Equations of plane and spherical waves. Equation of a plane traveling wave Excerpt characterizing a plane wave

An oscillatory process propagating in a medium in the form of a wave, the front of which is plane, called plane sound wave. In practice, a plane wave can be formed by a source whose linear dimensions are large compared to the long wavelength it emits, and if the wave field zone is located at a sufficiently large distance from it. But this is the case in an unconstrained environment. If the source fenced any obstacle, then a classic example of a plane wave is oscillations excited by a rigid, unbendable piston in a long pipe (waveguide) with rigid walls, if the diameter of the piston is significantly less than the length of the emitted waves. Due to the rigid walls, the front surface in the pipe does not change as the wave propagates along the waveguide (see Fig. 3.3). We neglect the losses of sound energy due to absorption and dissipation in the air.

If the emitter (piston) oscillates according to the harmonic law with a frequency
, and the dimensions of the piston (waveguide diameter) are significantly less than the sound wavelength, then the pressure created near its surface
. Obviously, from a distance X the pressure will be
, Where
– travel time of the wave from the emitter to the pointx. It is more convenient to write this expression as:
, Where
- wave number of wave propagation. Work
- determined phase shift of the oscillatory process at a point removed by a distance X from the emitter.

Substituting the resulting expression into the equation of motion (3.1), we integrate the latter with respect to the oscillatory velocity:

(3.8)

In general, for an arbitrary moment in time it turns out that:

. (3.9)

The right side of expression (3.9) is the characteristic, wave, or specific acoustic resistance of the medium (impedance). Equation (3.) itself is sometimes called the acoustic “Ohm’s law”. As follows from the solution, the resulting equation is valid in the field of a plane wave. Pressure and vibrational speed in phase, which is a consequence of the purely active resistance of the medium.

Example: Maximum pressure in a plane wave
Pa. Determine the amplitude of displacement of air particles by frequency?

Solution: Since , then:

From expression (3.10) it follows that the amplitude of sound waves is very small, at least in comparison with the size of the sound sources themselves.

In addition to the scalar potential, pressure and vibrational speed, the sound field is also characterized by energy characteristics, the most important of which is intensity - the vector of energy flux density transferred by the wave per unit time. A-priory
- is the result of the product of sound pressure and vibrational speed.

In the absence of losses in the medium, a plane wave, theoretically, can propagate without attenuation over arbitrarily large distances, because preservation of the flat front shape indicates the absence of “divergence” of the wave, and, therefore, the absence of attenuation. The situation is different if the wave has a curved front. Such waves include, first of all, spherical and cylindrical waves.

3.1.3. Models of waves with a non-plane front

For a spherical wave, the surface of equal phases is a sphere. The source of such a wave is also a sphere, all points of which oscillate with the same amplitudes and phases, and the center remains motionless (see Fig. 3.4, a).

A spherical wave is described by a function that is the solution of the wave equation in a spherical coordinate system for the potential of the wave propagating from the source:

. (3.11)

Working by analogy with a plane wave, it can be shown that at distances from the sound source the length of the waves being studied is significantly greater:
. This means that the acoustic “Ohm’s law” is also true in this case. In practical conditions, spherical waves are excited mainly by compact sources of arbitrary shape, the dimensions of which are significantly smaller than the length of the excited sound or ultrasonic waves. In other words, a “point” source emits predominantly spherical waves. At large distances from the source, or, as they say, in the “far” zone, a spherical wave, in relation to limited-size sections of the wave front, behaves like a plane wave, or, as they say: “degenerates into a plane wave.” The requirements for a small area are determined not only by frequency, but
- the difference in distances between compared points. Note that this function
has a feature:
at
. This causes certain difficulties in the rigorous solution of diffraction problems associated with the radiation and scattering of sound.

In turn, cylindrical waves (the surface of the wave front is a cylinder) are emitted by an infinitely long pulsating cylinder (see Fig. 3.4).

In the far zone, the expression for the potential function of such a source asymptotically tends to the expression:


. (3.12)

It can be shown that in this case the relation holds as well
. Cylindrical waves, like spherical ones, in the far zone degenerate into plane waves.

The weakening of elastic waves during propagation is associated not only with a change in the curvature of the wave front (“divergence” of the wave), but also with the presence of “attenuation,” i.e. sound weakening. Formally, the presence of attenuation in a medium can be described by representing the wave number as complex
. Then, for example, for a plane pressure wave one can obtain: R(x, t) = P Max
=
.

It can be seen that the real part of the complex wave number describes the spatial traveling wave, and the imaginary part characterizes the attenuation of the wave in amplitude. Therefore, the value  is called the attenuation (attenuation) coefficient,  is a dimensional value (Neper/m). One “Naper” corresponds to a change in wave amplitude by “e” times when the wave front moves per unit length. In the general case, attenuation is determined by absorption and scattering in the medium:  =  absorb +  diss. These effects are determined by different reasons and can be considered separately.

In general, absorption is associated with irreversible losses of sound energy when it is converted into heat.

Scattering is associated with the reorientation of part of the energy of the incident wave to other directions that do not coincide with the incident wave.

This function must be periodic both with respect to time and coordinates (a wave is a propagating oscillation, therefore a periodically repeating movement). In addition, points located at a distance l from each other vibrate in the same way.

Plane wave equation

Let us find the form of the function x in the case of a plane wave, assuming that the oscillations are harmonic in nature.

Let us direct the coordinate axes so that the axis x coincided with the direction of wave propagation. Then the wave surface will be perpendicular to the axis x. Since all points of the wave surface oscillate equally, the displacement x will depend only on X And t: . Let the oscillation of points lying in the plane have the form (at the initial phase)

Let us find the type of vibration of particles in a plane corresponding to an arbitrary value x. To go the way x, it takes time.

Hence, vibrations of particles in a planexwill be behind in time bytfrom vibrations of particles in the plane, i.e.

,

(5.2.3)

- This plane wave equation.

So x There is bias any of the points with coordinatexat a point in timet. In the derivation, we assumed that the amplitude of the oscillation is . This will happen if the wave energy is not absorbed by the medium.

Equation (5.2.3) will have the same form if the vibrations propagate along the axis y or z.

In general plane wave equation is written like this:

Expressions (5.2.3) and (5.2.4) are traveling wave equations .

Equation (5.2.3) describes a wave propagating in the direction of increasing x. A wave propagating in the opposite direction has the form:

.

The wave equation can be written in another form.

Let's introduce wave number , or in vector form:

where is the wave vector and is the normal to the wave surface.

Since then . From here. Then plane wave equation will be written like this:

.

(5.2.6)

Spherical wave equation

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 the 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)

Its general solution 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”).

PLATE WAVE

PLATE WAVE

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

u(z, t)=Aeiwt±ikz, (1)

where A 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 linear motions, which allow a general representation of the form:

u(z, t)=f(z±vt), (2)

where f is an arbitrary function. In nonlinear media with dispersion, stationary 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 form

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 .

: 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

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

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 This was the short Bonaparte speaking, looking directly into Alexander’s eyes from below, listening attentively to what was being said to him, and bowing his head, he smiled 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, the Adjutant threw it, hastily rushing forward from behind, and 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.