You can imagine what mechanical waves are by throwing a stone into the water. The circles that appear on it and are alternating troughs and ridges are an example of mechanical waves. What is their essence? Mechanical waves are the process of propagation of vibrations in elastic media.
Waves on liquid surfaces
Such mechanical waves exist due to the influence of intermolecular forces and gravity on the particles of the liquid. People have been studying this phenomenon for a long time. The most notable are the ocean and sea waves. As the wind speed increases, they change and their height increases. The shape of the waves themselves also becomes more complicated. In the ocean, they can reach frightening proportions. One of the most obvious examples of force is the tsunami, sweeping away everything in its path.
Energy of sea and ocean waves
Reaching the shore, sea waves increase with a sharp change in depth. They sometimes reach a height of several meters. At such moments, a colossal mass of water is transferred to coastal obstacles, which are quickly destroyed under its influence. The strength of the surf sometimes reaches grandiose values.
elastic waves
In mechanics, not only oscillations on the surface of a liquid are studied, but also the so-called elastic waves. These are perturbations that propagate in different media under the action of elastic forces in them. Such a perturbation is any deviation of the particles of a given medium from the equilibrium position. A good example of elastic waves is a long rope or rubber tube attached at one end to something. If you pull it tight, and then create a disturbance at its second (unfixed) end with a lateral sharp movement, you can see how it “runs” along the entire length of the rope to the support and is reflected back.
The initial perturbation leads to the appearance of a wave in the medium. It is caused by the action of some foreign body, which in physics is called the source of the wave. It can be the hand of a person swinging a rope, or a pebble thrown into the water. In the case when the action of the source is short-lived, a single wave often appears in the medium. When the “disturber” makes long waves, they begin to appear one after another.
Conditions for the occurrence of mechanical waves
Such oscillations are not always formed. A necessary condition for their appearance is the occurrence at the moment of disturbance of the medium of forces preventing it, in particular, elasticity. They tend to bring neighboring particles closer together when they move apart and push them away from each other when they approach each other. Elastic forces, acting on particles far from the source of perturbation, begin to unbalance them. Over time, all particles of the medium are involved in one oscillatory motion. The propagation of such oscillations is a wave.
Mechanical waves in an elastic medium
In an elastic wave, there are 2 types of motion simultaneously: particle oscillations and perturbation propagation. A longitudinal wave is a mechanical wave whose particles oscillate along the direction of its propagation. A transverse wave is a wave whose medium particles oscillate across the direction of its propagation.
Properties of mechanical waves
Perturbations in a longitudinal wave are rarefaction and compression, and in a transverse wave they are shifts (displacements) of some layers of the medium in relation to others. The compression deformation is accompanied by the appearance of elastic forces. In this case, it is associated with the appearance of elastic forces exclusively in solids. In gaseous and liquid media, the shift of the layers of these media is not accompanied by the appearance of the mentioned force. Due to their properties, longitudinal waves are able to propagate in any medium, and transverse waves - only in solid ones.
Features of waves on the surface of liquids
Waves on the surface of a liquid are neither longitudinal nor transverse. They have a more complex, so-called longitudinal-transverse character. In this case, the fluid particles move in a circle or along elongated ellipses. particles on the surface of the liquid, and especially with large fluctuations, are accompanied by their slow but continuous movement in the direction of wave propagation. It is these properties of mechanical waves in the water that cause the appearance of various seafood on the shore.
Frequency of mechanical waves
If in an elastic medium (liquid, solid, gaseous) vibration of its particles is excited, then due to the interaction between them, it will propagate with a speed u. So, if an oscillating body is in a gaseous or liquid medium, then its movement will begin to be transmitted to all particles adjacent to it. They will involve the next ones in the process and so on. In this case, absolutely all points of the medium will begin to oscillate with the same frequency, equal to the frequency of the oscillating body. It is the frequency of the wave. In other words, this quantity can be characterized as points in the medium where the wave propagates.
It may not be immediately clear how this process occurs. Mechanical waves are associated with the transfer of energy of oscillatory motion from its source to the periphery of the medium. As a result, so-called periodic deformations arise, which are carried by the wave from one point to another. In this case, the particles of the medium themselves do not move along with the wave. They oscillate near their equilibrium position. That is why the propagation of a mechanical wave is not accompanied by the transfer of matter from one place to another. Mechanical waves have different frequencies. Therefore, they were divided into ranges and created a special scale. Frequency is measured in hertz (Hz).
Basic formulas
Mechanical waves, whose calculation formulas are quite simple, are an interesting object for study. The wave speed (υ) is the speed of movement of its front (geometrical place of all points to which the oscillation of the medium has reached at a given moment):
where ρ is the density of the medium, G is the modulus of elasticity.
When calculating, one should not confuse the speed of a mechanical wave in a medium with the speed of movement of the particles of the medium that are involved in So, for example, a sound wave in air propagates with an average vibrational speed of its molecules of 10 m/s, while the speed of a sound wave in normal conditions is 330 m/s.
The wave front can be of different types, the simplest of which are:
Spherical - caused by fluctuations in a gaseous or liquid medium. In this case, the wave amplitude decreases with distance from the source in inverse proportion to the square of the distance.
Flat - is a plane that is perpendicular to the direction of wave propagation. It occurs, for example, in a closed piston cylinder when it oscillates. A plane wave is characterized by an almost constant amplitude. Its slight decrease with distance from the disturbance source is associated with the degree of viscosity of the gaseous or liquid medium.
Wavelength
Under understand the distance over which its front will move in a time that is equal to the period of oscillation of the particles of the medium:
λ = υT = υ/v = 2πυ/ ω,
where T is the oscillation period, υ is the wave speed, ω is the cyclic frequency, ν is the oscillation frequency of the medium points.
Since the propagation velocity of a mechanical wave is completely dependent on the properties of the medium, its length λ changes during the transition from one medium to another. In this case, the oscillation frequency ν always remains the same. Mechanical and similar in that during their propagation, energy is transferred, but no matter is transferred.
Lecture - 14. Mechanical waves.
2. Mechanical wave.
3. Source of mechanical waves.
4. Point source of waves.
5. Transverse wave.
6. Longitudinal wave.
7. Wave front.
9. Periodic waves.
10. Harmonic wave.
11. Wavelength.
12. Speed of distribution.
13. Dependence of the wave velocity on the properties of the medium.
14. Huygens' principle.
15. Reflection and refraction of waves.
16. The law of wave reflection.
17. The law of refraction of waves.
18. Equation of a plane wave.
19. Energy and intensity of the wave.
20. The principle of superposition.
21. Coherent vibrations.
22. Coherent waves.
23. Interference of waves. a) interference maximum condition, b) interference minimum condition.
24. Interference and the law of conservation of energy.
25. Diffraction of waves.
26. Huygens-Fresnel principle.
27. Polarized wave.
29. Sound volume.
30. Pitch of sound.
31. Sound timbre.
32. Ultrasound.
33. Infrasound.
34. Doppler effect.
1.Wave - this is the process of propagation of oscillations of any physical quantity in space. For example, sound waves in gases or liquids represent the propagation of pressure and density fluctuations in these media. An electromagnetic wave is the process of propagation in space of fluctuations in the strength of electric magnetic fields.
Energy and momentum can be transferred in space by transferring matter. Any moving body has kinetic energy. Therefore, it transfers kinetic energy by transferring matter. The same body, being heated, moving in space, transfers thermal energy, transferring matter.
Particles of an elastic medium are interconnected. Perturbations, i.e. deviations from the equilibrium position of one particle are transferred to neighboring particles, i.e. energy and momentum are transferred from one particle to neighboring particles, while each particle remains near its equilibrium position. Thus, energy and momentum are transferred along the chain from one particle to another, and there is no transfer of matter.
So, the wave process is the process of transfer of energy and momentum in space without the transfer of matter.
2. Mechanical wave or elastic wave is a perturbation (oscillation) propagating in an elastic medium. The elastic medium in which mechanical waves propagate is air, water, wood, metals and other elastic substances. Elastic waves are called sound waves.
3. Source of mechanical waves- a body that performs an oscillatory motion, being in an elastic medium, for example, vibrating tuning forks, strings, vocal cords.
4. Point source of waves - a source of a wave whose dimensions can be neglected compared to the distance over which the wave propagates.
5. transverse wave - a wave in which the particles of the medium oscillate in a direction perpendicular to the direction of wave propagation. For example, waves on the surface of water are transverse waves, because vibrations of water particles occur in a direction perpendicular to the direction of the water surface, and the wave propagates along the surface of the water. A transverse wave propagates along a cord, one end of which is fixed, the other oscillates in a vertical plane.
A transverse wave can propagate only along the interface between the spirit of different media.
6. Longitudinal wave - a wave in which vibrations occur in the direction of wave propagation. A longitudinal wave occurs in a long helical spring if one of its ends is subjected to periodic perturbations directed along the spring. The elastic wave running along the spring is a propagating sequence of compression and tension (Fig. 88)
A longitudinal wave can propagate only inside an elastic medium, for example, in air, in water. In solids and liquids, both transverse and longitudinal waves can propagate simultaneously, because a solid body and a liquid are always limited by a surface - the interface between two media. For example, if a steel rod is hit on the end with a hammer, then elastic deformation will begin to propagate in it. A transverse wave will run along the surface of the rod, and a longitudinal wave will propagate inside it (compression and rarefaction of the medium) (Fig. 89).
7. Wave front (wave surface) is the locus of points oscillating in the same phases. On the wave surface, the phases of the oscillating points at the considered moment of time have the same value. If a stone is thrown into a calm lake, then transverse waves in the form of a circle will begin to propagate along the surface of the lake from the place of its fall, with the center at the place where the stone fell. In this example, the wavefront is a circle.
In a spherical wave, the wave front is a sphere. Such waves are generated by point sources.
At very large distances from the source, the curvature of the front can be neglected and the wave front can be considered flat. In this case, the wave is called a plane wave.
8. Beam - straight line is normal to the wave surface. In a spherical wave, the rays are directed along the radii of the spheres from the center, where the wave source is located (Fig.90).
In a plane wave, the rays are directed perpendicular to the surface of the front (Fig. 91).
9. Periodic waves. When talking about waves, we meant a single perturbation propagating in space.
If the source of waves performs continuous oscillations, then elastic waves traveling one after one arise in the medium. Such waves are called periodic.
10. harmonic wave- a wave generated by harmonic oscillations. If the wave source makes harmonic oscillations, then it generates harmonic waves - waves in which particles oscillate according to a harmonic law.
11. Wavelength. Let a harmonic wave propagate along the OX axis and oscillate in it in the direction of the OY axis. This wave is transverse and can be represented as a sinusoid (Fig.92).
Such a wave can be obtained by causing vibrations in the vertical plane of the free end of the cord.
Wavelength is the distance between two nearest points. A and B oscillating in the same phases (Fig. 92).
12. Wave propagation speed– physical quantity numerically equal to the speed of propagation of oscillations in space. From Fig. 92 it follows that the time for which the oscillation propagates from point to point BUT to the point AT, i.e. by a distance of a wavelength equal to the period of oscillation. Therefore, the propagation speed of the wave is
13. Dependence of the wave propagation velocity on the properties of the medium. The frequency of oscillations when a wave occurs depends only on the properties of the wave source and does not depend on the properties of the medium. The speed of wave propagation depends on the properties of the medium. Therefore, the wavelength changes when crossing the interface between two different media. The speed of the wave depends on the bond between the atoms and molecules of the medium. The bond between atoms and molecules in liquids and solids is much more rigid than in gases. Therefore, the speed of sound waves in liquids and solids is much greater than in gases. In air, the speed of sound under normal conditions is 340, in water 1500, and in steel 6000.
The average speed of thermal motion of molecules in gases decreases with decreasing temperature, and as a result, the speed of wave propagation in gases decreases. In a denser medium, and therefore more inert, the wave speed is lower. If sound propagates in air, then its speed depends on the density of the air. Where the density of air is higher, the speed of sound is lower. Conversely, where the density of air is less, the speed of sound is greater. As a result, when sound propagates, the wave front is distorted. Over a swamp or over a lake, especially in the evening, the air density near the surface due to water vapor is greater than at a certain height. Therefore, the speed of sound near the surface of the water is less than at a certain height. As a result, the wave front turns in such a way that the upper part of the front bends more and more towards the lake surface. It turns out that the energy of a wave traveling along the lake surface and the energy of a wave traveling at an angle to the lake surface add up. Therefore, in the evening, the sound is well distributed over the lake. Even a quiet conversation can be heard standing on the opposite bank.
14. Huygens principle- each point of the surface that the wave has reached at a given moment is a source of secondary waves. Drawing a surface tangent to the fronts of all secondary waves, we get the wave front at the next time.
Consider, for example, a wave propagating over the surface of water from a point O(Fig.93) Let at the moment of time t the front had the shape of a circle of radius R centered on a point O. At the next moment of time, each secondary wave will have a front in the form of a circle of radius , where V is the speed of wave propagation. Drawing a surface tangent to the fronts of the secondary waves, we get the wave front at the moment of time (Fig. 93)
If the wave propagates in a continuous medium, then the wave front is a sphere.
15. Reflection and refraction of waves. When a wave falls on the interface between two different media, each point of this surface, according to the Huygens principle, becomes a source of secondary waves propagating on both sides of the section surface. Therefore, when crossing the interface between two media, the wave is partially reflected and partially passes through this surface. Because different media, then the speed of the waves in them is different. Therefore, when crossing the interface between two media, the direction of wave propagation changes, i.e. wave breaking occurs. Consider, on the basis of the Huygens principle, the process and the laws of reflection and refraction are complete.
16. Wave reflection law. Let a plane wave fall on a flat interface between two different media. Let's select in it the area between the two rays and (Fig. 94)
The angle of incidence is the angle between the incident beam and the perpendicular to the interface at the point of incidence.
Reflection angle - the angle between the reflected beam and the perpendicular to the interface at the point of incidence.
At the moment when the beam reaches the interface at the point , this point will become a source of secondary waves. The wave front at this moment is marked by a straight line segment AC(Fig.94). Consequently, the beam still has to go to the interface at this moment, the path SW. Let the beam travel this path in time . The incident and reflected rays propagate on the same side of the interface, so their velocities are the same and equal v. Then .
During the time the secondary wave from the point BUT will go the way. Hence . Right triangles and are equal, because - common hypotenuse and legs. From the equality of triangles follows the equality of angles 
. But also , i.e. .
Now we formulate the law of wave reflection: incident beam, reflected beam , the perpendicular to the interface between two media, restored at the point of incidence, lie in the same plane; the angle of incidence is equal to the angle of reflection.
17. Wave refraction law. Let a plane wave pass through a plane interface between two media. And the angle of incidence is different from zero (Fig.95).
The angle of refraction is the angle between the refracted beam and the perpendicular to the interface, restored at the point of incidence.
Denote and the wave propagation velocities in media 1 and 2. At the moment when the beam reaches the interface at the point BUT, this point will become a source of waves propagating in the second medium - the ray , and the ray still has to go the way to the surface of the section. Let be the time it takes the beam to travel the path SW, then . During the same time in the second medium, the beam will travel the path . Because , then and .
Triangles and right angles with a common hypotenuse , and = , are like angles with mutually perpendicular sides. For the angles and we write the following equalities
.
Taking into account that , , we get
Now we formulate the law of wave refraction: The incident beam, the refracted beam and the perpendicular to the interface between two media, restored at the point of incidence, lie in the same plane; the ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant value for two given media and is called the relative refractive index for the two given media.
18. Plane wave equation. Particles of the medium that are at a distance S from the source of the waves begin to oscillate only when the wave reaches it. If a V is the speed of wave propagation, then the oscillations will begin with a delay for a time
If the wave source oscillates according to the harmonic law, then for a particle located at a distance S from the source, we write the law of oscillations in the form
.
Let's introduce the value 
called the wave number. It shows how many wavelengths fit at a distance equal to units of length. Now the law of oscillations of a particle of a medium located at a distance S from the source we write in the form
.
This equation defines the displacement of the oscillating point as a function of time and distance from the wave source and is called the plane wave equation.
19. Wave Energy and Intensity. Each particle that the wave has reached oscillates and therefore has energy. Let a wave propagate in some volume of an elastic medium with an amplitude BUT and cyclic frequency. This means that the average energy of oscillations in this volume is equal to
Where m- the mass of the allocated volume of the medium.
The average energy density (average over volume) is the wave energy per unit volume of the medium
, where is the density of the medium.
Wave intensity is a physical quantity numerically equal to the energy that a wave transfers per unit of time through a unit area of the plane perpendicular to the direction of wave propagation (through a unit area of the wave front), i.e.
.
The average power of a wave is the average total energy transferred by a wave per unit time through a surface with an area S. We obtain the average wave power by multiplying the wave intensity by the area S
20.The principle of superposition (overlay). If waves from two or more sources propagate in an elastic medium, then, as observations show, the waves pass one through the other without affecting each other at all. In other words, the waves do not interact with each other. This is explained by the fact that within the limits of elastic deformation, compression and tension in one direction in no way affect the elastic properties in other directions.
Thus, each point of the medium where two or more waves come takes part in the oscillations caused by each wave. In this case, the resulting displacement of a particle of the medium at any time is equal to the geometric sum of the displacements caused by each of the emerging oscillatory processes. This is the essence of the principle of superposition or superposition of oscillations.
The result of the addition of oscillations depends on the amplitude, frequency and phase difference of the emerging oscillatory processes.
21. Coherent oscillations - oscillations with the same frequency and a constant phase difference in time.
22.coherent waves- waves of the same frequency or the same wavelength, the phase difference of which at a given point in space remains constant in time.
23.Wave interference- the phenomenon of an increase or decrease in the amplitude of the resulting wave when two or more coherent waves are superimposed.
a) . interference maximum conditions. Let waves from two coherent sources and meet at a point BUT(Fig.96).
Displacements of medium particles at a point BUT, caused by each wave separately, we write according to the wave equation in the form
where and , , 
- amplitudes and phases of oscillations caused by waves at a point BUT, and - point distances, 
- the difference between these distances or the difference in the course of the waves.
Due to the difference in the course of the waves, the second wave is delayed compared to the first. This means that the phase of oscillations in the first wave is ahead of the phase of oscillations in the second wave, i.e. . Their phase difference remains constant over time.
To the point BUT particles oscillated with maximum amplitude, the crests of both waves or their troughs should reach the point BUT simultaneously in identical phases or with a phase difference equal to , where n- integer, and - is the period of the sine and cosine functions,
Here , therefore, the condition of the interference maximum can be written in the form
Where is an integer.
So, when coherent waves are superimposed, the amplitude of the resulting oscillation is maximum if the difference in the path of the waves is equal to an integer number of wavelengths.
b) Interference minimum condition. The amplitude of the resulting oscillation at a point BUT is minimal if the crest and trough of two coherent waves arrive at this point simultaneously. This means that one hundred waves will come to this point in antiphase, i.e. their phase difference is equal to or 
, where is an integer.
The interference minimum condition is obtained by performing algebraic transformations:
Thus, the amplitude of oscillations when two coherent waves are superimposed is minimal if the difference in the path of the waves is equal to an odd number of half-waves.
24. Interference and the law of conservation of energy. When waves interfere in places of interference minima, the energy of the resulting oscillations is less than the energy of the interfering waves. But in the places of interference maxima, the energy of the resulting oscillations exceeds the sum of the energies of the interfering waves by as much as the energy has decreased in the places of interference minima.
When waves interfere, the energy of oscillations is redistributed in space, but the conservation law is strictly observed.
25.Wave diffraction- the phenomenon of wave wrapping around the obstacle, i.e. deviation from rectilinear wave propagation.
Diffraction is especially noticeable when the size of the obstacle is less than or comparable to the wavelength. Let a screen with a hole, the diameter of which is comparable with the wavelength (Fig. 97), be located on the path of propagation of a plane wave.
According to the Huygens principle, each point of the hole becomes a source of the same waves. The size of the hole is so small that all sources of secondary waves are located so close to each other that they can all be considered one point - one source of secondary waves.
If an obstacle is placed in the path of the wave, the size of which is comparable to the wavelength, then the edges, according to the Huygens principle, become a source of secondary waves. But the size of the gap is so small that its edges can be considered coinciding, i.e. the obstacle itself is a point source of secondary waves (Fig.97).
The phenomenon of diffraction is easily observed when waves propagate over the surface of water. When the wave reaches the thin, motionless stick, it becomes the source of the waves (Fig. 99).
25. Huygens-Fresnel principle. If the size of the hole significantly exceeds the wavelength, then the wave, passing through the hole, propagates in a straight line (Fig. 100).
If the size of the obstacle significantly exceeds the wavelength, then a shadow zone is formed behind the obstacle (Fig. 101). These experiments contradict Huygens' principle. The French physicist Fresnel supplemented Huygens' principle with the idea of the coherence of secondary waves. Each point at which a wave has arrived becomes a source of the same waves, i.e. secondary coherent waves. Therefore, waves are absent only in those places where the conditions of the interference minimum are satisfied for the secondary waves.
26. polarized wave is a transverse wave in which all particles oscillate in the same plane. If the free end of the filament oscillates in one plane, then a plane-polarized wave propagates along the filament. If the free end of the filament oscillates in different directions, then the wave propagating along the filament is not polarized. If an obstacle in the form of a narrow slit is placed on the path of an unpolarized wave, then after passing through the slit, the wave becomes polarized, because the slot passes the oscillations of the cord occurring along it.
If a second slot parallel to the first one is placed on the path of a polarized wave, then the wave will freely pass through it (Fig. 102).
If the second slot is placed at right angles to the first, then the wave will stop spreading. A device that separates vibrations occurring in one specific plane is called a polarizer (first slot). The device that determines the plane of polarization is called an analyzer.
27.Sound - this is the process of propagation of compressions and rarefactions in an elastic medium, for example, in a gas, liquid or metals. The propagation of compressions and rarefaction occurs as a result of the collision of molecules.
28. Sound volume is the force of the impact of a sound wave on the eardrum of the human ear, which is from sound pressure.
Sound pressure - This is the additional pressure that occurs in a gas or liquid when a sound wave propagates. Sound pressure depends on the amplitude of the oscillation of the sound source. If we make the tuning fork sound with a light blow, then we get one volume. But, if the tuning fork is hit harder, then the amplitude of its oscillations will increase and it will sound louder. Thus, the loudness of the sound is determined by the amplitude of the oscillation of the sound source, i.e. amplitude of sound pressure fluctuations.
29. Sound pitch determined by the oscillation frequency. The higher the frequency of the sound, the higher the tone.
Sound vibrations occurring according to the harmonic law are perceived as a musical tone. Usually sound is a complex sound, which is a combination of vibrations with close frequencies.
The root tone of a complex sound is the tone corresponding to the lowest frequency in the set of frequencies of the given sound. Tones corresponding to other frequencies of a complex sound are called overtones.
30. Sound timbre. Sounds with the same basic tone differ in timbre, which is determined by a set of overtones.
Each person has his own unique timbre. Therefore, we can always distinguish the voice of one person from the voice of another person, even if their fundamental tones are the same.
31.Ultrasound. The human ear perceives sounds whose frequencies are between 20 Hz and 20,000 Hz.
Sounds with frequencies above 20,000 Hz are called ultrasounds. Ultrasounds propagate in the form of narrow beams and are used in sonar and flaw detection. Ultrasound can determine the depth of the seabed and detect defects in various parts.
For example, if the rail has no cracks, then the ultrasound emitted from one end of the rail, reflected from its other end, will give only one echo. If there are cracks, then the ultrasound will be reflected from the cracks and the instruments will record several echoes. With the help of ultrasound, submarines, schools of fish are detected. The bat navigates in space with the help of ultrasound.
32. infrasound– sound with a frequency below 20 Hz. These sounds are perceived by some animals. Their source is often vibrations of the earth's crust during earthquakes.
33. Doppler effect- this is the dependence of the frequency of the perceived wave on the movement of the source or receiver of the waves.
Let a boat rest on the surface of the lake and waves beat against its side with a certain frequency. If the boat starts moving against the direction of wave propagation, then the frequency of wave impacts on the side of the boat will become greater. Moreover, the greater the speed of the boat, the greater the frequency of wave impacts on board. Conversely, when the boat moves in the direction of wave propagation, the frequency of impacts will become less. These considerations are easy to understand from Fig. 103.
The greater the speed of the oncoming movement, the less time is spent on passing the distance between the two nearest ridges, i.e. the shorter the period of the wave and the greater the frequency of the wave relative to the boat.
If the observer is motionless, but the source of waves is moving, then the frequency of the wave perceived by the observer depends on the movement of the source.
Let a heron walk along a shallow lake towards the observer. Every time she puts her foot in the water, waves ripple out from that spot. And each time the distance between the first and last waves decreases, i.e. at a smaller distance, a greater number of ridges and troughs fit. Therefore, for a stationary observer towards which the heron is walking, the frequency increases. And vice versa for a motionless observer who is in a diametrically opposite point at a greater distance, there are the same number of ridges and troughs. Therefore, for this observer, the frequency decreases (Fig. 104).
A mechanical or elastic wave is the process of propagation of oscillations in an elastic medium. For example, air begins to oscillate around a vibrating string or speaker cone - the string or speaker has become sources of a sound wave.
For the occurrence of a mechanical wave, two conditions must be met - the presence of a wave source (it can be any oscillating body) and an elastic medium (gas, liquid, solid).
Find out the cause of the wave. Why do the particles of the medium surrounding any oscillating body also come into oscillatory motion?
The simplest model of a one-dimensional elastic medium is a chain of balls connected by springs. Balls are models of molecules, the springs connecting them model the forces of interaction between molecules.
Suppose the first ball oscillates with a frequency ω. Spring 1-2 is deformed, an elastic force arises in it, which changes with frequency ω. Under the action of an external periodically changing force, the second ball begins to perform forced oscillations. Since forced oscillations always occur at the frequency of the external driving force, the oscillation frequency of the second ball will coincide with the oscillation frequency of the first. However, the forced oscillations of the second ball will occur with some phase delay relative to the external driving force. In other words, the second ball will begin to oscillate somewhat later than the first ball.
The vibrations of the second ball will cause a periodically changing deformation of the spring 2-3, which will make the third ball oscillate, and so on. Thus, all the balls in the chain will alternately be involved in an oscillatory motion with the oscillation frequency of the first ball.
Obviously, the cause of wave propagation in an elastic medium is the presence of interaction between molecules. The oscillation frequency of all particles in the wave is the same and coincides with the oscillation frequency of the wave source.
According to the nature of particle oscillations in a wave, waves are divided into transverse, longitudinal and surface waves.
AT longitudinal wave particles oscillate along the direction of wave propagation.
The propagation of a longitudinal wave is associated with the occurrence of tensile-compressive deformation in the medium. In the stretched areas of the medium, a decrease in the density of the substance is observed - rarefaction. In compressed areas of the medium, on the contrary, there is an increase in the density of the substance - the so-called thickening. For this reason, a longitudinal wave is a movement in space of areas of condensation and rarefaction.
Tensile-compressive deformation can occur in any elastic medium, so longitudinal waves can propagate in gases, liquids and solids. An example of a longitudinal wave is sound.
AT shear wave particles oscillate perpendicular to the direction of wave propagation.
The propagation of a transverse wave is associated with the occurrence of shear deformation in the medium. This kind of deformation can only exist in solids, so transverse waves can only propagate in solids. An example of a shear wave is the seismic S-wave.
surface waves occur at the interface between two media. Oscillating particles of the medium have both transverse, perpendicular to the surface, and longitudinal components of the displacement vector. During their oscillations, the particles of the medium describe elliptical trajectories in a plane perpendicular to the surface and passing through the direction of wave propagation. An example of surface waves are waves on the water surface and seismic L - waves.
The wave front is the locus of points reached by the wave process. The shape of the wave front can be different. The most common are plane, spherical and cylindrical waves.
Note that the wavefront is always located perpendicular direction of the wave! All points of the wavefront will begin to oscillate in one phase.
To characterize the wave process, the following quantities are introduced:
1. Wave frequencyν is the oscillation frequency of all the particles in the wave.
2. Wave amplitude A is the oscillation amplitude of the particles in the wave.
3. Wave speedυ is the distance over which the wave process (perturbation) propagates per unit time.
Pay attention - the speed of the wave and the speed of oscillation of particles in the wave are different concepts! The speed of a wave depends on two factors: the type of wave and the medium in which the wave propagates.
The general pattern is as follows: the speed of a longitudinal wave in a solid is greater than in liquids, and the speed in liquids, in turn, is greater than the speed of a wave in gases.
It is not difficult to understand the physical reason for this regularity. The cause of wave propagation is the interaction of molecules. Naturally, the perturbation propagates faster in the medium where the interaction of molecules is stronger.
In the same medium, the regularity is different - the speed of the longitudinal wave is greater than the speed of the transverse wave.
For example, the speed of a longitudinal wave in a solid, where E is the elastic modulus (Young's modulus) of the substance, ρ is the density of the substance.
Shear wave velocity in a solid, where N is the shear modulus. Since for all substances , then . One of the methods for determining the distance to the source of an earthquake is based on the difference in the velocities of longitudinal and transverse seismic waves.
The speed of a transverse wave in a stretched cord or string is determined by the tension force F and the mass per unit length μ:
4. Wavelengthλ is the minimum distance between points that oscillate equally.
For waves traveling on the surface of water, the wavelength is easily defined as the distance between two adjacent humps or adjacent depressions.
For a longitudinal wave, the wavelength can be found as the distance between two adjacent concentrations or rarefactions.
5. In the process of wave propagation, sections of the medium are involved in an oscillatory process. An oscillating medium, firstly, moves, therefore, it has kinetic energy. Secondly, the medium through which the wave runs is deformed, therefore, it has potential energy. It is easy to see that wave propagation is associated with the transfer of energy to unexcited parts of the medium. To characterize the energy transfer process, we introduce wave intensity I.
When in any place of a solid, liquid or gaseous medium, particle vibrations are excited, the result of the interaction of the atoms and molecules of the medium is the transmission of vibrations from one point to another with a finite speed.
Definition 1
Wave is the process of propagation of vibrations in the medium.
There are the following types of mechanical waves:
Definition 2
transverse wave: particles of the medium are displaced in a direction perpendicular to the direction of propagation of a mechanical wave.
Example: waves propagating along a string or a rubber band in tension (Figure 2.6.1);
Definition 3
Longitudinal wave: the particles of the medium are displaced in the direction of propagation of the mechanical wave.
Example: waves propagating in a gas or an elastic rod (Figure 2.6.2).
Interestingly, the waves on the liquid surface include both transverse and longitudinal components.
Remark 1
We point out an important clarification: when mechanical waves propagate, they transfer energy, form, but do not transfer mass, i.e. in both types of waves, there is no transfer of matter in the direction of wave propagation. While propagating, the particles of the medium oscillate around the equilibrium positions. In this case, as we have already said, waves transfer energy, namely, the energy of oscillations from one point of the medium to another.
Figure 2. 6. one . Propagation of a transverse wave along a rubber band in tension.
Figure 2. 6. 2. Propagation of a longitudinal wave along an elastic rod.
A characteristic feature of mechanical waves is their propagation in material media, unlike, for example, light waves, which can also propagate in a vacuum. For the occurrence of a mechanical wave impulse, a medium is needed that has the ability to store kinetic and potential energies: i.e. the medium must have inert and elastic properties. In real environments, these properties are distributed over the entire volume. For example, each small element of a solid body has mass and elasticity. The simplest one-dimensional model of such a body is a set of balls and springs (Figure 2.6.3).
Figure 2. 6. 3 . The simplest one-dimensional model of a rigid body.
In this model, inert and elastic properties are separated. The balls have mass m, and springs - stiffness k . Such a simple model makes it possible to describe the propagation of longitudinal and transverse mechanical waves in a solid. When a longitudinal wave propagates, the balls are displaced along the chain, and the springs are stretched or compressed, which is a stretching or compression deformation. If such deformation occurs in a liquid or gaseous medium, it is accompanied by compaction or rarefaction.
Remark 2
A distinctive feature of longitudinal waves is that they are able to propagate in any medium: solid, liquid and gaseous.
If in the specified model of a rigid body one or several balls receive a displacement perpendicular to the entire chain, we can speak of the occurrence of a shear deformation. Springs that have received deformation as a result of displacement will tend to return the displaced particles to the equilibrium position, and the nearest undisplaced particles will begin to be influenced by elastic forces tending to deflect these particles from the equilibrium position. The result will be the appearance of a transverse wave in the direction along the chain.
In a liquid or gaseous medium, elastic shear deformation does not occur. Displacement of one liquid or gas layer at some distance relative to the neighboring layer will not lead to the appearance of tangential forces at the boundary between the layers. The forces that act on the boundary of a liquid and a solid, as well as the forces between adjacent layers of a fluid, are always directed along the normal to the boundary - these are pressure forces. The same can be said about the gaseous medium.
Remark 3
Thus, the appearance of transverse waves is impossible in liquid or gaseous media.
In terms of practical applications, simple harmonic or sine waves are of particular interest. They are characterized by particle oscillation amplitude A, frequency f and wavelength λ. Sinusoidal waves propagate in homogeneous media with some constant speed υ.
Let us write an expression showing the dependence of the displacement y (x, t) of the particles of the medium from the equilibrium position in a sinusoidal wave on the coordinate x on the O X axis along which the wave propagates, and on time t:
y (x, t) = A cos ω t - x υ = A cos ω t - k x .
In the above expression, k = ω υ is the so-called wave number, and ω = 2 π f is the circular frequency.
Figure 2. 6. 4 shows "snapshots" of a shear wave at time t and t + Δt. During the time interval Δ t the wave moves along the axis O X at a distance υ Δ t . Such waves are called traveling waves.
Figure 2. 6. 4 . "Snapshots" of a traveling sine wave at a moment in time t and t + ∆t.
Definition 4
Wavelengthλ is the distance between two adjacent points on the axis O X oscillating in the same phases.
The distance, the value of which is the wavelength λ, the wave travels in a period T. Thus, the formula for the wavelength is: λ = υ T, where υ is the wave propagation speed.
With the passage of time t, the coordinate changes x any point on the graph displaying the wave process (for example, point A in Figure 2 . 6 . 4), while the value of the expression ω t - k x remains unchanged. After a time Δ t point A will move along the axis O X some distance Δ x = υ Δ t . Thus:
ω t - k x = ω (t + ∆ t) - k (x + ∆ x) = c o n s t or ω ∆ t = k ∆ x .
From this expression it follows:
υ = ∆ x ∆ t = ω k or k = 2 π λ = ω υ .
It becomes obvious that a traveling sinusoidal wave has a double periodicity - in time and space. The time period is equal to the oscillation period T of the particles of the medium, and the spatial period is equal to the wavelength λ.
Definition 5
wave number k = 2 π λ is the spatial analogue of the circular frequency ω = - 2 π T .
Let us emphasize that the equation y (x, t) = A cos ω t + k x is a description of a sinusoidal wave propagating in the direction opposite to the direction of the axis O X, with the speed υ = - ω k .
When a traveling wave propagates, all particles of the medium oscillate harmonically with a certain frequency ω. This means that, as in a simple oscillatory process, the average potential energy, which is the reserve of a certain volume of the medium, is the average kinetic energy in the same volume, proportional to the square of the oscillation amplitude.
Remark 4
From the foregoing, we can conclude that when a traveling wave propagates, an energy flux appears that is proportional to the speed of the wave and the square of its amplitude.
Traveling waves move in a medium with certain velocities, which depend on the type of wave, inert and elastic properties of the medium.
The speed with which transverse waves propagate in a stretched string or rubber band depends on the linear mass μ (or mass per unit length) and the tension force T:
The speed with which longitudinal waves propagate in an infinite medium is calculated with the participation of such quantities as the density of the medium ρ (or the mass per unit volume) and the bulk modulus B(equal to the coefficient of proportionality between the change in pressure Δ p and the relative change in volume Δ V V , taken with the opposite sign):
∆ p = - B ∆ V V .
Thus, the propagation velocity of longitudinal waves in an infinite medium is determined by the formula:
Example 1
At a temperature of 20 ° C, the propagation velocity of longitudinal waves in water is υ ≈ 1480 m / s, in various grades of steel υ ≈ 5 - 6 km / s.
If we are talking about longitudinal waves propagating in elastic rods, the formula for the wave velocity contains not the compression modulus, but Young's modulus:
For steel difference E from B insignificantly, but for other materials it can be 20 - 30% or more.
Figure 2. 6. 5 . Model of longitudinal and transverse waves.
Suppose that a mechanical wave propagating in a certain medium encounters some obstacle on its way: in this case, the nature of its behavior will change dramatically. For example, at the interface between two media with different mechanical properties, the wave is partially reflected, and partially penetrates into the second medium. A wave running along a rubber band or string will be reflected from the fixed end, and a counter wave will arise. If both ends of the string are fixed, complex oscillations will appear, which are the result of the superimposition (superposition) of two waves propagating in opposite directions and experiencing reflections and re-reflections at the ends. This is how the strings of all stringed musical instruments “work”, fixed at both ends. A similar process occurs with the sound of wind instruments, in particular, organ pipes.
If the waves propagating along the string in opposite directions have a sinusoidal shape, then under certain conditions they form a standing wave.
Suppose a string of length l is fixed in such a way that one of its ends is located at the point x \u003d 0, and the other at the point x 1 \u003d L (Figure 2.6.6). There is tension in the string T.
Picture 2 . 6 . 6 . The emergence of a standing wave in a string fixed at both ends.
Two waves with the same frequency run simultaneously along the string in opposite directions:
- y 1 (x, t) = A cos (ω t + k x) is a wave propagating from right to left;
- y 2 (x, t) = A cos (ω t - k x) is a wave propagating from left to right.
The point x = 0 is one of the fixed ends of the string: at this point the incident wave y 1 creates a wave y 2 as a result of reflection. Reflecting from the fixed end, the reflected wave enters antiphase with the incident one. In accordance with the principle of superposition (which is an experimental fact), the vibrations created by counterpropagating waves at all points of the string are summed up. It follows from the above that the final fluctuation at each point is defined as the sum of the fluctuations caused by the waves y 1 and y 2 separately. Thus:
y \u003d y 1 (x, t) + y 2 (x, t) \u003d (- 2 A sin ω t) sin k x.
The above expression is a description of a standing wave. Let us introduce some concepts applicable to such a phenomenon as a standing wave.
Definition 6
Knots are points of immobility in a standing wave.
antinodes– points located between the nodes and oscillating with the maximum amplitude.
If we follow these definitions, for a standing wave to arise, both fixed ends of the string must be nodes. The above formula meets this condition at the left end (x = 0) . For the condition to be satisfied at the right end (x = L) , it is necessary that k L = n π , where n is any integer. From what has been said, we can conclude that a standing wave does not always appear in a string, but only when the length L string is equal to an integer number of half-wavelengths:
l = n λ n 2 or λ n = 2 l n (n = 1 , 2 , 3 , . . .) .
The set of values λ n of wavelengths corresponds to the set of possible frequencies f
f n = υ λ n = n υ 2 l = n f 1 .
In this notation, υ = T μ is the speed with which transverse waves propagate along the string.
Definition 7
Each of the frequencies f n and the type of string vibration associated with it is called a normal mode. The lowest frequency f 1 is called the fundamental frequency, all others (f 2 , f 3 , ...) are called harmonics.
Figure 2. 6. 6 illustrates the normal mode for n = 2.
A standing wave has no energy flow. The energy of vibrations, "locked" in the segment of the string between two neighboring nodes, is not transferred to the rest of the string. In each such segment, a periodic (twice per period) T) conversion of kinetic energy into potential energy and vice versa, similar to an ordinary oscillatory system. However, there is a difference here: if a weight on a spring or a pendulum has a single natural frequency f 0 = ω 0 2 π , then the string is characterized by the presence of an infinite number of natural (resonant) frequencies f n . Figure 2. 6. 7 shows several variants of standing waves in a string fixed at both ends.
Figure 2. 6. 7. The first five normal vibration modes of a string fixed at both ends.
According to the principle of superposition, standing waves of different types (with different values n) are able to simultaneously be present in the vibrations of the string.
Figure 2. 6. eight . Model of normal modes of a string.
If you notice a mistake in the text, please highlight it and press Ctrl+Enter
