Spherical and plane waves. Propagation of a plane wave What is a plane wave

PLATE WAVE

PLATE WAVE

A wave whose direction of propagation is the same at all points in space. The simplest example- homogeneous monochromatic undamped P.v.:

u(z, t)=Aeiwt±ikz, (1)

where A is the amplitude, j= wt±kz - , w=2p/T - circular frequency, T - oscillation period, k - . Constant phase surfaces (phase fronts) j=const P.v. are planes.

In the absence of dispersion, when vph and vgr are identical and constant (vgr = vph = v), there are stationary (i.e., moving as a whole) running P. v., which allow general idea type:

u(z, t)=f(z±vt), (2)

where f is an arbitrary function. In nonlinear media with dispersion, stationary running PVs are also possible. type (2), but their shape is no longer arbitrary, but depends both on the parameters of the system and on the nature of the movement. In absorbing (dissipative) media P. v. decrease their amplitude as they spread; with linear damping, this can be taken into account by replacing k in (1) with the complex wave number kd ± ikм, where km is the coefficient. attenuation of P. v.

A homogeneous PV that occupies the entire infinite is an idealization, but any wave concentrated in a finite region (for example, directed by transmission lines or waveguides) can be represented as a superposition of PV. with one space or another. spectrum k. In this case, the wave may still have a flat phase front, but non-uniform amplitude. Such P. v. called plane inhomogeneous waves. Some areas are spherical. and cylindrical waves that are small compared to the radius of curvature of the phase front behave approximately like a phase wave.

Physical encyclopedic Dictionary. - M.: Soviet Encyclopedia. . 1983 .

PLATE WAVE

- wave, the direction of propagation is the same at all points in space.

Where A - amplitude, - phase, - circular frequency, T - period of oscillation k- wave number. = const P.v. are planes.
In the absence of dispersion, when the phase velocity v f and group v gr are identical and constant ( v gr = v f = v) there are stationary (i.e., moving as a whole) running P. c., which can be represented in general view

Where f- arbitrary function. In nonlinear media with dispersion, stationary running PVs are also possible. type (2), but their shape is no longer arbitrary, but depends both on the parameters of the system and on the nature of the wave motion. In absorbing (dissipative) media, P. k on the complex wave number k d ik m, where k m - coefficient attenuation of P. v. A homogeneous wave field that occupies the entire infinity is an idealization, but any wave field concentrated in a finite region (for example, directed transmission lines or waveguides), can be represented as a superposition P. V. with one or another spatial spectrum k. In this case, the wave may still have a flat phase front, with a non-uniform amplitude distribution. Such P. v. called plane inhomogeneous waves. Dept. areasspherical or cylindrical waves that are small compared to the radius of curvature of the phase front behave approximately like PT.

Lit. see under Art. Waves.

M. A. Miller, L. A. Ostrovsky.

Physical encyclopedia. In 5 volumes. - M.: Soviet Encyclopedia. Editor-in-chief A. M. Prokhorov. 1988 .

For most problems involving waves, it is important to know the state of the oscillations various points environment at one time or another. The states of points in 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 polarization. For a plane linearly polarized wave, it is enough to have an expression that allows you to determine the displacement c(x, t) from the equilibrium position of any point in the medium with coordinate X, at any time t. This expression is called wave equation.

Rice. 2.21.

Let's consider the so-called running wave, those. a wave with a plane wavefront propagating in one specific direction (for example, along the x-axis). Let the particles of the medium immediately adjacent to the source of plane waves oscillate according to the harmonic law; %(0, /) = = LsobsoG (Fig. 2.21). In Figure 2.21, A through ^(0, t) indicates the displacement of particles of the medium lying in a plane perpendicular to the drawing and having a coordinate in the selected coordinate system X= 0 at time t. The time reference point is chosen so that initial phase oscillations defined through the cosine function was 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 one phase. The wave front itself in a given medium moves along the axis X with speed And wave propagation in a given medium.

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

in time Consequently, the oscillations of particles lying in a plane distant from the source at a distance X, will lag in time by an amount 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 A oscillations (in the case of a plane wave) will not depend on the x coordinate, i.e.

This is the required equation the melancholy of a 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 phase of oscillation 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 different particles will, generally speaking, be different, but it is possible to identify particles whose oscillations will occur in the same phase (in phase). We can also assume that the phase difference between the 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 wavelength relationship 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 wavelength: The wavelength is the distance over which the vibrations of the particles of the medium have time to propagate in a time equal to the period of the vibrations.

The wave equation reveals double periodicity: in coordinate and time: ^(x, t) = Z,(x + nk, t) = l,(x, t + mT) = ​​Tx + pX, ml), Where pete - any integers. You can, for example, fix the coordinates of particles (put x = const) and consider their displacement as a function of time. Or, conversely, fix a moment in time (accept t = const) and consider the displacement of particles as a function of coordinates (the instantaneous state of displacements is an instantaneous photograph of a wave). So, while on the pier you can use a camera at a moment in time t photograph the sea surface, but you can by throwing a chip into the sea (i.e. fixing the coordinate X), monitor 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 relationship is denoted To and is called wave number

Because , That

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

Let us find an expression relating the phase difference Der of vibrations 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 the general case by the equation

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

It is also possible complex form writing the wave equation. 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 arbitrary direction

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

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

Bearing in mind relation (2.129), we write

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

In general terms for?, = c(x, y, z,/) the wave equation in Cartesian coordinates looks like this

or in a more compact form:

where D is the Laplace differential operator

Phase speed is the speed of propagation of wave points 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 therefore any wave surface) moves along the axis Oh with speed And. Consequently, the speed of propagation of oscillations in the medium coincides with the speed of movement of a given phase of oscillations. Therefore the speed And, determined by relation (2.129), i.e.

usually called phase speed.

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

which coincides with (2.142).

Plane traveling wave propagating in the negative axis direction Oh, 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 phase velocity on frequency is called dispersion, and the environments in which this dependence occurs are called dispersing media. One should not think, however, that expression (2.142) is the indicated dependence. The point is that in the absence of dispersion the wave number To in direct ratio

with and therefore . Dispersion occurs only when ω depends on To nonlinear).

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

For a monochromatic wave, the angular frequency co and amplitude A do not depend on time. This means that a monochromatic wave is limitless 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 to monochromatic this wave is. Often in practice, a monochromatic wave is called a sufficiently large segment of the wave, within which the frequency and amplitude do not change, just as a segment of a sine wave is depicted in the figure, and it is called a sine wave.

Waves depending on one spatial coordinate

Animation

Description

In a plane wave, all points of the medium lying in any plane perpendicular to the direction of propagation of the wave correspond at each moment of time to the same displacements and velocities of particles of the medium. Thus, all quantities characterizing a plane wave are functions of time and only one coordinate, for example, x, if the Ox axis coincides with the direction of wave propagation.

The wave equation for a longitudinal plane wave has the form:

d 2 j / dx 2 = (1/c 2 ) d 2 j / dt 2 . (1)

His common decision is expressed as follows:

j = f 1 (ct - x)+f 2 (ct + x), (2)

where j is potential or other quantity characterizing the wave motion of the medium (displacement, displacement speed, etc.);

c is the speed of wave propagation;

f 1 and f 2 are arbitrary functions, with the first term (2) describing a plane wave propagating in the positive direction of the Ox axis, and the second in the opposite direction.

Wave surfaces or geometric locations of points in the medium where, at a given moment in time, the wave phase has the same value, for PVs they represent a system of parallel planes (Fig. 1).

Wave surfaces of a plane wave

Rice. 1

In a homogeneous isotropic medium, the wave surfaces of a plane wave are perpendicular to the direction of wave propagation (the direction of energy transfer), called the ray.

Timing characteristics

Initiation time (log to -10 to 1);

Lifetime (log tc from -10 to 3);

Degradation time (log td from -10 to 1);

Time of optimal development (log tk from -3 to 1).

Diagram:

Technical implementations of the effect

Technical implementation of the effect

Strictly speaking, no real wave is a plane wave, because A plane wave propagating along the x axis must cover the entire region of space along the y and z coordinates from -Ґ to +Ґ. However, in many cases it is possible to indicate a section of the wave limited in y, z, where it practically coincides with a plane wave. First of all, this is possible in a homogeneous isotropic medium at sufficiently large distances R from the source. Thus, for a harmonic plane wave, the phase at all points of the plane perpendicular to the direction of its propagation is the same. It can be shown that any harmonic wave can be considered a plane wave over a section of width r<< (2R l )1/2 .

Applying an effect

Some wave technologies are most effective in approximating plane waves. In particular, it is shown that during seismoacoustic impacts (in order to increase oil and gas recovery) on oil and gas formations represented by layered geological structures, the interaction of direct and plane wave fronts reflected from the boundaries of layers leads to the appearance of standing waves, initiating the gradual movement and concentration of hydrocarbon fluids at the antinodes of a standing wave (see description of the FE “Standing Waves”).

Plane wave

The front of a plane wave is a plane. According to the definition of a wave front, sound rays intersect it at right angles, so in a plane wave they are parallel to each other. Since the flow of energy does not diverge, the intensity of the sound should not decrease with distance from the sound source. Nevertheless, it decreases due to molecular attenuation, viscosity of the medium, dust content, scattering, etc. losses. However, these losses are so small that they can be ignored when the wave propagates over short distances. Therefore, it is usually believed that the intensity of sound in a plane wave does not depend on the distance to the sound source.

Since the amplitudes of sound pressure and vibration speed also do not depend on this distance

Let us derive the basic equations for a plane wave. Equation (1.8) looks like this: A particular solution to the wave equation for a plane wave propagating in the positive direction has the form

where is the amplitude of sound pressure; - angular frequency of oscillations; - wave number.

Substituting sound pressure into the equation of motion (1.5) and integrating over time, we obtain the oscillation speed

where is the amplitude of the oscillation speed.

From these expressions we find the specific acoustic resistance (1.10) for a plane wave:

For normal atmospheric pressure and temperature, acoustic impedance

Acoustic resistance for a plane wave is determined only by the speed of sound and the density of the medium and is active, as a result of which the pressure and vibration speed are in the same phase, i.e., therefore, the sound intensity

where and are the effective values ​​of sound pressure and vibration speed. Substituting (1.17) into this expression, we obtain the most commonly used expression for determining sound intensity

Spherical wave

The front of such a wave is a spherical surface, and the sound rays, according to the definition of the wave front, coincide with the radii of the sphere. As a result of the divergence of waves, the sound intensity decreases with distance from the source. Since energy losses in the medium are small, as in the case of a plane wave, when the wave propagates over short distances, they can be ignored. Therefore, the average energy flow through a spherical surface will be the same as through any other spherical surface with a large radius, if there is no source or energy sink in between.

Cylindrical wave

For a cylindrical wave, the sound intensity can be determined provided that the energy flow does not diverge along the generatrix of the cylinder. For a cylindrical wave, the sound intensity is inversely proportional to the distance from the cylinder axis.

Phase shift occurs only when sound beams diverge or converge. In the case of a plane wave, the sound rays travel parallel, so each layer of the medium, enclosed between adjacent wave fronts spaced at the same distance from each other, has the same mass. The masses of these layers can be represented as a chain of identical balls. If you push the first ball, it will reach the second and give it forward motion, and it will stop, then the third ball will also be set in motion, and the second will stop, and so on, i.e., the energy imparted to the first ball will be transferred sequentially to all farther and farther. There is no reactive component of the sound wave power. Let us consider the case of a diverging wave, when each subsequent layer has a large mass. The mass of the ball will increase with increasing its number, quickly at first, and then more and more slowly. After the collision, the first ball gives only part of the energy to the second and moves backward, the second will set the third in motion, but then will also move backward. Thus, part of the energy will be reflected, i.e., a reactive component of power appears, which determines the reactive component of acoustic impedance and the appearance of a phase shift between pressure and oscillation speed. The balls further away from the first one will transfer almost all the energy to the balls in front, since their masses will be almost the same.

If the mass of each ball is taken equal to the mass of air contained between the wave fronts located at a distance of half a wave from each other, then the longer the wavelength, the more sharply the mass of the balls will change as their numbers increase, the greater part of the energy will be reflected when the balls collide and the greater the phase shift will be.

For short wavelengths, the masses of neighboring balls differ slightly, so the reflection of energy will be less.

Basic properties of hearing

The ear consists of three parts: outer, middle and inner. The first two parts of the ear serve as a transmission device for bringing sound vibrations to the auditory analyzer located in the inner ear - the cochlea. This transmission device serves as a lever system that converts air vibrations with a large amplitude of vibration speed and low pressure into mechanical vibrations with a small amplitude of speed and high pressure. The transformation coefficient is on average 50-60. In addition, the transmission device makes a correction to the frequency response of the next perception link - the cochlea.

The boundaries of the frequency range perceived by hearing are quite wide (20-20000 Hz). Due to the limited number of nerve endings located along the main membrane, a person remembers no more than 250 frequency gradations in the entire frequency range, and the number of these gradations decreases with decreasing sound intensity and averages about 150, i.e., neighboring gradations on average differ from each other from each other in frequency by at least 4%, which on average is approximately equal to the width of the critical hearing strips. The concept of pitch has been introduced, which refers to a subjective assessment of the perception of sound across the frequency range. Since the width of the critical hearing band at medium and high frequencies is approximately proportional to frequency, the subjective scale of perception in frequency is close to the logarithmic law. Therefore, an octave is taken as an objective unit of sound pitch, approximately reflecting subjective perception: a double frequency ratio (1; 2; 4; 8; 16, etc.). The octave is divided into parts: half octaves and third octaves. For the latter, the following range of frequencies is standardized: 1; 1.25; 1.6; 2; 2.5; 3.15; 4; 5; 6.3; 8; 10, which are the boundaries of one-third octaves. If these frequencies are placed at equal distances along the frequency axis, you get a logarithmic scale. Based on this, to get closer to the subjective scale, all frequency characteristics of sound transmission devices are plotted on a logarithmic scale. To more accurately correspond to the auditory perception of sound in frequency, a special, subjective scale has been adopted for these characteristics - almost linear up to a frequency of 1000 Hz and logarithmic above this frequency. Units of pitch called “chalk” and “bark” () were introduced. In general, the pitch of a complex sound cannot be calculated accurately.

: such a wave does not exist in nature, since the front of a plane wave begins at $-\backslash mathcal(1)$ and ends at $+\backslash mathcal(1)$, which obviously cannot be. Also, a plane wave would carry infinite power, and it would take infinite energy to create a plane wave. A wave with a complex (real) front can be represented as a spectrum of plane waves using the Fourier transform in spatial variables.

Quasi-plane wave- a wave whose front is close to flat in a limited area. If the dimensions of the region are large enough for the problem under consideration, then the quasi-plane wave can be approximately considered plane. A wave with a complex front can be approximated by a set of local quasi-plane waves, the phase velocity vectors of which are normal to the real front at each of its points. Examples of sources of quasi-plane electromagnetic waves are laser, mirror and lens antennas: the distribution of the phase of the electromagnetic field in a plane parallel to the aperture (emitting hole) is close to uniform. As it moves away from the aperture, the wave front takes on a complex shape.

Definition

The equation of any wave is a solution to a differential equation called wave. Wave equation for the function $A$ written in the form

$\backslash Delta\; A(\backslash vec(r),t)\; =\; \backslash frac\; (1)\; (v^2)\; \backslash ,\; \backslash frac\; (\backslash partial^2\; A(\backslash vec(r),t))\; (\backslash partial\; t^2)$ Where

• $\backslash Delta$- Laplace operator;
• $A(\backslash vec(r),t)$- the required function;
• $r$- radius vector of the desired point;
• $v$- wave speed;
• $t$- time.

One-dimensional case

$\backslash Delta\; W\_k\; =\; \backslash cfrac\; (\backslash rho)\; (2)\; \backslash left(\backslash cfrac\; (\backslash partial\; A)\; (\backslash partial\; t)\; \backslash right)^2\; \backslash Delta\; V$ $\backslash Delta\; W\_p\; =\; \backslash cfrac\; (E)\; (2)\; \backslash left(\backslash cfrac\; (\backslash partial\; A)\; (\backslash partial\; x)\; \backslash right)^2\; \backslash Delta\; V\; =\; \backslash cfrac\; (\backslash rho\; v^2)\; (2)\; \backslash left\; (\backslash cfrac\; (\backslash partial\; A)\; (\backslash partial\; x)\; \backslash right)^2\; \backslash Delta\; V\; .$

Total energy is

$W\; =\; \backslash Delta\; W\_k\; +\; \backslash Delta\; W\_p\; =\; \backslash cfrac(\backslash rho)(2)\; \backslash bigg[\; \backslash left(\backslash cfrac\; (\backslash partial\; A)\; (\backslash partial\; t)\; \backslash right)^2\; +\; v^2\; \backslash left(\backslash \; cfrac(\backslash partial\; A)(\backslash partial\; (x))\; \backslash right)^2\; \backslash bigg]\; \backslash Delta\; V\; .$

The energy density is, accordingly, equal to

$\backslash omega\; =\; \backslash cfrac\; (W)\; (\backslash Delta\; V)\; =\; \backslash cfrac(\backslash rho)(2)\; \backslash bigg[\; \backslash left(\backslash cfrac\; (\backslash partial\; A)\; (\backslash partial\; t)\; \backslash right)^2\; +\; v^2\; \backslash left(\backslash cfrac\; (\backslash partial\; A)\; (\backslash partial\; (x))\; \backslash right)^2\; \backslash bigg]\; =\; \backslash rho\; A^2\; \backslash omega^2\; \backslash sin^2\; \backslash left(\backslash omega\; t\; -\; k\; x\; +\; \backslash varphi\_0\; \backslash right)\; .$

Polarization

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Literature

• Savelyev I.V.[Part 2. Waves. Elastic waves.] // Course of general physics / Edited by Gladnev L.I., Mikhalin N.A., Mirtov D.A.. - 3rd ed. - M.: Nauka, 1988. - T. 2. - P. 274-315. - 496 s. - 220,000 copies.

Notes

see also

An excerpt characterizing a plane wave

- It’s a pity, it’s a pity for the fellow; give me a letter.
Rostov barely had time to hand over the letter and tell Denisov’s whole business when quick steps with spurs began to sound from the stairs and the general, moving away from him, moved towards the porch. The gentlemen of the sovereign's retinue ran down the stairs and went to the horses. Bereitor Ene, the same one who was in Austerlitz, brought the sovereign's horse, and a light creaking of steps was heard on the stairs, which Rostov now recognized. Forgetting the danger of being recognized, Rostov moved with several curious residents to the porch itself and again, after two years, he saw the same features he adored, the same face, the same look, the same gait, the same combination of greatness and meekness... And the feeling of delight and love for the sovereign was resurrected with the same strength in Rostov’s soul. The Emperor in the Preobrazhensky uniform, in white leggings and high boots, with a star that Rostov did not know (it was legion d'honneur) [star of the Legion of Honor] went out onto the porch, holding his hat at hand and putting on a glove. He stopped, looking around and that's it illuminating the surroundings with his gaze. He said a few words to some of the generals. He also recognized the former chief of the division, Rostov, smiled at him and called him over.
The entire retinue retreated, and Rostov saw how this general said something to the sovereign for quite a long time.
The Emperor said a few words to him and took a step to approach the horse. Again the crowd of the retinue and the crowd of the street in which Rostov was located moved closer to the sovereign. Stopping by the horse and holding the saddle with his hand, the sovereign turned to the cavalry general and spoke loudly, obviously with the desire for everyone to hear him.
“I can’t, general, and that’s why I can’t because the law is stronger than me,” said the sovereign and raised his foot in the stirrup. The general bowed his head respectfully, the sovereign sat down and galloped down the street. Rostov, beside himself with delight, ran after him with the crowd.

On the square where the sovereign went, a battalion of Preobrazhensky soldiers stood face to face on the right, and a battalion of the French Guard in bearskin hats on the left.
While the sovereign was approaching one flank of the battalions, which were on guard duty, another crowd of horsemen jumped up to the opposite flank and ahead of them Rostov recognized Napoleon. It couldn't be anyone else. He rode at a gallop in a small hat, with a St. Andrew's ribbon over his shoulder, in a blue uniform open over a white camisole, on an unusually thoroughbred Arabian gray horse, on a crimson, gold embroidered saddle cloth. Having approached Alexander, he raised his hat and with this movement, Rostov’s cavalry eye could not help but notice that Napoleon was sitting poorly and not firmly on his horse. The battalions shouted: Hurray and Vive l "Empereur! [Long live the Emperor!] Napoleon said something to Alexander. Both emperors got off their horses and took each other's hands. There was an unpleasantly feigned smile on Napoleon's face. Alexander said something to him with an affectionate expression .
Rostov, without taking his eyes off, despite the trampling of the horses of the French gendarmes besieging the crowd, followed every move of Emperor Alexander and Bonaparte. He was struck as a surprise by the fact that Alexander behaved as an equal with Bonaparte, and that Bonaparte was completely free, as if this closeness with the sovereign was natural and familiar to him, as an equal, he treated the Russian Tsar.
Alexander and Napoleon with a long tail of their retinue approached the right flank of the Preobrazhensky battalion, directly towards the crowd that stood there. The crowd suddenly found itself so close to the emperors that Rostov, who was standing in the front rows, became afraid that they would recognize him.
“Sire, je vous demande la permission de donner la legion d"honneur au plus brave de vos soldats, [Sire, I ask your permission to give the Order of the Legion of Honor to the bravest of your soldiers,] said a sharp, precise voice, finishing each letter It was the short Bonaparte who spoke, looking straight into Alexander's eyes from below. Alexander listened attentively to what was being said to him, and bowed his head, smiling pleasantly.
“A celui qui s"est le plus vaillament conduit dans cette derieniere guerre, [To the one who showed himself bravest during the war],” Napoleon added, emphasizing each syllable, with a calm and confidence outrageous for Rostov, looking around the ranks of Russians stretched out in front of there are soldiers, keeping everything on guard and motionlessly looking into the face of their emperor.
“Votre majeste me permettra t elle de demander l"avis du colonel? [Your Majesty will allow me to ask the colonel’s opinion?] - said Alexander and took several hasty steps towards Prince Kozlovsky, the battalion commander. Meanwhile, Bonaparte began to take off his white glove, small hand and, tearing it apart, threw it in. The adjutant, hastily rushing forward from behind, picked it up.
- Who should I give it to? – Emperor Alexander asked Kozlovsky not loudly, in Russian.
- Whom do you order, Your Majesty? “The Emperor winced with displeasure and, looking around, said:
- But you have to answer him.
Kozlovsky looked back at the ranks with a decisive look and in this glance captured Rostov as well.
“Isn’t it me?” thought Rostov.
- Lazarev! – the colonel commanded with a frown; and the first-ranked soldier, Lazarev, smartly stepped forward.
-Where are you going? Stop here! - voices whispered to Lazarev, who did not know where to go. Lazarev stopped, looked sideways at the colonel in fear, and his face trembled, as happens with soldiers called to the front.
Napoleon slightly turned his head back and pulled back his small chubby hand, as if wanting to take something. The faces of his retinue, having guessed at that very second what was going on, began to fuss, whisper, passing something on to one another, and the page, the same one whom Rostov saw yesterday at Boris’s, ran forward and respectfully bent over the outstretched hand and did not make her wait either one second, he put an order on a red ribbon into it. Napoleon, without looking, clenched two fingers. The Order found itself between them. Napoleon approached Lazarev, who, rolling his eyes, stubbornly continued to look only at his sovereign, and looked back at Emperor Alexander, thereby showing that what he was doing now, he was doing for his ally. A small white hand with an order touched the button of soldier Lazarev. It was as if Napoleon knew that in order for this soldier to be happy, rewarded and distinguished from everyone else in the world forever, it was only necessary for him, Napoleon’s hand, to be worthy of touching the soldier’s chest. Napoleon just put the cross to Lazarev's chest and, letting go of his hand, turned to Alexander, as if he knew that the cross should stick to Lazarev's chest. The cross really stuck.