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 sound pressure amplitude; - 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 low velocity amplitude 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.

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 presented 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 this or that 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. Chief Editor A. M. Prokhorov. 1988 .

> Spherical and plane waves

Learn to differentiate spherical and plane waves. Read what wave is called plane or spherical, source, role of the wave front, characteristics.

Spherical waves arise from a point source in a spherical pattern, and flat– infinite parallel planes normal to the phase velocity vector.

Learning Objective

• Calculate the sources of spherical and plane wave patterns.

Main points

• Waves create constructive and destructive interference.
• Sphericals arise from a single point source in a spherical shape.
• Flat water is frequency water, the wave fronts of which act as infinite parallel planes with a stable amplitude.
• In reality, it will not be possible to obtain a perfect plane wave, but many are approaching this state.

Terms

• Destructive interference - waves interfere with each other, and the points do not coincide.
• Constructive - the waves interfere and the points are located in identical phases.
• A wave front is an imaginary surface extending through oscillating points in the phase of the medium.

Spherical waves

What wave is called spherical? Christian Huygens managed to develop a method for determining the method and location of wave propagation. In 1678, he proposed that every point that encounters a light disturbance becomes a source of a spherical wave. Summation of secondary waves calculates the appearance at any time. This principle showed that upon contact, waves create destructive or constructive interference.

Constructive ones are formed if the waves are completely in phase with each other, and the final one intensifies. In destructive waves, they do not correspond in phases and the final one is simply shortened. The waves originate from a single point source, so they form in a spherical pattern.

If the waves are generated from a point source, they appear spherical

This principle applies the law of refraction. Each point on the wave creates waves that interfere with each other either constructively or destructively

Plane waves

Now let's understand what kind of wave is called plane. The flat one displays a frequency wave, the fronts of which appear as infinite parallel planes with a stable amplitude located perpendicular to the phase velocity vector. In reality, it is impossible to obtain a true plane wave. Only a flat one with infinite extension can match it. True, many waves approach this state. For example, an antenna produces a field that is approximately flat.

Planar displays an infinite number of wavefronts normal to the side of propagation

: 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 spectrum plane waves using Fourier transform by 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 quasi-planar sources electromagnetic waves are laser, mirror and lens antennas: phase distribution electromagnetic field in a plane parallel to the aperture (emitting hole), 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 differential equation, called wave. Wave equation for 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 ;
• $A(\backslash vec(r),t)$ - required function;
• $r$ - radius vector 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.] // Well 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 directly into Alexander’s eyes, Alexander listened attentively to what was being said, 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, 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.