Waves in a discrete chain. Polarization of waves. The speed of the transverse wave. The density of the kinetic energy of running water.

Waves.

For a long time, the visual image of the wave has always been associated with the waves on the surface of the water. But water waves are a much more complex phenomenon than many other wave processes are such as the spread of sound in a homogeneous isotropic medium. Therefore, it is natural to begin to study the wave movement not with waves on water, but with simpler cases.

Waves in a discrete chain.

The easiest way to imagine a wave spreading on an infinite chain of bound pendulums (Fig. 192). From the infinite chain, we begin to consider the wave spreading in one direction, and do not think about it possible to reflect it from the end of the chain.

Fig. 192. Wave in a chain of bound pendulums If the pendulum, which is at the beginning of the chain, lead to a harmonic oscillatory movement with some frequency of CO and amplitude A, then the oscillatory movement will spread over the chain. This spread of oscillations from one place to another is called a wave process or wave. In the absence of attenuation, any other pendulum in the chain will repeat the forced oscillations of the first pendulum with a certain phase lag. This delay is due to the fact that the spread of oscillations on the chain occurs at a certain final speed. The speed of propagation of oscillations and depends on the stiffness of the springs connecting pendulum, on how strong the connection between the pendulums is. If the first pendulum in the chain is moving on a specific law, its mixing from the equilibrium position is a given time function, then the displacement of the pendulum, separated from the beginning of the chain to the distance, at any time will be exactly the same as the mixture of the first pendulum in the earlier time will be Describe a function. Let with the harmonic oscillations of the first pendulum, its displacement from the equilibrium position is given by expression. Each of the chain pendulums is characterized by the distance to which he will be from the beginning of the chain. Therefore, its displacement from the equilibrium position during the passage of the wave naturally designate through. Then, in accordance with what was said above, we have the wave described by the equation is called monochromatic. A characteristic feature of a monochromatic wave is that each of the pendulums makes a sinusoidal oscillation of a certain frequency. The spread of the wave along the chain of pendulum is accompanied by the transfer of energy and momentum. But no transfer of mass is happening at the same time: every pendulum, making oscillations near the position of the equilibrium, on average remains in place.

Polarization of waves.Depending on whether the pendulum oscillations occur in the waves of different polarization. If the oscillations of the pendulum occur along the direction of the wave propagation, as in Fig. 192, the wave is called longitudinal, if the cross is cross. Typically, waves of different polarization are distributed with different speeds. The considered chain of bound pendulums is an example of a mechanical system with concentrated parameters.

Another example of a system with concentrated parameters in which waves can be distributed is a chain of balls associated with light springs (Fig. 193). In such a system, the inert properties focus on the balls, and the elastic springs. When the wave is propagated, the kinetic energy of oscillations is localized on the balls, and the potential on the springs. It is easy to figure out that such a chain of balls connected by balls can be considered as a model of a one-dimensional system with distributed parameters, such as an elastic string. In the string, each element of length has simultaneously mass, inert properties, and rigidity, elastic properties. Waves in a stretched string. Consider a transverse monochromatic wave propagating in an infinite stretched string. The preliminary tension of the string is necessary because a non-tight flexible string, unlike a solid rod, has elasticity only with respect to stretching deformation, but not compression. The monochromatic wave in the string is described by the same expression as the wave in the chain of pendulums. However, now the role of a separate pendulum plays each element of the string, so the variable in the equation, which characterizes the equilibrium position of the pendulum, takes continuous values. The displacement of any element of the string from the equilibrium position during the passage of the wave is the function of two times and the equilibrium position of this element. If in the formula to submit to consider a specific element of the string, then the function with a fixed manifests the dedicated element of the string depending on the time. This mixing is a harmonic oscillation with a frequency of CO and amplitude. The initial phase of the oscillations of this element of the string depends on its equilibrium position. All elements of the string when passing the monochromatic wave make harmonic oscillations of the same frequency and amplitude, but differing in phase.

Wavelength.

If in the formula is fixed, consider the entire string into the same point in time, then the function with a fixed gives an instantaneous picture of the displacements of all the elements of the string as it were for an instant photo of the wave. On this "Photos" we will see the frozen sinusoid (Fig. 194). The period of this sinusoid, the distance between adjacent humps or depadies is called a wavelength. From the formula, it can be found that the wavelength is associated with the frequency of C and the speed of the wave and the ratio of the oscillation period. The picture of the wave propagation can be imagined if this "frozen" sinusoid lead in motion along the axis at speeds.

Fig. 194. Displacement of different points of the string in the same point in time. Fig. 195. Pictures of shifts of the string points at the time of time. Two consecutive "instant photos" of the waves at the time of time are shown in Fig. 195. It can be seen that the wavelength is equal to the distance passing by any hump over a period of oscillations in accordance with the formula.

The speed of the transverse wave.

We define the rate of propagation of monochromatic transverse wave in the string. We assume that the amplitude is small compared with the wavelength. Let the wave run to the right at speed and. Let us turn into a new reference system moving along the string at a speed equal to the wave velocity and. This reference system is also inertial and, consequently, Newton's laws are fair. From this reference system, the wave seems to be frozen sinusoid, and the string substance slides along this sinusoid to the left: any pre-painted element of the string will seem running along the sinusoids to the left at speed.

Fig. 196. To calculate the speed of spreading the wave in the string. Consider in this reference system an element of a string of length, which is much less than the wavelength, at the moment when it is on the crest of sinusoids (Fig. 196). Apply to this element the second law Newton. Forces acting on the element on the side of the neighboring sections of the string are shown in the highlighted circle in Fig. 196. Since a transverse wave is considered, in which the displacements of the elements of the string are perpendicular to the direction of propagation of the wave, the horizontal component of the tension force. Stories constantly along the whole string. Since the length of the section under consideration, the directions of the tension forces acting on the dedicated element are almost horizontal, and their module can be considered equal. Equality of these forces is directed down and equal. The rate of the element under consideration is equal to the left, and the small portion of its sinusoidal trajectory near the hump can be considered an arc of the circle of the radius. Therefore, the acceleration of this element string is directed down and equal. The mass of the element of the string can be represented as the density of the material of the string, and the cross-sectional area, which, in view of the smallness of the deformations, can be considered as in the absence of a wave. Based on the second law of Newton. This is the desired transition rate of the transverse monochromatic wave of a small amplitude in a stretched string. It can be seen that it depends only on the mechanical stress of the stretched string and its density and does not depend on the amplitude and wavelength. This means that transverse waves of any lengths spread in a stretched string at the same speed. If the string is simultaneously distributed, for example, two monochromatic waves with the same amplitudes and close frequencies of CO, then the "instant photos" of these monochromatic waves and the resulting waves will be viewed in Fig. 197.

Where the hump of one wave coincides with the hump of another, in the resulting wave the mixture as possible. Since the sinusoids corresponding to individual waves run along the z axis at the same speed and, then the resulting curve runs at the same speed without changing its form. It turns out that this is true for the wave perturbation of any form: transverse waves of arbitrary species are distributed in a stretched string without changing their form. On dispersion of waves. If the speed of propagation of monochromatic waves does not depend on the wave or frequency length, they say that there is no dispersion. The preservation of the form of any wave during its distribution is a consequence of the lack of dispersion. Dispersion is missing for waves of any kind propagating in solid elastic media. This circumstance makes it very easy to find the speed of longitudinal waves.

The speed of longitudinal waves.

Consider, for example, a long elastic rod area in which a longitudinal perturbation with a steep front front is distributed. Let this front at some time, moving at speed, reached the point with the coordinate to the right of the front all the points of the rod still rest. After a period of time, the front will move to the right to the distance (Fig. 198). Within this layer, all particles are moving at the same speed. After this period of time, the rod particles, which were at the time on the front of the waves, move along the rod to the distance. Applicable to the mass of the rod involved in the wave process, the law of the impulse conservation law. Existing on the mass express through the deformation of the element of the rod using the bike law. The length of the selected element of the rod is equal, and the change in its length under the action of force is equal. Therefore, with the help of finding this value in, we obtain the speed of longitudinal sound waves in the elastic rod depends only on the Jung module and density. It is easy to make sure that in most metals this speed is approximately. The speed of longitudinal waves in an elastic medium is always more transverse. Compare, for example, the rates of longitudinal and transverse waves and (in a stretched flexible string. Since with small deformations, the elastic constants do not depend on the applied forces, the speed of longitudinal waves in the stretched string does not depend on its preliminary tension and is determined by the formula. In order to compare this The velocity of the previously found transverse wave-wave velocity I will express the strength of the string in the formula, through the relative deformation of the string caused by this preliminary tension. Substituting the value in the formula, we obtain the speed of the transverse waves in the stretched string UT is significantly less than the speed of longitudinal waves, so As a relative stretching string E a lot less than a unit. The energy of the wave. When the wave is propagated, energy is transmitted without transfer of the substance. The wave energy in the elastic medium consists of the kinetic energy of the fluctuations of the substance particles and from the potential energy of the elastic environmental deformation. Consider, for example, Wave in an elastic rod. At a fixed point in time, the kinetic energy is unevenly distributed over the volume of the rod, since some of the rod points at this moment rest, others, on the contrary, are moving at maximum speed. The same is true for potential energy, since at this moment some elements of the rod are not deformed, others are deformed as much as possible. Therefore, when considering the energy of the wave, it is natural to introduce the density of kinetic and potential energies. The density of the energy of the wave at each point of the medium does not remain constant, and periodically changes during the passage of the wave: the energy spreads along with the wave.

Why, when propagating a transverse wave in a stretched string, the longitudinal component of the strength of the string of the string is the same along the entire string and does not change during the passage of the wave?

What are monochromatic waves? How is the length of monochromatic wave connected with the frequency and speed of distribution? In which cases are the waves are called longitudinal and in what transverse? Show using high-quality reasoning that the speed of the wave propagation is the greater, the more power seeking to return the perturbed area of \u200b\u200bthe environment to the equilibrium state, and the smaller the more the inertness of this site. What characteristics of the medium is determined by the speed of longitudinal waves and the speed of transverse waves? How are the speed of such waves in the stretched string?

The density of the kinetic energy of a running wave.

Consider the density of kinetic energy in a monochromatic elastic wave described by the equation. We highlight the small element in the rod between the planes such that its length in the undeformed state is much less than the wavelength. Then the rates of all the rod particles in this element can be considered the same when the wave propagation can be considered the same. With the help of formulas, we find the speed, considering as a function of time and counting the value characterizing the position of the considered element of the rod fixed. The mass of the selected element of the rod, so its kinetic energy at the time of time is using the expression we find the density of kinetic energy at the point at the time of time. Potential energy density. Let us turn to the calculation of the density of the potential energy of the wave. Since the length of the selected element of the rod is small compared with the wavelength, the caused wave of deformation of this element can be considered homogeneous. Therefore, the potential energy of deformation can be written in the form of elongation of the considered element of the rod caused by the passing wave. To find this elongation, it is necessary to consider the position of the planes that limit the dedicated element at some point in time. The instantaneous position of any plane, the equilibrium position of which is characterized by the coordinate, is determined by the function considered as a function with a fixed one. Therefore, the elongation of the considered element of the rod, as can be seen from fig. 199, equal to the relative elongation of this element. If in this expression, go to the limit when it turns into a derivative function in a variable with a fixed. With the help of the formula we get

Fig. 199. To the calculation of the relative elongation of the rod, the expression for potential energy takes the form and the density of potential energy at the point at the time of time is the energy of the running wave. Since the speed of propagation of longitudinal waves, the right parts in the formulas coincide. This means that in the running longitudinal elastic wave of the density of kinetic and potential energies is equal at any time at any point of the medium. The dependence of the density of the energy of the wave from the coordinate at a fixed point in time is shown in Fig. 200. We draw attention to the fact that, unlike localized oscillations (oscillator), where kinetic and potential energy varies in antiphase, in the running wave of oscillations of kinetic and potential energy occur in the same phase. The kinetic and potential energy at each point of the medium simultaneously reaches the maximum values \u200b\u200band simultaneously appeal to zero. Equality of instantaneous values \u200b\u200bof the density of kinetic and potential energies is the total property of traveling waves propagating in a certain direction. You can make sure that it is true for transverse waves in a stretched flexible string. Fig. 200. Displacement of the medium particles and energy density in the running wave

So far, we have considered waves spreading in a system that has an infinite length of only one direction: in the chain of pendulum, in the string, in the rod. But the waves can spread in a medium having endless dimensions in all directions. In such a solid medium, the waves are of different types depending on the method of their excitation. Flat wave. If, for example, the wave occurs as a result of harmonic oscillations of an infinite plane, then in a homogeneous medium it spreads in the direction perpendicular to this plane. In such a wave, the displacement of all points of the medium lying on any plane perpendicular to the direction of propagation occurs exactly the same. If the wave energy is absorbed in the medium, the amplitude of the oscillations of the points of the medium is the same everywhere and their displacement is given by the formula. Such a wave is called flat.

Spherical wave.

The wave of another type of spherical creates in a homogeneous isotropic elastic medium a pulsating ball. Such a wave applies at the same speed in all directions. Its wave surfaces, the surface of the constant phase, are concentric spheres. In the absence of energy absorption in the medium it is easy to determine the dependence of the amplitude of the spherical wave from the distance to the center. Since the flow of the wave energy proportional to the square amplitude is the same through any sphere, the wave amplitude decreases inversely in proportion to the distance from the center. The equation of a longitudinal spherical wave has the form where the amplitude of oscillations at a distance from the center of the wave.

How depends on the transfaying energy of the energy from the frequency and from the amplitude of the wave?

What is a flat wave? Spherical wave? How depend on the distance amplitude of flat and spherical waves?

Explain why in the running wave kinetic energy and potential energy change in the same phase.

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. Distribution rate.

13. The dependence of the wave velocity from the properties of the medium.

14. Guiggens principle.

15. Reflection and refraction of waves.

16. The law of reflection of waves.

17. The law of refraction of the waves.

18. Flat wave equation.

19. Energy and intensity of the wave.

20. The principle of superposition.

21. Coherent oscillations.

22. Coherent waves.

23. Wave interference. a) The condition of the interference maximum, b) the condition of the interference minimum.

24. Interference and the law of conservation of energy.

25. Diffraction of waves.

26. Guygens - Fresnel principle.

27. Polarized wave.

29. Sound volume.

30. Sound tone height.

31. Sound timbre.

32. Ultrasound.

33. Infrasubuk.

34. Doppler effect.

1.Wave -this is the process of distribution of oscillations of any physical value in space. For example, sound waves in gases or liquids are the propagation of pressure oscillations and density in these environments. The electromagnetic wave is the process of distribution in the space of oscillations of electric magnetic fields.

Energy and impulse can be transferred in space by transferring a substance. Any moving body has kinetic energy. Consequently, it transfers kinetic energy, carrying a substance. This body being heated, moving in space tolerates energy thermal, carrying substance.

Particles of elastic medium are interconnected. Indignation, i.e. Deviations from the equilibrium position of one particle are transmitted to neighboring particles, i.e. Energy and impulse are transmitted from one particle to neighboring particles, with each particle remains near its equilibrium position. Thus, the energy and impulse are transmitted along the chain from one particle to the other and the transfer of the substance does not occur.

So, the wave process is the process of transferring energy and momentum in space without transferring a substance.

2. Mechanical wave or elastic wave - indignation (oscillation) spreading in an elastic environment. An elastic medium in which mechanical waves are spread is air, water, metal wood and other elastic substances. Elastic waves are called sound waves.

3. Source of mechanical waves - The body that makes the oscillatory movement while in an elastic medium, such as oscillating tubes, strings, voice ligaments.

4. Point Source Waves -the source of the wave, the sizes of which can be neglected compared with the distance to which the wave applies.

5. Transverse wave -a wave in which particles of the medium fluctuate in the direction perpendicular to the direction of propagation of the wave. For example, waves on the surface of the water - transverse waves, because The oscillations of water particles occur in the direction perpendicular to the surface of the water, and the wave spreads over the surface of the water. The transverse wave spreads along the cord, one end of which is fixed, the other performs oscillations in the vertical plane.

The transverse wave can be distributed only along the border of the spirit of different environments.

6. Longitudinal wave -a wave in which oscillations occur in the direction of the wave propagation. The longitudinal wave occurs in a long spiral spring if one end is subjected to periodic perturbations directed along the spring. Elastic wave, running along the spring, is a propagating sequence of compression and stretching (Fig. 88)

The longitudinal wave can be distributed only inside the elastic medium, for example, in the air, in water. In solid bodies and in liquids, both transverse and longitudinal waves can be distributed simultaneously. The solid body and fluid are always limited to the surface - the surface of the section of two media. For example, if the steel rod is to hit the elder with a hammer, then elastic deformation will begin to spread. A transverse wave will run through the surface of the rod, and the wave of longitudinal (compression and permitting medium) will be spread inside it (Fig.89).

7. Front Waves (Wave Surface)- geometric location of points, fluctuating in the same phases. On the wave surface of the phase of the oscillating points in the considered point in time they have the same value. If you throw a stone into a relaxed lake, then transverse waves in the form of a circle will begin along the surface of the lake from the place of its fall, with the center in the place of falling stone. In this example, the front of the wave is a circle.

In a spherical wave, the wave front is a sphere. Such waves are generated by point sources.

At very long distances from the source, you can neglected the curvature front and consider the wave front flat. In this case, the wave is called flat.

8. Ray is straightdelimal to the wave surface. In a spherical wave, rays are directed along the radii of spheres from the center, where the source of the waves is located (Fig. 90).

In a flat wavelength, rays are directed perpendicular to the surface of the front (Fig. 91).

9. Periodic waves. Arguing about the waves, we meant a single outrage that spreads in space.

If the source of the waves makes continuous fluctuations, then in the medium running one after one elastic wave. Such waves are called periodic.

10. Harmonic wave - A wave generated by harmonic oscillations. If the source of the wave performs harmonic oscillations, it generates harmonic waves - the waves in which particles fluctuate in harmonic law.

11. Wavelength.Let the harmonic wave spread along the OX axis, and the oscillations in it occur in the direction of the Oy axis. This wave is transverse and can be depicted in the form of sinusoids (Fig. 92).

Such a wave can be obtained, causing oscillations in the vertical plane of the free end of the cord.

Wavelength call the distance between the two closest points A and B,fluctuating in the same phases (Fig. 92).

12. Wave propagation rate - The physical value is numerically equal to the rate of propagation of oscillations in space. From fig. 92 It follows that the time for which the oscillation extends from the point to the point BUT to the point IN. The distance of the wavelength is equal to the oscillation period. Therefore, the wave propagation rate is equal

13. The dependence of the wave propagation rate from the properties of the environment. The frequency of oscillations during the occurrence of the wave depends only on the properties of the source of the wave and does not depend on the properties of the medium. The wave propagation rate depends on the properties of the medium. Therefore, the wavelength varies when the border crossing the sections of two different environments. The wave speed depends on the connection between atoms and molecules of the medium. The relationship between atoms and molecules in liquids and solid bodies is much more rigid than in gases. Therefore, the velocities of sound waves in liquids and solid bodies are much more than in gases. In the air, the sound speed under normal conditions is 340, in water 1500, and in steel 6000.

The average speed of thermal motion of molecules in gases with a decrease in temperature decreases and as a result, the speed of propagation of the wave in the gases is reduced. In the medium, more dense, and therefore more inert, the speed of the wave is less. If the sound propagates in the air, its speed depends on the air density. Where the air density is more, there is less sound. And on the contrary, where the air density is less there the speed of sound is greater. As a consequence, when the sound is spreading the front, the wave is distorted. Above the swamp or over the lake especially in the evening, the density of air near the surface of water vapors is greater than at some height. Therefore, the speed of sound near the surface of the water is less than at some height. As a result, the wave front turns in such a way that the upper part of the front is increasingly bent towards the surface of the lake. It turns out that the energy of the wave going along the surface of the lake and the energy of the waves going at an angle to the surface of the lake is folded. Therefore, in the evening, the sound spreads well on the lake. Even a quiet mortgage can be heard, standing on the opposite shore.

14. Guygens principle - Each point of the surface that the wave reached at the moment is a source of secondary waves. After spending the surface tangent to the fronts of all secondary waves, we get the front of the waves in the next point in time.

Consider for example the wave spreading on the surface of the water from the point ABOUT (Fig. 93) Let at the time of time t. The front had the shape of the circle of the radius R. with center at point ABOUT. The next point in time, each secondary wave will have a front in the form of a radius circle, where V. - The speed of the wave propagation. Having a surface tangent to the fronts of secondary waves, we obtain the front of the wave at the time of time (Fig. 93)

If the wave spreads in a solid medium, the front of the wave is a sphere.

15. Reflection and refraction of waves. When the wave falls on the surface of the section of two different environments, each point of this surface according to the Guigens principle becomes a source of secondary waves propagating on both sides of Rada. Therefore, when switching the boundary of the section of two media environments, the wave is partially reflected and partially passes through this surface. Because The mediums are different, the speed of the waves in them is different. Therefore, when switching the boundary of the section of two environments, the direction of propagation of the ox is changed, i.e. There is a refraction of the wave. Consider on the basis of the principle of the Guygens process and the laws of reflection and refraction of full.

16. Law of reflection of waves. Let a flat wave fall on the flat surface of the section of two different environments. Select a plot between the two rays and (Fig.94)

The angle of the fall is an angle - between the beam of the incident and perpendicular to the surface of the section at the point of the fall.

The reflection angle is the angle between the beam reflected and perpendicular to the surface of the section at the point of the fall.

At the moment when the beam reaches the surface of the section at the point, this point will become the source of secondary waves. The front of the waves at this point marked by a straight line AC(Fig. 94). Consequently, the beam still has to go to the surface of the section St.. Let the beam goes through this path during the time. Falling and reflected rays apply on one side of the surface of the section therefore their speed is the same and equal V. Then.

During the secondary wave from the point BUTwalk the way. Hence . Rectangular triangles and are equal, because - General hypotenuse and cathets. From the equality of triangles and should be equal to the corners. But, i.e. .

Now we formulate the law of reflection of the waves: ray falling, ray reflected , perpendictor to the border of the section of two environments, the duct dwelled at the point in the same plane; The angle of the fall is equal to the corner of the reflection.

17. Law of refraction of waves. Let a flat wave passes through the flat boundary of the section of the two environments. And The angle of the fall is different from zero (Fig. 95).

The refractive angle is the angle between the ray refracted and perpendicular to the interface, restored at the fall point.

Denote and the speed of propagation of waves in media 1 and 2. At the moment when the beam reaches the boundary of the partition at the point BUT This dot will become a source of waves propagating in the second environment - the beam, and the beam still has to go through the surface of Rada. Let - the time for which the beam passes the way Sv,then. During the same time in the second medium, the path will pass. Because , then.

Triangles and rectangular with general hypotenuse, and \u003d, as angles with mutually perpendicular sides. For corners and write the following equalities

Considering that,, we get

Now we formulate the law of refraction of the waves: The ray falling, the beam is refracted and perpendicular to the border of the section of two environments, subscribed at the point of the fall, lie in the same plane; The ratio of the sine angle of falling to the sinus of the refractive angle is the value constant for two data environments and is called relative refractive index for two media data.

18. Equation of a flat wave.Medium particles in distances S. From the source of the waves begin to fluctuate only when the wave reaches it. If a V.there is a wave propagation rate, the oscillations will begin late for the time

If the source of the wave fluctuates on the harmonic law, then for a particle that is at a distance S. from the source, the law of oscillations write in the form

We introduce a value called a wave number. It shows how many wavelengths fit at a distance equal to units of length. Now the law oscillations of the particle medium at a distance S. from the source we write in the form

This equation determines the displacement of the oscillating point, as the function of the time and distance from the source of the waves and is called the equation of a flat wave.

19. Wave Energy and Intensity. Each particle that has reached the wave hesitates and therefore possesses energy. Let a wave with an amplitude applies in some volume of the elastic medium BUTand cyclic frequency. This means that the average energy of oscillations in this volume is equal to

Where m -the mass of the highlighted volume of the medium.

The average energy density (average by volume) is the energy of the wave per unit of the volume of the medium

Where the density of the medium.

Intensity of the wave - The physical value, numerically equal to the energy that transfers the wave per unit of time through the unit of the plane area perpendicular to the direction of the wave propagation (through the unit of the wave front area), i.e.

The average power of the wave is the average total energy, carrying the wave per unit of time through the surface with an area S. . The average power of the wave is obtained, multiplies the intensity of the wave to the area S.

20.The principle of superposition (overlay). If the elastic medium spreads the waves from two or more sources, then as observations show, the waves pass one through another completely without affecting each other. In other words, the waves do not interact with each other. This is explained by the fact that within the limits of elastic deformation of compression and stretching in one direction in no way affect the elastic properties in other areas.

Thus, each point of the medium where two and more waves come accepted in oscillations caused by each wave. In this case, the resulting displacement of the medium particle at any time is equal to the geometric sum of displacements caused by each of the developing vibrational processes. This is the essence of the principle of superposition or the imposition of oscillations.

The result of the addition of oscillations depends on the amplitude, frequency and difference of phases of the developing vibrational processes.

21. Coherent oscillations - oscillations with the same frequency and constant in time difference phase.

22.Coherent waves - waves of the same frequency or the same wavelength, the phase difference in this point of space remains constant in time.

23.Interference waves - phenomenon of increase or reduce the amplitude of the resulting wave when two or more coherent waves are applied.

but) . Conditions of the interference maximum. Let waves from two coherent sources and are found at the point BUT (Fig.96).

Displacement of particles medium at point BUTcaused by each wave separately write according to the wave equation

Where and, - amplitudes and phases of oscillations caused by waves at the point BUT, and - distance distances - the difference is these distances or the difference in the way of the waves.

Due to the difference in the movement of waves, the second wave is delayed compared to the first. This means that the oscillation phase in the first wave is ahead of the oscillation phase in the second wave, i.e. . Their phase difference remains constant in time.

In order to at the point BUTparticles made oscillations with a maximum amplitude, crests of both waves or their depressions should reach the point BUT simultaneously in the same phases or a phase difference equal to where n - an integer, and - there is a period of functions of sinus and cosine,

Here, therefore, the condition of the interference maximum will write down in the form

Where is an integer.

So, when applying coherent waves, the amplitude of the resulting oscillation is maximal if the difference of the waves is equal to an integer number of wavelengths.

b) The condition of the interference minimum. The amplitude of the resulting oscillation at the point BUT It is minimal if the comb and collar of two coherent waves will come to this point at the same time. This means one hundred waves will come to this point in antiphase, i.e. The difference of their phases is equal to or, where the integer.

The condition of the interference minimum is obtained by conducting algebraic transformations:

Thus, the amplitude of oscillations when applying two coherent waves is minimal if the difference of the waves is equal to the odd number of half-breaves.

24. Interference and energy conservation law.In the interference of the waves in the places of interference minima, the energy of the resulting oscillations is less than the energy of interfering waves. But in the places of interference maxima, the energy of the resulting oscillations exceeds the amount of the energy of interfering waves as much as the energy has decreased in the places of interference minima.

In the interference waves, the energy of oscillations is redistributed in space, but the conservation law is strictly executed.

25.Diffraction of waves - the phenomenon of the envelope of the wave of the prefirmation, i.e. Deviation from rectilinear wave propagation.

The diffraction is particularly noticeable in the case when the size of the tenure is less than the wavelength or comparable to it. Suppose on the path of propagation of a flat wave there is a screen with a hole, the diameter of which is comparable to a wavelength (Fig. 97).

According to the principle of Guigens, each point of the opening becomes the source of the same waves. The size of the opening is so small that all sources of secondary waves are located as close to each other, that everything can be considered one point - one source of secondary waves.

If on the path of the wave put a rest, the size of which is comparable to the wavelength, then the edges on the principle of the Guiggens become the source of secondary waves. But the dimensions of the tenant are so small that the edges can be considered coinciding, i.e. The presence itself is a point source of secondary waves (Fig. 97).

The diffraction phenomenon is easily observed when the waves are propagated along the water surface. When the wave reaches a thin, motionless stick, it becomes the source of the waves (Fig. 99).

25. Guiggens-Fresnel principle. If the dimensions are constantly exceeding the wavelength, the wave, passing the hole spreads straightforwardly (Fig. 100).

If the dimensions dimensions significantly exceed the wavelength, then the shadow zone is formed for the prefiguration (Fig. 101). These experiments contradict the Guigens principle. French physicist Frenel supplemented the Guygens principle by the presentation of the coherers of secondary waves. Each point in which the wave comes becomes the source of the same waves, i.e. Secondary coherent waves. Therefore, the waves are not only in those places in which the conditions of the interference minimum are performed for the secondary waves.

26. Polarized wave - transverse wave in which all particles are oscillations in the same plane. If the free end of the cord makes oscillations in the same plane, a flat-polarized wave is distributed over the cord. If the free end of the cord makes oscillations in different directions, then the wave is not overcoorzed by the cord. If on the way of a non-polarized wave to put a prefirmation in the form of a narrow slit, then after passing the crack, the wave becomes polarized, because The gap misses the cord fluctuations that occur along it.

If on the path of the polarized wave to put the second slit parallel first, the wave will pass freely through it (Fig.102).

If the second gap is located at a right angle with respect to the first, then the spread of the ox will stop. A device that highlights oscillations occurring in one defined plane is called the polarizer (first gap). A device that determines the polarization plane is called the analyzer.

27.Sound -this is the process of propagation of compression and resolutions in an elastic medium, for example, in gas, liquid or in metals. The propagation of compression and resolutions occurs as a result of the collision of molecules.

28. Sound volume This is the power of the impact of the sound wave on the eardrum of the human ear, which is from sound pressure.

Sound Pressure - this is an additional pressure arising in gas or liquid when the sound wave is propagated.Sound pressure depends on the amplitude of the sound source fluctuations. If it is forced to sound a slight blow, then we get one volume. But, if the camera is stronger to strike, then the amplitude of its oscillations will increase and it will sound louder. Thus, the sound volume is determined by the amplitude of the oscillation of the sound source, i.e. amplitude of sound pressure oscillations.

29. Height of tone sounddetermined by the frequency of oscillations. The greater the sound frequency, the higher the tone.

Sound oscillations occurring in harmonic law are perceived as a musical tone. Usually sound is a complex sound, which is a combination of oscillations with close frequencies.

The main tone of the complex sound is the tone corresponding to the smallest frequency in the set of frequencies of this sound. Tones corresponding to the other frequencies of complex sound are called overallones.

30. Sound timbre. The sounds of the same main tone differ in the timbre, which is determined by the set of overtones.

Each person has its own timbre. Therefore, we can always distinguish the voice of one person from the voice of another person, even if their main tones are the same.

31.Ultrasound. The human ear perceives the sounds whose frequencies are in the range from 20 Hz to 20000Hz.

Sounds with frequencies of more than 200ggz are called ultrasounds. Ultrasounds are distributed in the form of narrow beams and are used in hydrolycations and flaw detection. Using ultrasound, you can determine the depth of the seabed and detect defects in different parts.

For example, if the rail has no cracks, then the ultrasound emitted from one end of the rail, reflected from the other end of its end will give only one echo. If there are cracks, the ultrasound will be reflected from cracks and the devices will fix several echoes. Ultrasound, submarines are detected using ultrasound, fish shoals. Bat oriented in space with ultrasound.

32. Infrase- Sound with a frequency below 20Hz. These sounds are perceived by some animals. Their source is often oscillations of earthquakes during earthquakes.

33. Doppler effect - This is the dependence of the frequency of the perceived wave from the movement of the source or receiving the waves.

Let the boat be resting on the surface of the lake and waves be afraid of her board with some frequency. If the boat starts to move against the direction of the wave propagation, the frequency of shocks of the waves on the board of the boat will become more. Moreover, the greater the speed of the boat, the greater the frequency of shocks of the waves on the board. And on the contrary, when the boat moves in the direction of the spread of waves, the frequency of beats will become less. These arguments are easy to understand from fig. 103.

The greater the speed of the counter movement, the mortar time is spent on the distance between the two nearest ridges, i.e. The less period of the wave and the greater the frequency of the wave relative to the boat.

If the observer is immobile, but the source of the waves is moving, the frequency of the wave perceived by the observer depends on the movement of the source.

Let the Heron go to the observer in the direction of the observer. Every time she lowers the leg into the water from this place circles the waves are diverged. And each time the distance between the first and the last waves decreases, i.e. At a smaller distance, a greater number of ridges and depressions are stacked. Therefore, for a fixed observer in the direction of which the Heron goes the frequency increases. And on the contrary, for a fixed observer located in a diametrically opposite point at a greater distance of as many ridges and depressions. Therefore, for this observer, the frequency decreases (Fig.104).

If several electromagnetic waves are distributed in some spaces of space, then in the overlay area at each point, vectors and Waves geometrically fold. This is the essence of the principle of superposition in wave processes. In the case of applying coherent waves (waves with the same frequencies or with a constant phase difference of oscillations at each point), the interference phenomenon is observed - the redistribution of the wave energy between the interface points in the overlay area with maxima and minima of the oscillation energy. A special case of interference is a wave process, called a standing wave, which occurs when applying oncoming plane waves with the same frequency (usually waves - running and reflected). Standing wave is formed in a limited area of \u200b\u200bspace.

Running wave (10)

Reflected Wave (11)

then the equation is for the vector has appearance

where
- amplitude of the standing wave,
- its phase,
- wave vector,
- Length of a running wave.

At points where
(n \u003d 0,1,2, ...) amplitude in a standing wave the largest. This is her whims. At points where
(n \u003d 0.1.2, ....), The amplitude of the standing wave turn into zero. These are a standing wave nodes. The distance between adjacent whites, as well as between adjacent nodes, is .

From equation (12) it follows that the phase of oscillations
x does not depend, the adjacent points should simultaneously achieve maximum and minimum deviations. However, during the transition through the phase node, it changes to the opposite, because Multiplier 2e 0 COSKX When switching through zero changes its sign.

Polarized waves

The wave depicted in Figure 1 is called linearly or flat-polarized, because Direction (plane) vibration vibrations and relative to the velocity vector In the process of propagation, the wave remains unchanged. There are other, more complex forms of polarization of electromagnetic wave-elliptical (or circular). In this case, in the distribution process in space. Vector and changes its direction of oscillation relatively But in such a way that its end describes in the Ellipse space (or circle). In a polarized wave, there is always some kind of definite orientation. relative to the direction of the wave propagation (axial symmetry).

However, such waves can be implemented in real conditions, where the above position is violated - vector in a wave, any directions of oscillations can have, and in some directions it can have a greater amplitude, in the other one. That is, there may be non-polarized waves. Such waves may arise due to the lack of axial symmetry in the emitter, during the refraction and reflection of the waves at the boundaries of the two media, when the waves are propagated in an anisotropic medium.

The presence or absence of polarization can be checked by special analyzers devices. For radiodia waves (centimeter and millimeter radio waves), for example, a lattice with parallel metal spots can be used as an analyzer polarization machine.For the electromagnetic wave of the optical range, the role of the analyzer (polarizer) is performed by natural anisotropic crystals or plates cut from transparent anisotropic crystals.

Consider what happens when electromagnetic waves pass through the polarization grid (Fig. 3). Suppose that the wave of a centimeter range, propagating along the z axis, has x and y components of the vector . What action do the wires on them when the wave pass through the grille? Let's start the SY-component. The electric field of the wave will cause the movement of electrons in the metal along the wires. During the time, the smaller wave period, the electrons reach the steady speed. The wavefield will work on electrons, will give them a part of their energy. In turn, the electrons are partially transmitted in part with collisions with a crystal conductor grille, which will go into heat. This is first. Secondly, because Electrons, experiencing an alternating electric field, perform oscillatory movements along the wires, then they are elementary emitters of secondary electromagnetic waves. Most of the electron energy is emitted. The calculation shows that when the secondary wave is added with the Z axis falling in the positive direction. These waves mutually repay each other, i.e. The electron wave destroys the falling wave. In the opposite direction (-z), the radiation caused by the movement of electrons along the Y axis gives a reflected wave. So, the fence from the wire excludes - component in the last wave. And what happens to the X-component of the vector? Metal electrons can not freely move along this direction due to limited wire size. Therefore, they do not reach a certain ultimate speed, as it was in the case of moving alongy, and form, the surface charge along the surfaces of the wire addressed to the axes + x and x. When the field of this surface charge becomes sufficient to compensate for the external field inside the conductor, electrons Welts will stop moving. This state is achieved during the time, the smaller period of oscillations of the incident wave. That is, in this case, electrons are in static equilibrium. They do not emit and do not absorb energy. Therefore, when passing through the wire fence, the X-component will not change. Thus, the polarization lattice has a selective (selective) bandwidth for waves with different direction of the vibrations of the vector .

International Scientific and Practical Conference

"First steps in science"

Research

"Waves on the surface of the water."

Dichchenkova Anastasia,

Safronova Alena,

Leader:

Educational institution:

MBOU SOSH No. 52 Bryansk.

Div_adblock252 "\u003e

The main properties of the waves are:

1) absorption;

2) scattering;

3) reflection;

4) refraction;

5) interference;

8) polarization.

It should be noted that the wave nature of any process proves the phenomena of interference and diffraction.

Consider some properties of waves in more detail:

The formation of standing waves.

When the straight and reflected running waves, there is a standing wave. It is called standing, since, firstly, knots and beacons in space are not moved, secondly, it does not tolerate energy in space.

Standing wave is formed stable if the whole number of half-mounted is fit at the length L.

Any elastic body (for example, string) with free oscillations has a basic tone and overtone. The more overtones have an elastic body, the more beautiful it sounds.

Examples of standing waves:

Wind musical instruments (organ, pipe)

String Musical Instruments (Guitar, Piano, Violin)

Tambleton

Wave interference.

The interference of the waves is a steady distribution over time the amplitude of oscillations in space when applying coherent waves.

They have the same frequencies;

The shift in the phase of the waves, which came to this point, the value is constant, that is, it does not depend on time.

At this point, the interference is observed at least if the difference in the movement of the waves is equal to an odd number of semi-fell.

At this point, the interference is observed maximum if the difference in the movement of the waves is equal to the even number of half-filled or an integer number of wavelengths.

In the interference, the redistribution of the energy of the waves occurs, that is, it almost does not come to the point of minimum, and it comes to the point maximum.

Diffraction of waves.

Waves are capable of riding obstacles. So, sea waves are freely enveloped by a stone protruding from the water if its size is less than wavelengths or comparable to it. Behind the stone, the waves apply as if it were not at all. Similarly, the wave from the stone abandoned in the pond is envelopes sticking out a twist. Only for the obstacle, compared with the wavelength, size is formed "Shadow": the waves for the obstacle do not penetrate.

Sound waves have the ability to overtake obstacles. You can hear the signal of the car around the corner of the house when the machine itself is not visible. In the forest trees obscure your comrades. To not lose them, you start shouting. Sound waves, in contrast to the light, freely envelop the trunks of trees and coming to your voice to comrades.

The diffraction is the phenomenon of violation of the law of the straight-line propagation of waves in a homogeneous environment or crossing obstacles to the waves.

On the way of the wave screen with a slit:

The length of the slit is much larger than the wavelength. Diffraction is not observed.

The length of the slit is commensurate with a wavelength. Diffraction is observed.

On the way of the wave of the barrier:

The size of the barrier is much greater than the wavelength. Diffraction is not observed.

The size of the barrier is commensurate with a wavelength. The diffraction is observed (the wave envelopes an obstacle).

Diffraction Condition: Wavelength Size Size with dimensions of obstacles, cracks or barriers

Practical part.

For experiments, we used the device "Wave Bath"

Interference of two circular waves.

Pour water in the bath. Lower the nozzle to the formation of two circular waves.

https://pandia.ru/text/78/151/images/image008_25.jpg "width \u003d" 295 "height \u003d" 223 src \u003d "\u003e

Alternation of bright and dark stripes. At those points where the phases are the same, an increase in the amplitude of oscillations;

Sources are coherent.

Circular wave.

Interference of the falling and reflected wave.

https://pandia.ru/text/78/151/images/image010_18.jpg "width \u003d" 285 "height \u003d" 214 src \u003d "\u003e

Conclusion: To observe interference, the sources of waves must be coherent.

Interference flat waves.

https://pandia.ru/text/78/151/images/image012_16.jpg "width \u003d" 302 "height \u003d" 226 src \u003d "\u003e

Standing waves.

https://pandia.ru/text/78/151/images/image014_13.jpg "width \u003d" 196 "height \u003d" 263 src \u003d "\u003e

1. Posted in the vibrator nozzle to create a flat wave and get a steady picture of flat waves on the screen.

2. Installed the barrier reflector parallel to the wave front.

3. Collected from two obstacles an analogue of the corner reflector and immerse it in the cuvette. You will see a standing wave in the form of a two-dimensional (mesh) structure.

4. The criterion for obtaining a standing wave is the transition of the surface shape at points where it is a piggy, from convex (light points) into concave (dark points) without any displacement of these points.

Diffraction of the wave per obesic.

Received a steady picture of the radiation of a flat wave. At a distance of about 50 mm from the emitter, place an obstacle - eraser.

Reducing the size of the eraser, we get the following: (A - length of the lasty)

https://pandia.ru/text/78/151/images/image016_10.jpg "width \u003d" 262 "height \u003d" 198 src \u003d "\u003e

a \u003d 8 cm a \u003d 7mm

https://pandia.ru/text/78/151/images/image018_8.jpg "width \u003d" 274 "height \u003d" 206 src \u003d "\u003e

a \u003d 4.5 mm a \u003d 1.5 mm

Pin: diffraction is not observed if, a\u003e λ, diffraction is observed,

if A.< λ, следовательно, волна огибает препятствия.

Determination of wavelength.

https://pandia.ru/text/78/151/images/image020_5.jpg "width \u003d" 290 "height \u003d" 217 src \u003d "\u003e

The wavelength λ is the distance between adjacent ridges or depressions. The image on the screen is increased by 2 times compared with the real object.

λ \u003d 6 mm / 2 \u003d 3mm.

The wavelength does not depend on the emitter configuration (the wave is flat or round). λ \u003d 6 mm / 2 \u003d 3mm.

https://pandia.ru/text/78/151/images/image022_5.jpg "width \u003d" 278 "height \u003d" 208 src \u003d "\u003e

The wavelength λ depends on the frequency of the vibrator, increasing the frequency of the vibrator - the wavelength will decrease.

λ \u003d 4 mm / 2 \u003d 2mm.

Conclusions.

1. To observe interference, the sources of waves must be coherent.

2. The diffraction is not observed if the width of the obstacle is greater than the wavelength, diffraction is observed if the width of the obstacle is less than the wavelength, therefore, the wave envelopes obstacles.

3. The wavelength does not depend on the emitter configuration (flat or round wave).

4. The wavelength depends on the frequency of the vibrator, increasing the frequency of the vibrator - the wavelength will decrease.

5. This work can be used when studying wave phenomena in grade 9 and grade 11.

Bibliography:

1. Landsberg Textbook Physics. M.: Science, 1995.

2., CIKOIN 9 CL. M.: Enlightenment, 1997.

3. Encyclopedia for children. Avanta +. T.16, 2000.

4. Saveliev General Physics. Book 1.M.: Science, 2000.

5. Internet - Resources:

http: // en. Wikipedia. ORG / WIKI / WAVE

http: // www. / article / index. php? id_article \u003d 1898.

http: // www. / NODE / 1785

Surface acoustic waves (Surfactant) - elastic waves propagating along the surface of a solid body or along the border with other environments. Pav is divided into two types: with vertical polarization and with horizontal polarization ( waves Lyava).

The most commonly encountered special cases of surface waves include the following:

• Railey waves (or Rayleigh), in a classical understanding that extending along the boundaries of an elastic half-space with a vacuum or a fairly rarefied gas medium.
• On the border of the solid with liquid.
• running along the border of liquid and solid
• Wave Stonellipropagating along the flat boundary of two hard media, the modulus of elasticity and density of which are not widely different.
• Waves Lyava - Surface waves with horizontal polarization (SH type), which can be distributed in the structure of the elastic layer on an elastic half-space.

Encyclopedic YouTube.

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✪ Seismic waves

✪ Longitudinal and transverse waves. Sound waves. Lesson 120.

✪ Lecture Seventh: Waves

Subtitles

In this video, I want to discuss seismic waves a little. We write the topic. First, they are very interesting in themselves and, secondly, very important to understand the structure of the Earth. You have already seen my video about the layers of the Earth, and it is thanks to the seismic waves we made a conclusion from which layers our planet consists. And, although the seismic waves are usually associated with earthquakes, in fact these are any waves traveling on the ground. They may arise from an earthquake, a strong explosion, anything, which is able to send a lot of energy directly to the ground and stone. So, there are two main types of seismic waves. And we will focus more on one of them. First - surface waves. We write. The second is bulk waves. Surface waves are just waves spread over the surface of something. In our case, on the surface of the Earth. Here, on the illustration, it can be seen how the surface waves look like. They look like ripples that can be seen on the surface of the water. Surface waves are two types: Rayleigh waves and waves of Lyava. I will not spread, but here it can be seen that Rayleigh waves move up and down. Here the earth moves up and down. There is moving down. Here - up. And here - again down. Looks like a wave running around the ground. Waves of Lyava, in turn, move to the sides. That is, here the wave is not moving up and down, but if you look in the direction of the wave, it moves to the left. It moves here to the right. Here - left. Here - again right. In both cases, the wave movement is perpendicular to the direction of its movement. Sometimes such waves are called transverse. And they, as I said, look like waves in the water. Much more interesting volume waves, because, firstly, these are the fastest waves. And, moreover, it is these waves that are used to study the structure of the Earth. Volga waves are two types. There are P-waves, or primary waves. And S-waves, or secondary. They can be seen here. Such waves are the energy moving inside the body. And not just on its surface. So, in this picture, which I downloaded from Wikipedia, it can be seen how the hammer beaten by the big stone. And when the hammer hits the stone ... Let me redraw more more. Here I will have a stone, and I beat him with a hammer. He will be squeezed there, where he fell. Then the energy from the blow will push the molecule that will stay in the molecules in the neighborhood. And these molecules will die into the molecules behind them, and those, in turn, in the molecules nearby. It turns out that this compressed part of the stone moves the wave. These are compressed molecules, they will die in the molecules nearby and then here the stone will become denser. The first molecules, those that started all the movement will return to the place. Therefore, the compression has shifted, and further will move. It turns out a compression wave. You beat the hammer here and get a changing density that moves towards the wave. In our case, the molecules move forward and backward along one axis. Parallel to the direction of the wave. This is a r-waves. R-waves can spread in the air. Essentially, sound waves are the compression waves. They can move both in liquids and in solids. And, depending on the medium, they move with different speeds. In the air, they move at a speed of 330 m / s, which is not so slow for everyday life. In the liquid, they move at a speed of 1,500 m / s. And in granite, from which most of the earth's surface consists, they move at a speed of 5,000 m / s. Let me write it. 5,000 meters, or 5 km / s in granite. And the S-waves, now I will draw, because this is too small. If you hit the hammer here, the blow strength will temporarily move the stone to the side. It is a bit deformed and pulls behind the neighboring area of \u200b\u200bthe stone. Then this stone will be pulled down from above, and the stone, which was originally hit, will return up. And approximately through the millisecond layer of stone from above, a little deform to the right. And on, over time, the deformation will move up. Note that in this case the wave is also moving up. But the movement of the material is now not parallel to the axis, as in the R-waves, and perpendicularly. These perpendicular waves are also called transverse oscillations. Movement of particles perpendicular to the axis of the wave movement. This is the S-waves. They move slightly slower p-waves. Therefore, if the earthquake suddenly happens, you first feel the r-wave. And then, approximately 60% of the s-waves will come. So, for understanding the structure of the Earth it is important to remember that the S-waves can only move in solids. We write it. You could say that we saw lateral waves on the water. But there were superficial waves. And we are discussing bulk waves. Waves that pass inside the volume of water. To make it easier to imagine, I paint a little water, say, here is the pool here. In context. Something like that. Yes, I could also draw better. So, there will be a swimming pool in the context, and I hope that you will understand what happens in it. And if I squeeze a part of the water, for example, by hitting it with something very large so that the water is quickly clenched. The R-wave can move, because the water molecules will die in the molecules in the neighborhood, which will stay in the molecules behind them. And this compression, this r-wave will move towards my blow. It can be seen that the p-wave can move both in liquids and, for example, in the air. Okay. And remember that we are talking about underwater waves. Not about surfaces. Our waves are moving in the volume of water. Suppose we took the hammer and hit this volume of water from the side. And this will only arise a compression wave in this direction. And nothing more. The transverse wave will not arise, because the wave does not have that elasticity that allows its parts to range from side to side. For the S-wave, such elasticity is needed, which only happens in solids. In the future, we will use the properties of p-waves that can move in the air, liquids and solids, and the properties of the S-waves to find out what the land consists of. Subtitles by The Amara.org Community

Railey waves

Flowing waves of Rayleev type

Flowing waves of Rayleev type on the boundary of the solid with liquid.

Impeachment wave with vertical polarization

Impeachment wave with vertical polarization, running along the boundary of the liquid and solid at the speed of sound in this environment.