Propagation of vibrations in a medium. Waves

We present to your attention a video lesson on the topic “Propagation of vibrations in an elastic medium. Longitudinal and transverse waves. In this lesson, we will study issues related to the propagation of oscillations in an elastic medium. You will learn what a wave is, how it appears, how it is characterized. Let us study the properties and differences between longitudinal and transverse waves.

We turn to the study of issues related to waves. Let's talk about what a wave is, how it appears and what it is characterized by. It turns out that in addition to just an oscillatory process in a narrow region of space, it is also possible to propagate these oscillations in a medium, and it is precisely such propagation that is wave motion.

Let's move on to a discussion of this distribution. To discuss the possibility of the existence of oscillations in a medium, we must define what a dense medium is. A dense medium is a medium that consists of a large number particles whose interaction is very close to elastic. Imagine the following thought experiment.

Rice. 1. Thought experiment

Let us place a sphere in an elastic medium. The ball will shrink, decrease in size, and then expand like a heartbeat. What will be observed in this case? In this case, the particles that are adjacent to this ball will repeat its movement, i.e. move away, approach - thereby they will oscillate. Since these particles interact with other particles more distant from the ball, they will also oscillate, but with some delay. Particles that are close to this ball, oscillate. They will be transmitted to other particles, more distant. Thus, the oscillation will propagate in all directions. Note that in this case, the oscillation state will propagate. This propagation of the state of oscillations is what we call a wave. It can be said that the process of propagation of vibrations in an elastic medium over time is called a mechanical wave.

Please note: when we talk about the process of occurrence of such oscillations, we must say that they are possible only if there is an interaction between particles. In other words, a wave can exist only when there is an external perturbing force and forces that oppose the action of the perturbing force. In this case, these are elastic forces. The propagation process in this case will be related to the density and strength of interaction between the particles of this medium.

Let's note one more thing. The wave does not carry matter. After all, particles oscillate near the equilibrium position. But at the same time, the wave carries energy. This fact can be illustrated by tsunami waves. Matter is not carried by the wave, but the wave carries such energy that brings great disasters.

Let's talk about the types of waves. There are two types - longitudinal and transverse waves. What longitudinal waves? These waves can exist in all media. And the example with a pulsating ball inside a dense medium is just an example of the formation of a longitudinal wave. Such a wave is a propagation in space over time. This alternation of compaction and rarefaction is a longitudinal wave. I repeat once again that such a wave can exist in all media - liquid, solid, gaseous. A longitudinal wave is a wave, during the propagation of which the particles of the medium oscillate along the direction of wave propagation.

Rice. 2. Longitudinal wave

As for the transverse wave, transverse wave can only exist in solids and on the surface of the liquid. A wave is called a transverse wave, during the propagation of which the particles of the medium oscillate perpendicular to the direction of wave propagation.

Rice. 3. Shear wave

The propagation speed of longitudinal and transverse waves is different, but this is the topic of the next lessons.

List of additional literature:

Are you familiar with the concept of a wave? // Quantum. - 1985. - No. 6. - S. 32-33. Physics: Mechanics. Grade 10: Proc. for in-depth study of physics / M.M. Balashov, A.I. Gomonova, A.B. Dolitsky and others; Ed. G.Ya. Myakishev. - M.: Bustard, 2002. Elementary textbook of physics. Ed. G.S. Landsberg. T. 3. - M., 1974.

waves are any perturbations of the state of matter or field, propagating in space over time.

Mechanical called waves that arise in elastic media, i.e. in media in which forces arise that prevent:

1) tensile (compression) deformations;

2) shear deformations.

In the first case, there longitudinal wave, in which the oscillations of the particles of the medium occur in the direction of propagation of the oscillations. Longitudinal waves can propagate in solid, liquid and gaseous bodies, because they are associated with the appearance of elastic forces when changing volume.

In the second case, there exists in space transverse wave, in which the particles of the medium oscillate in directions perpendicular to the direction of propagation of vibrations. Transverse waves can only propagate in solids, because associated with the emergence of elastic forces when changing forms body.

If a body oscillates in an elastic medium, then it acts on the particles of the medium adjacent to it, and makes them perform forced oscillations. The medium near the oscillating body is deformed, and elastic forces arise in it. These forces act on particles of the medium that are more and more distant from the body, taking them out of equilibrium. Everything over time large quantity particles of the medium is involved in oscillating motion.

Mechanical wave phenomena are of great importance for Everyday life. For example, thanks to the sound waves caused by the elasticity of the environment, we can hear. These waves in gases or liquids are pressure fluctuations propagating in a given medium. As examples of mechanical waves, one can also cite: 1) waves on the water surface, where the connection of adjacent sections of the water surface is due not to elasticity, but to gravity and surface tension forces; 2) blast waves from shell explosions; 3) seismic waves - fluctuations in earth's crust propagating from the earthquake.

The difference between elastic waves and any other ordered motion of the particles of the medium is that the propagation of oscillations is not associated with the transfer of the substance of the medium from one place to another over long distances.

The locus of points to which oscillations reach a certain point in time is called front waves. The wave front is the surface that separates the part of space already involved in the wave process from the area in which oscillations have not yet arisen.

The locus of points oscillating in the same phase is called wave surface. The wave surface can be drawn through any point in the space covered by the wave process. Consequently, there are an infinite number of wave surfaces, while there is only one wave front at any moment of time, it moves all the time. The shape of the front can be different depending on the shape and dimensions of the oscillation source and the properties of the medium.

In the case of a homogeneous and isotropic medium, spherical waves propagate from a point source, i.e. the wave front in this case is a sphere. If the source of oscillations is a plane, then near it any section of the wave front differs little from a part of the plane, therefore waves with such a front are called plane waves.

Let us assume that during the time some section of the wave front has moved to . Value

is called the propagation speed of the wave front or phase speed waves at this location.

A line whose tangent at each point coincides with the direction of the wave at that point, i.e. with the direction of energy transfer is called beam. In a homogeneous isotropic medium, the beam is a straight line perpendicular to the wave front.

Oscillations from the source can be either harmonic or non-harmonic. Accordingly, waves run from the source monochromatic and non-monochromatic. A non-monochromatic wave (containing vibrations of different frequencies) can be decomposed into monochromatic waves (each of which contains vibrations of the same frequency). A monochromatic (sinusoidal) wave is an abstraction: such a wave must be infinitely extended in space and time.

Let the oscillating body be in a medium, all particles of which are interconnected. The particles of the medium in contact with it will begin to oscillate, as a result of which periodic deformations (for example, compression and tension) occur in the areas of the medium adjacent to this body. During deformations, elastic forces appear in the medium, which tend to return the particles of the medium to their original state of equilibrium.

Thus, periodic deformations that have appeared in some place of the elastic medium will propagate at a certain speed, depending on the properties of the medium. In this case, the particles of the medium are not involved by the wave in translational motion, but perform oscillatory motions around their equilibrium positions, only elastic deformation is transmitted from one part of the medium to another.

The process of propagation of oscillatory motion in a medium is called wave process or just wave. Sometimes this wave is called elastic because it is caused by the elastic properties of the medium.

Depending on the direction of particle oscillations in relation to the direction of wave propagation, longitudinal and transverse waves are distinguished.Interactive demonstration of transverse and longitudinal waves

Longitudinal wave – it is a wave in which the particles of the medium oscillate along the direction of wave propagation.

Longitudinal wave can be observed on a long soft spring large diameter. By hitting one of the ends of the spring, one can notice how successive condensations and rarefaction of its coils will spread along the spring, running one after another. In the figure, the dots show the position of the coils of the spring at rest, and then the positions of the coils of the spring at successive intervals equal to a quarter of the period.

Thus, aboutLongitudinal wave in the case under consideration is an alternating cluster (Sg) and rarefaction (Once) spring coils.
Longitudinal Wave Propagation Demonstration

transverse wave - This is a wave in which the particles of the medium oscillate in directions perpendicular to the direction of wave propagation.

Let us consider in more detail the process of formation of transverse waves. Let us take as a model of a real cord a chain of balls (material points) connected to each other by elastic forces. The figure shows the process of propagation of a transverse wave and shows the positions of the balls at successive time intervals equal to a quarter of the period.

At the initial moment of time (t0 = 0) all points are in equilibrium. Then we cause a perturbation by deviating point 1 from the equilibrium position by the value A and the 1st point begins to oscillate, the 2nd point, elastically connected to the 1st, comes into oscillatory motion a little later, the 3rd - even later, etc. . After a quarter period of oscillation ( t 2 = T 4 ) spread to the 4th point, the 1st point will have time to deviate from its equilibrium position by a maximum distance equal to the amplitude of oscillations A. After half a period, the 1st point, moving down, will return to the equilibrium position, the 4th deviated from the equilibrium position by a distance equal to the amplitude of oscillations A, the wave propagated to the 7th point, etc.

By the time t5 = T The 1st point, having made a complete oscillation, passes through the equilibrium position, and the oscillatory movement will spread to the 13th point. All points from the 1st to the 13th are located so that they form a complete wave consisting of hollows and comb.

Demonstration of shear wave propagation

The type of wave depends on the type of deformation of the medium. Longitudinal waves are caused by compressive - tensile deformation, transverse waves - by shear deformation. Therefore, in gases and liquids, in which elastic forces arise only during compression, the propagation of transverse waves is impossible. In solids, elastic forces arise both during compression (tension) and shear, therefore, the propagation of both longitudinal and transverse waves is possible in them.

As the figures show, in both transverse and longitudinal waves, each point of the medium oscillates around its equilibrium position and shifts from it by no more than an amplitude, and the state of deformation of the medium is transferred from one point of the medium to another. An important difference between elastic waves in a medium and any other ordered motion of its particles is that the propagation of waves is not associated with the transfer of matter in the medium.

Consequently, during the propagation of waves, the energy of elastic deformation and momentum are transferred without the transfer of matter. The energy of a wave in an elastic medium consists of the kinetic energy of the oscillating particles and the potential energy of the elastic deformation of the medium.

A medium is called elastic if there are interaction forces between its particles that prevent any deformation of this medium. When a body oscillates in an elastic medium, it acts on the particles of the medium adjacent to the body and causes them to perform forced oscillations. The medium near the oscillating body is deformed, and elastic forces arise in it. These forces act on particles of the medium that are more and more distant from the body, taking them out of their equilibrium position. Gradually, all particles of the medium are involved in oscillatory motion.

The bodies that cause elastic waves propagating in the medium are wave sources(oscillating tuning forks, strings of musical instruments).

elastic waves called mechanical perturbations (deformations) produced by sources that propagate in an elastic medium. Elastic waves cannot propagate in a vacuum.

When describing the wave process, the medium is considered to be continuous and continuous, and its particles are infinitesimal volume elements (sufficiently small compared to the wavelength) in which the a large number of molecules. When a wave propagates in a continuous medium, the particles of the medium participating in the oscillations have certain oscillation phases at each moment of time.

The locus of points of the medium, oscillating in the same phases, forms wave surface.

The wave surface that separates the oscillating particles of the medium from particles that have not yet begun to oscillate is called the wave front. Depending on the shape of the wave front, waves are plane, spherical, etc.

A line drawn perpendicular to the wave front in the direction of wave propagation is called a beam. The beam indicates the direction of wave propagation.;;

AT plane wave wave surfaces are planes perpendicular to the direction of wave propagation (Fig. 15.1). Plane waves can be obtained on the surface of water in a flat bath by means of vibrations of a flat rod.

In a spherical wave, the wave surfaces are concentric spheres. A spherical wave can be created by a ball pulsating in a homogeneous elastic medium. Such a wave propagates with the same speed in all directions. The rays are the radii of the spheres (Fig. 15.2).

Repetitive movements or changes in state are called oscillations (alternating electric current, the movement of a pendulum, the work of the heart, etc.). All oscillations, regardless of their nature, have certain general patterns. Oscillations propagate in the medium in the form of waves. This chapter deals with mechanical vibrations and waves.

7.1. HARMONIC OSCILLATIONS

Among various kinds fluctuations the simplest form is harmonic Oscillation, those. one in which the oscillating value changes with time according to the law of sine or cosine.

Let, for example, a material point with mass t suspended on a spring (Fig. 7.1, a). In this position, the elastic force F 1 balances the force of gravity mg. If the spring is pulled a distance X(Fig. 7.1, b), then on material point there will be a large elastic force. The change in elastic force, according to Hooke's law, is proportional to the change in the length of the spring or displacement X points:

F = -kh,(7.1)

where to- spring stiffness; the minus sign indicates that the force is always directed towards the equilibrium position: F< 0 at X> 0, F > 0 at X< 0.

Another example.

The mathematical pendulum is deviated from the equilibrium position by a small angle α (Fig. 7.2). Then the trajectory of the pendulum can be considered a straight line coinciding with the axis OH. In this case, the approximate equality

where X- displacement of a material point relative to the equilibrium position; l is the length of the pendulum string.

A material point (see Fig. 7.2) is affected by the tension force F H of the thread and the force of gravity mg. Their resultant is:

Comparing (7.2) and (7.1), we see that in this example the resultant force is similar to elastic, since it is proportional to the displacement of the material point and is directed towards the equilibrium position. Such forces, which are inelastic in nature, but similar in properties to forces arising from minor deformations of elastic bodies, are called quasi-elastic.

Thus, a material point suspended on a spring (spring pendulum) or a thread (mathematical pendulum) performs harmonic oscillations.

7.2. KINETIC AND POTENTIAL ENERGY OF VIBRATIONAL MOTION

The kinetic energy of an oscillating material point can be calculated from well-known formula, using expression (7.10):

7.3. ADDITION OF HARMONIC OSCILLATIONS

A material point can simultaneously participate in several oscillations. In this case, to find the equation and the trajectory of the resulting motion, one should add the vibrations. The simplest is the addition harmonic vibrations.

Let's consider two such problems.

Addition of harmonic oscillations directed along one straight line.

Let the material point simultaneously participate in two oscillations occurring along one line. Analytically, such fluctuations are expressed by the following equations:

those. the amplitude of the resulting oscillation is equal to the sum of the amplitudes of the terms of the oscillations, if the difference in the initial phases is equal to an even number π (Fig. 7.8, a);

those. the amplitude of the resulting oscillation is equal to the difference in the amplitudes of the terms of the oscillations, if the difference in the initial phases is equal to an odd number π (Fig. 7.8, b). In particular, for A 1 = A 2 we have A = 0, i.e. there is no fluctuation (Fig. 7.8, c).

This is quite obvious: if a material point simultaneously participates in two oscillations that have the same amplitude and occur in antiphase, the point is motionless. If the frequencies of the added oscillations are not the same, then the complex oscillation will no longer be harmonic.

An interesting case is when the frequencies of the oscillation terms differ little from each other: ω 01 and ω 02

The resulting oscillation is similar to a harmonic one, but with a slowly changing amplitude (amplitude modulation). Such fluctuations are called beats(Fig. 7.9).

Addition of mutually perpendicular harmonic vibrations. Let the material point simultaneously participate in two oscillations: one is directed along the axis OH, the other is along the axis OY. Oscillations are given by the following equations:

Equations (7.25) define the trajectory of a material point in a parametric form. If we substitute into these equations different meanings t, coordinates can be determined X and y, and the set of coordinates is the trajectory.

Thus, with simultaneous participation in two mutually perpendicular harmonic oscillations of the same frequency, a material point moves along an elliptical trajectory (Fig. 7.10).

Some special cases follow from expression (7.26):

7.4. DIFFICULT VIBRATION. HARMONIC SPECTRUM OF A COMPLEX OSCILLATION

As can be seen from 7.3, the addition of vibrations results in more complex waveforms. For practical purposes, the opposite operation may be necessary: ​​the decomposition of a complex oscillation into simple, usually harmonic, oscillations.

Fourier showed that a periodic function of any complexity can be represented as a sum of harmonic functions whose frequencies are multiples of the frequency of a complex periodic function. Such a decomposition of a periodic function into harmonic ones and, consequently, the decomposition of various periodic processes (mechanical, electrical, etc.) into harmonic oscillations is called harmonic analysis. There are mathematical expressions that allow you to find the components of harmonic functions. Automatic harmonic analysis of oscillations, including for medical purposes, is carried out by special devices - analyzers.

The set of harmonic oscillations into which a complex oscillation is decomposed is called harmonic spectrum of a complex oscillation.

It is convenient to represent the harmonic spectrum as a set of frequencies (or circular frequencies) of individual harmonics together with their corresponding amplitudes. The most visual representation of this is done graphically. As an example, in fig. 7.14, a graphs of a complex oscillation are shown (curve 4) and its constituent harmonic oscillations (curves 1, 2 and 3); in fig. 7.14b shows the harmonic spectrum corresponding to this example.

Rice. 7.14b

Harmonic analysis allows you to describe and analyze any complex oscillatory process in sufficient detail. It finds application in acoustics, radio engineering, electronics and other fields of science and technology.

7.5. DAMPING OSCILLATIONS

When studying harmonic oscillations, the forces of friction and resistance that exist in real systems were not taken into account. The action of these forces significantly changes the nature of the motion, the oscillation becomes fading.

If, in addition to the quasi-elastic force, the resistance forces of the medium (friction forces) act in the system, then Newton's second law can be written as follows:

The rate of decrease in the oscillation amplitude is determined by attenuation factor: the larger β, the stronger the retarding effect of the medium and the faster the amplitude decreases. In practice, however, the degree of attenuation is often characterized by logarithmic damping decrement, meaning by this the value equal to natural logarithm the ratio of two successive oscillation amplitudes separated by a time interval equal to the oscillation period:

With strong damping (β 2 >> ω 2 0), it is clear from formula (7.36) that the oscillation period is an imaginary quantity. The movement in this case is already called aperiodic 1 . Possible aperiodic movements are presented in the form of graphs in fig. 7.16. This case applies to electrical phenomena discussed in more detail in Chap. eighteen.

Undamped (see 7.1) and damped oscillations are called own or free. They arise as a result of the initial displacement or initial velocity and occur in the absence of external influence due to the initially accumulated energy.

7.6. FORCED VIBRATIONS. RESONANCE

Forced vibrations are called oscillations that occur in the system with the participation of an external force that changes according to a periodic law.

Let us assume that, in addition to the quasi-elastic force and the friction force, an external driving force acts on the material point:

1 Note that if some physical quantity takes imaginary values, then this means some kind of unusual, extraordinary nature of the corresponding phenomenon. In the considered example, the extraordinary thing lies in the fact that the process ceases to be periodic.

From (7.43) it can be seen that in the absence of resistance (β=0) the amplitude of forced oscillations at resonance is infinitely large. Moreover, from (7.42) it follows that ω res = ω 0 - resonance in the system without damping occurs when the frequency of the driving force coincides with the frequency of natural oscillations. The graphical dependence of the amplitude of forced oscillations on the circular frequency of the driving force for different values ​​of the damping coefficient is shown in Fig. 7.18.

Mechanical resonance can be both beneficial and detrimental. The harmful effect of resonance is mainly due to the destruction it can cause. So, in technology, taking into account different vibrations, it is necessary to provide for the possible occurrence of resonant conditions, otherwise there may be destruction and catastrophes. Bodies usually have several natural vibration frequencies and, accordingly, several resonant frequencies.

If the attenuation coefficient of the internal organs of a person were small, then the resonant phenomena that arose in these organs under the influence of external vibrations or sound waves could lead to tragic consequences: rupture of organs, damage to ligaments, etc. However, such phenomena are practically not observed under moderate external influences, since the attenuation coefficient of biological systems is quite large. Nevertheless, resonant phenomena under the action of external mechanical vibrations occur during internal organs. This, apparently, is one of the reasons for the negative impact of infrasonic vibrations and vibrations on the human body (see 8.7 and 8.8).

7.7. AUTO OSCILLATIONS

As shown in 7.6, oscillations can be maintained in a system even in the presence of drag forces, if the system is periodically subjected to an external influence (forced oscillations). This external influence does not depend on the oscillating system itself, while the amplitude and frequency of forced oscillations depend on this external influence.

However, there are also such oscillatory systems that themselves regulate the periodic replenishment of wasted energy and therefore can fluctuate for a long time.

The undamped oscillations that exist in any system in the absence of a variable external influence are called self-oscillations, and the systems themselves are called self-oscillatory.

The amplitude and frequency of self-oscillations depend on the properties of the self-oscillating system itself; in contrast to forced oscillations, they are not determined by external influences.

In many cases, self-oscillatory systems can be represented by three main elements:

1) the actual oscillatory system;

2) energy source;

3) a regulator of energy supply to the actual oscillatory system.

Oscillating system by channel feedback(Fig. 7.19) acts on the regulator, informing the regulator about the state of this system.

A classic example of a mechanical self-oscillating system is a clock, in which a pendulum or balance is an oscillatory system, a spring or a raised weight is a source of energy, and an anchor is a regulator of the energy supply from the source to the oscillatory system.

Many biological systems (heart, lungs, etc.) are self-oscillatory. A typical example of an electromagnetic self-oscillating system is generators electromagnetic oscillations(see ch. 23).

7.8. EQUATION OF MECHANICAL WAVES

A mechanical wave is a mechanical disturbance propagating in space and carrying energy.

There are two main types of mechanical waves: elastic waves - the propagation of elastic deformations - and waves on the surface of a liquid.

Elastic waves arise due to the bonds that exist between the particles of the medium: the movement of one particle from the equilibrium position leads to the movement of neighboring particles. This process propagates in space with a finite speed.

The wave equation expresses the dependence of the displacement s oscillating point participating in wave process, on the coordinate of its equilibrium position and time.

For a wave propagating along a certain direction OX, this dependence is written in the general form:

If a s and X directed along one straight line, then the wave longitudinal, if they are mutually perpendicular, then the wave transverse.

Let us derive the plane wave equation. Let the wave propagate along the axis X(Fig. 7.20) without damping so that the oscillation amplitudes of all points are the same and equal to A. Let us set the oscillation of a point with coordinate X= 0 (oscillation source) by the equation

Solving partial differential equations is beyond the scope of this course. One of the solutions (7.45) is known. However, it is important to note the following. If a change in any physical quantity: mechanical, thermal, electrical, magnetic, etc., corresponds to equation (7.49), then this means that the corresponding physical quantity propagates in the form of a wave with a speed υ.

7.9. WAVE ENERGY FLOW. UMOV VECTOR

The wave process is associated with the transfer of energy. The quantitative characteristic of the transferred energy is the energy flow.

The wave energy flux is equal to the ratio of the energy carried by waves through a certain surface to the time during which this energy was transferred:

The unit of the wave energy flux is watt(W). Let us find the connection between the flow of wave energy and the energy of oscillating points and the speed of wave propagation.

We select the volume of the medium in which the wave propagates in the form of a rectangular parallelepiped (Fig. 7.21), the area cross section which S, and the length of the edge is numerically equal to the velocity υ and coincides with the direction of wave propagation. In accordance with this, for 1 s through the area S the energy that oscillating particles possess in the volume of a parallelepiped will pass Sυ. This is the flow of wave energy:

7.10. SHOCK WAVES

One common example mechanical wave - sound wave(see ch. 8). In this case maximum speed vibrations of an individual air molecule is several centimeters per second even for a sufficiently high intensity, i.e. it is much less than the wave speed (the speed of sound in air is about 300 m/s). This corresponds, as they say, to small perturbations of the medium.

However, with large disturbances (explosion, supersonic motion of bodies, powerful electric discharge, etc.), the speed of oscillating particles of the medium can already become comparable with the speed of sound, and a shock wave arises.

During the explosion, highly heated products with a high density expand and compress the layers of the surrounding air. Over time, the volume of compressed air increases. The surface that separates compressed air from unperturbed air is called in physics shock wave. Schematically, the jump in the gas density during the propagation of a shock wave in it is shown in Fig. 7.22 a. For comparison, the same figure shows the change in the density of the medium during the passage sound wave(Fig. 7.22, b).

Rice. 7.22

The shock wave can have significant energy, so in a nuclear explosion, the formation of a shock wave in environment about 50% of the energy of the explosion is expended. Therefore, the shock wave, reaching biological and technical objects, is capable of causing death, injury and destruction.

7.11. DOPPLER EFFECT

The Doppler effect is a change in the frequency of the waves perceived by the observer (wave receiver) due to the relative motion of the wave source and the observer.