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 of 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 exist only in solids and on the surface of a 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.
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 the various types of oscillations, 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 a large elastic force will act on the material point. 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 using the 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 of harmonic oscillations.
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 different values into these equations 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 a value equal to the natural logarithm of 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, as applied to electrical phenomena, is considered 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 on 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 in the 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.
The oscillatory system through the feedback channel (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 the generators of electromagnetic oscillations (see Chap. 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 the 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 single out the volume of the medium in which the wave propagates in the form of a rectangular parallelepiped (Fig. 7.21), the cross-sectional area of \u200b\u200bwhich is 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 of a mechanical wave is sound wave(see ch. 8). In this case, the maximum speed of oscillations 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 of a sound wave (Fig. 7.22, b).
Rice. 7.22
A shock wave can have significant energy, so in a nuclear explosion about 50% of the explosion energy is spent on the formation of a shock wave in the environment. 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.
Lecture No. 9
mechanical waves
6.1. Propagation of vibrations in an elastic medium.
6.2. Plane wave equation.
6.3. wave equation.
6.4. Wave propagation speed in various media.
Mechanical vibrations propagating in an elastic medium (solid, liquid or gaseous) are called mechanical or elastic waves.
The process of propagation of oscillations in a continuous medium is commonly called a wave process or a wave. The particles of the medium in which the wave propagates are not involved by the wave in translational motion. they only oscillate around their equilibrium positions. Together with the wave, only the state of oscillatory motion and its energy are transmitted from particle to particle of the medium. For this reason the main property of all waves, regardless of their nature, is the transfer of energy without the transfer of matter.
Taking into account the dependence on the direction of particle oscillations with respect to the direction in which the wave propagates, we distinguish longitudinal and transverse waves.
longitudinal, if the oscillations of the particles of the medium occur in the direction of wave propagation. Longitudinal waves are associated with volumetric tensile-compressive deformation of the medium; therefore, they can propagate both in solids and in liquids and gaseous media.
An elastic wave is called transverse, if the oscillations of the particles of the medium occur in planes perpendicular to the direction of propagation of the wave. Transverse waves can occur only in a medium that has the elasticity of the form, i.e., is able to resist shear deformation. Only solid bodies have this property.
On fig. 6.1.1 shows a harmonic shear wave propagating along the 0 axis X. The wave graph gives the dependence of the displacement of all particles of the medium on the distance to the source of vibrations at a given time. The distance between the nearest particles oscillating in the same phase is called wavelength. The wavelength is also equal to that distance, a certain phase of the oscillation spreads over the ĸᴏᴛᴏᴩᴏᴇ during the oscillation period
Not only particles located along the 0 axis oscillate X, but a set of particles enclosed in a certain volume. The locus of points to which oscillations reach by the moment of time t, commonly called wave front. 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. Wave surfaces come in all shapes. In the simplest cases, they have the shape of a plane or sphere. Accordingly, the wave in these cases is called flat or spherical. In a plane wave, the wave surfaces are a set of planes parallel to each other, and in a spherical wave, they are a set of concentric spheres.
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 the particles of the medium that are more and more distant from the body, removing them from the equilibrium position. Over time, an increasing number of particles of the medium is involved in oscillatory 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 the earth's crust, propagating from the place of an 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 oscillations of different frequencies) can be decomposed into monochromatic waves (each of which contains oscillations of the same frequency). A monochromatic (sinusoidal) wave is an abstraction: such a wave must be infinitely extended in space and time.
Page 1
The process of propagation of vibrations in an elastic medium is called sound.
The process of propagation of oscillations in space is called a wave. The boundary separating oscillating particles from particles that have not yet begun to oscillate is called the water front. The propagation of a wave in a medium is characterized by a speed called the speed of an ultrasonic wave. The distance between the nearest particles oscillating in the same way (in the same phase) is called the wavelength. The number of waves passing through a given point in 1 second is called the frequency of the ultrasound.
The process of propagation of oscillations in an elastic medium is called wave motion, or an elastic wave.
The process of propagation of oscillations in space over time is called a wave. Waves propagating due to the elastic properties of the medium are called elastic. Elastic waves are transverse and longitudinal.
The process of vibration propagation in an elastic medium is called a wave. If the direction of oscillation coincides with the direction of wave propagation, then such a wave is called longitudinal, for example, a sound wave in air. If the direction of oscillation is perpendicular to the direction of wave propagation, then such a wave is called transverse.
The process of propagation of oscillations in space is called the wave process.
The process of propagation of oscillations in space is called a wave.
The process of vibration propagation in an elastic medium is called a wave. If the direction of oscillation coincides with the direction of wave propagation, then such a wave is called longitudinal, for example, a sound wave in air. If the direction of oscillation is perpendicular to the direction of wave propagation, then such a wave is called transverse.
The process of propagation of particle oscillations in an elastic medium is called a wave process or simply a wave.
The processes of propagation of fluctuations of liquid or gas particles in a pipe are complicated by the influence of its walls. Oblique reflections along the pipe walls create conditions for the formation of radial oscillations. Having set the task of studying axial vibrations of liquid or gas particles in narrow pipes, we must take into account a number of conditions under which radial vibrations can be neglected.
A wave is the process of propagation of oscillations in a medium. Each particle of the medium oscillates around the equilibrium position.
A wave is the process of propagation of vibrations.
The process of propagation of oscillations in an elastic medium considered by us is an example of wave motions, or, as they usually say, waves. So, for example, it turns out that electromagnetic waves (see § 3.1) can propagate not only in matter, but also in vacuum. The so-called gravitational waves (gravity waves) have the same property, with the help of which perturbations of the gravitational fields of bodies are transmitted, due to a change in the masses of these bodies or their positions in space. Therefore, in physics, waves are any perturbations of the state of matter or field propagating in space. So, for example, sound waves in gases or liquids are pressure fluctuations propagating in these media, and electromagnetic waves are fluctuations in the strengths E and H of the electromagnetic field propagating in space.
