickness.

When viewed in white light, the bands will be colored. Stripes of equal thickness are localized near the surface of the plate. Under normal incidence of light - on the surface.

In real conditions, when observing the coloring of soap and oil films, mixed-type bands are observed.

Diffraction of light.

27.1. Diffraction of light

Diffractioncalled a set of phenomena observed in a medium with sharp optical inhomogeneities and associated with deviations in the propagation of light from the laws of geometric optics .

To observe diffraction, an opaque barrier is placed in the path of a light wave from a certain source, covering part of the wave surface of the wave emitted by the source. emerging diffraction pattern observed on a screen located on the continuation of the beams.

There are two types of diffraction. If the rays coming from the source and from the obstacle to the point of observation can be considered almost parallel, then they say that there isFraunhofer diffraction, or diffraction in parallel beams. If the Fraunhofer diffraction conditions are not met,talking about Fresnel diffraction.

It must be clearly understood that there is no fundamental physical difference between interference and diffraction. Both phenomena are due to the redistribution of the energy of superimposed coherent light waves. Usually when considering a finite number discrete sources light, they talk about interference . If we consider the superposition of waves from coherent sources continuously distributed in space , then they talk about diffraction .

27.2. Huygens–Fresnel principle

Huygens' principle makes it possible in principle to explain the penetration of light into the region of a geometric shadow, but says nothing about the intensity of waves propagating in different directions. Fresnel supplemented the Huygens principle with an indication of how to calculate the intensity of radiation from an element of the wave surface in various directions, as well as an indication that the secondary waves are coherent, and when calculating the light intensity at a certain point, it is necessary to take into account the interference of secondary waves. .

Wave processes

Basic concepts and definitions

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 particles, the vibrations will, being transmitted from one particle of the medium to another, propagate in the medium with a certain speed. Process propagation of vibrations in space is called wave .

If the particles in the medium oscillate in the direction of wave propagation, then it is called longitudinal. If the oscillations of particles occur in a plane perpendicular to the direction of wave propagation, then the wave is called transverse . Transverse mechanical waves can only arise in a medium with a nonzero shear modulus. Therefore, 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 oscillations in a spring - see figure.

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

The region of space in which all particles of the medium oscillate 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 locus of points to which oscillations have reached a given point in time. In a homogeneous and isotropic medium, the direction of wave propagation perpendicular to the front of the wave.

As long as there is a wave 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 equal to T. Particles separated from each other 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 traveled by a wave in one period of oscillation .

The locus of points that oscillate in one phase is called wave surface . The wave front is a special case of the wave surface. Wavelength – minimum the distance between two wave surfaces in which the points oscillate 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 during oscillations of an infinite plane. 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, whose dimensions compared with the distance to the observation point can be neglected. 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.

Plane and spherical wave equations

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 does periodical fluctuations, then the 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 fluctuate in the same way

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

Consider a plane wave. We choose a coordinate system so that the axis Oh coincides 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 oscillate in the same way, the displacement of the points of the medium from the equilibrium positions will only depend on x and t:

Let oscillations of points lying in a plane have the form:

(22.4)

Oscillations in a plane at a distance X from the origin of coordinates, lag behind the oscillations in time by the time interval necessary for the wave to overcome 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 energy of the wave is not absorbed by the medium.

Consider some value of the phase in equation (22.5):

(22.6)

Equation (22.6) gives the relationship between time t and place - X, wherein specified value phase is currently underway. Determining from equation (22.6) , we find the speed with which the given phase value moves. Differentiating (22.6), we get:

Whence follows (22.7)

wave equation is an equation expressing the dependence of the displacement of an oscillating particle participating in the 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 that are at a distance l from each other, fluctuate in the same way.

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

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 oscillating motion particles of the medium depend only on time and coordinates . The offset will depend only on and : . Let the oscillation of the point with the coordinate (the source of oscillations) be given by the function . A task: find the type of fluctuation of points in the plane corresponding to an arbitrary value of . It takes time for a wave to travel from a plane to that plane. Consequently, oscillations of particles lying in the plane will lag behind in phase by a time from oscillations of particles in the plane . Then the equation of oscillations of particles in a plane will look like:

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

. (3)

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

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

Let us reduce this expression by and differentiate. As a result, we get:

or .

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

For a sine wave, the energy transfer rate is equal to the phase velocity. 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 velocity 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, mass cannot move faster than the speed of light c .

Usually, the plane wave equation is given a form that is symmetric with respect to and. To do this, enter the value , which is called wave number . Let's transform the expression for the wave number. We write it in the form (). Substitute this expression into the plane wave equation:

Finally we get

This is the equation of a plane wave propagating in the direction of increasing . 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 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 - initial phase waves.

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

Everything considered above referred 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 is the attenuation coefficient of the wave. A0 is the oscillation amplitude at the point with coordinates . This is the reciprocal of the distance at which the wave amplitude decreases in e once.

Let's find the equation of a spherical wave. We will consider the source of oscillations to be a point source. This is possible if we confine ourselves to considering the wave at a distance much greater than the dimensions 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 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 with distance from the source according to the law . Therefore, the spherical wave equation has the form:

or

By virtue of the assumptions made, the equation is valid only for , significantly exceeding the dimensions of the wave source. Equation (6) is not applicable for small values ​​of , because the amplitude would tend to infinity, which is absurd.

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

group speed

A strictly monochromatic wave is an endless sequence of "humps" and "troughs" in time and space.

The phase velocity of this wave, or (2)

With the help of such a wave it is impossible to transmit a signal, because. at any point of the wave, all "humps" are the same. The signal must be different. Be a sign (label) on the wave. But then the wave will no longer be harmonic, and will not be described by equation (1). The signal (impulse) can be represented according to the Fourier theorem as a superposition of harmonic waves with frequencies contained in a certain interval Dw . A superposition of waves that differ little from each other in frequency

called wave packet or wave group .

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, there is an increase in amplitude, and where the phases are opposite, there is a damping of the amplitude (the result of interference). Such a picture is shown in the figure. In order for the superposition of waves to be considered as a group of waves, it is necessary to fulfill next condition 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 Wave Optics section. In the absence of dispersion, the velocity of the wave packet travel coincides with the phase velocity v . In a dispersive medium, each wave disperses at its own speed. Therefore, the wave packet spreads over time, its width increases.

If the dispersion is small, then the spreading of the wave packet does not occur too quickly. Therefore, the movement of the entire packet can be assigned a certain speed U .

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

In a dispersive medium v¹ U . Along with the movement of the wave packet itself, there is a movement of "humps" inside the packet itself. "Humps" move in space at a speed v , and the package as a whole with the speed U .

Let us consider in more detail the motion 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 us write down the equations of two waves. Let us take for simplicity the initial phases j0 = 0.

Here

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

We add the fluctuations and carry out transformations using the trigonometric formula for the sum of cosines:

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

(4)

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

The cosine takes the maximum value modulo through p .

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

Resolving (5) with respect to xmax get.

Since the phase velocity called the group speed. The maximum amplitude of the wave packet moves with 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 from it that the wave packet spreads over time.
The expression for the group velocity can be given a different form.

In the absence of dispersion

The intensity maximum falls on the center of the wave group. Therefore, the energy transfer rate is equal to the group velocity.

The concept of group velocity is applicable only under the condition that the wave absorption in the medium is small. With a significant attenuation of the waves, 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.