Elastic waves (mechanical waves). Examples of longitudinal and transverse waves

Let the oscillating body be in a medium in which all the particles are interconnected. The particles of the medium in contact with it will begin to vibrate, 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 appear in some place in an 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 drawn into translational motion by the wave, but perform oscillatory movements around their equilibrium positions; only elastic deformation is transferred from one part of the medium to another.

The process of propagation of oscillatory motion in a medium is called wave process or simply 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 relative to the direction of wave propagation, longitudinal and transverse waves are distinguished.Interactive demonstration of transverse and longitudinal waves

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

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

Thus, aboutthe longitudinal wave in the case under consideration represents alternating condensations (Сг) and rarefaction (Once) spring coils.
Demonstration of longitudinal wave propagation

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

Let's take a closer look at the education process transverse waves. Let's take a chain of balls as a model of a real cord ( material points), connected to each other by elastic forces. The figure depicts 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 (t 0 = 0) all points are in a state of equilibrium. Then we cause a disturbance by deviating point 1 from the equilibrium position by an amount 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 of the oscillation period ( t 2 = T 4 ) will 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 has propagated to the 7th point, etc.

By the time t 5 = T The 1st point, having completed 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 depressions And ridge

Demonstration of shear wave propagation

The type of wave depends on the type of deformation of the medium. Longitudinal waves are caused by compression-tension 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, so 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 movement of its particles is that the propagation of waves is not associated with the transfer of matter in the medium.

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

Mechanical waves

If vibrations of particles are excited in any place in a solid, liquid or gaseous medium, then due to the interaction of atoms and molecules of the medium, the vibrations begin to be transmitted from one point to another with a finite speed. The process of propagation of vibrations in a medium is called wave .

Mechanical waves there are different types. If particles of the medium in a wave are displaced in a direction perpendicular to the direction of propagation, then the wave is called transverse . An example of a wave of this kind can be waves running along a stretched rubber band (Fig. 2.6.1) or along a string.

If the displacement of particles of the medium occurs in the direction of propagation of the wave, then the wave is called longitudinal . Waves in an elastic rod (Fig. 2.6.2) or sound waves in a gas are examples of such waves.

Waves on the surface of a liquid have both transverse and longitudinal components.

In both transverse and longitudinal waves, there is no transfer of matter in the direction of wave propagation. In the process of propagation, particles of the medium only oscillate around equilibrium positions. However, waves transfer vibration energy from one point in the medium to another.

Characteristic feature mechanical waves is that they propagate in material media (solid, liquid or gaseous). There are waves that can propagate in emptiness (for example, light waves). Mechanical waves necessarily require a medium that has the ability to store kinetic and potential energy. Therefore, the environment must have inert and elastic properties. In real environments, these properties are distributed throughout the entire volume. For example, any small element of a solid body has mass and elasticity. In the simplest one-dimensional model a solid body can be represented as a collection of balls and springs (Fig. 2.6.3).

Longitudinal mechanical waves can propagate in any media - solid, liquid and gaseous.

If in a one-dimensional model of a solid body one or more balls are displaced in a direction perpendicular to the chain, then deformation will occur shift. The springs, deformed by such a displacement, will tend to return the displaced particles to the equilibrium position. In this case, elastic forces will act on the nearest undisplaced particles, tending to deflect them from the equilibrium position. As a result, a transverse wave will run along the chain.

In liquids and gases, elastic shear deformation does not occur. If one layer of liquid or gas is displaced a certain distance relative to the adjacent layer, then no tangential forces will appear at the boundary between the layers. The forces acting at the boundary of a liquid and a solid, as well as the forces between adjacent layers of liquid, are always directed normal to the boundary - these are pressure forces. The same applies to gaseous media. Hence, transverse waves cannot exist in liquid or gaseous media.

Of significant practical interest are simple harmonic or sine waves . They are characterized amplitudeA particle vibrations, frequencyf And wavelengthλ. Sinusoidal waves propagate in homogeneous media with a certain constant speed v.

Bias y (x, t) particles of the medium from the equilibrium position in a sinusoidal wave depends on the coordinate x on the axis OX, along which the wave propagates, and on time t in law.

Longitudinal wave– this is a wave, during the propagation of which the particles of the medium are displaced in the direction of propagation of the wave (Fig. 1, a).

The cause of the longitudinal wave is compression/tension deformation, i.e. resistance of the medium to changes in its volume. In liquids or gases, such deformation is accompanied by rarefaction or compaction of the particles of the medium. Longitudinal waves can propagate in any media - solid, liquid and gaseous.

Examples of longitudinal waves are waves in an elastic rod or sound waves in gases.

Transverse wave– this is a wave, during the propagation of which the particles of the medium are displaced in the direction perpendicular to the propagation of the wave (Fig. 1, b).

The cause of the transverse wave is the shear deformation of one layer of the medium relative to another. When a transverse wave propagates through a medium, ridges and troughs are formed. Liquids and gases, unlike solids, do not have elasticity with respect to the shear of layers, i.e. do not resist changing shape. Therefore, transverse waves can propagate only in solids.

Examples of transverse waves are waves traveling along a stretched rope or string.

Waves on the surface of a liquid are neither longitudinal nor transverse. If you throw a float onto the surface of the water, you can see that it moves, swaying on the waves, along a circular path. Thus, a wave on the surface of a liquid has both transverse and longitudinal components. Waves of a special type can also appear on the surface of a liquid - the so-called surface waves. They arise as a result of gravity and surface tension.

Fig.1. Longitudinal (a) and transverse (b) mechanical waves

Question 30

Wavelength.

Each wave travels at a certain speed. Under wave speed understand the speed of propagation of disturbance. For example, a blow to the end of a steel rod causes local compression in it, which then propagates along the rod at a speed of about 5 km/s.

The speed of the wave is determined by the properties of the medium in which the wave propagates. When a wave passes from one medium to another, its speed changes.

In addition to speed, important characteristic wave is the wavelength. Wavelength is the distance over which a wave propagates in a time equal to the period of oscillation in it.

Since the speed of a wave is a constant value (for a given medium), the distance traveled by the wave is equal to the product of the speed and the time of its propagation. Thus, to find the wavelength, you need to multiply the speed of the wave by the period of oscillation in it:

v - wave speed; T is the period of oscillation in the wave; λ ( greek letter"lambda") - wavelength.

By choosing the direction of wave propagation as the direction of the x axis and denoting by y the coordinate of the particles oscillating in the wave, we can construct wave chart. A graph of a sine wave (at a fixed time t) is shown in Figure 45. The distance between adjacent crests (or troughs) in this graph coincides with the wavelength λ.

Formula (22.1) expresses the relationship between wavelength and its speed and period. Considering that the period of oscillation in a wave is inversely proportional to the frequency, i.e. T = 1/ν, we can obtain a formula expressing the relationship between the wavelength and its speed and frequency:

The resulting formula shows that the speed of the wave is equal to the product of the wavelength and the frequency of oscillations in it.

The frequency of oscillations in the wave coincides with the frequency of oscillations of the source (since the oscillations of the particles of the medium are forced) and does not depend on the properties of the medium in which the wave propagates. When a wave passes from one medium to another, its frequency does not change, only the speed and wavelength change.

Question 30.1

Wave equation

To obtain the wave equation, that is, an analytical expression for a function of two variables S = f (t, x) , Let's imagine that at some point in space harmonic oscillations with a circular frequency arise w and the initial phase, equal to zero for simplicity (see Fig. 8). Offset at a point M: S m = A sin w t, Where A- amplitude. Since the particles of the medium filling space are interconnected, vibrations from a point M spread along the axis X with speed v. After some time D t they reach the point N. If there is no attenuation in the medium, then the displacement at this point has the form: S N = A sin w(t- D t), i.e. oscillations are delayed by time D t relative to the point M. Since , then replacing an arbitrary segment MN coordinate X, we get wave equation in the form.

1. You already know that the process of propagation of mechanical vibrations in a medium is called mechanical wave.

Let's fasten one end of the cord, slightly stretch it and move the free end of the cord up and then down (let it oscillate). We will see that a wave will “run” along the cord (Fig. 84). Parts of the cord are inert, so they will shift relative to the equilibrium position not simultaneously, but with some delay. Gradually all sections of the cord will begin to vibrate. An oscillation will spread across it, in other words, a wave will be observed.

Analyzing the propagation of oscillations along the cord, one can notice that the wave “runs” in the horizontal direction, and the particles oscillate in the vertical direction.

Waves whose direction of propagation is perpendicular to the direction of vibration of the particles of the medium are called transverse.

Transverse waves represent an alternation humps And depressions.

In addition to transverse waves, longitudinal waves can also exist.

Waves, the direction of propagation of which coincides with the direction of vibration of the particles of the medium, are called longitudinal.

Let's fasten one end of a long spring suspended on threads and hit its other end. We will see how the condensation of turns that appears at the end of the spring “runs” along it (Fig. 85). Movement occurs thickenings And rarefaction.

2. Analyzing the process of formation of transverse and longitudinal waves, the following conclusions can be drawn:

- mechanical waves are formed due to the inertia of particles of the medium and the interaction between them, manifested in the existence of elastic forces;

- each particle of the medium performs forced oscillations, the same as the first particle brought into vibration; the vibration frequency of all particles is the same and equal to the frequency of the vibration source;

- the vibration of each particle occurs with a delay, which is due to its inertia; This delay is greater the further the particle is from the source of oscillations.

An important property of wave motion is that no substance is transferred along with the wave. This is easy to verify. If you throw pieces of cork onto the surface of the water and create a wave movement, you will see that the waves will “run” along the surface of the water. The pieces of cork will rise up at the crest of the wave and fall down at the trough.

3. Let's consider the medium in which longitudinal and transverse waves propagate.

The propagation of longitudinal waves is associated with a change in the volume of the body. They can propagate in both solid, liquid and gaseous bodies, since elastic forces arise in all these bodies when their volume changes.

The propagation of transverse waves is associated mainly with changes in the shape of the body. In gases and liquids, when their shape changes, elastic forces do not arise, so transverse waves cannot propagate in them. Transverse waves propagate only in solids.

An example of wave motion in a solid body is the propagation of vibrations during earthquakes. Both longitudinal and transverse waves propagate from the center of the earthquake. A seismic station receives longitudinal waves first, and then transverse ones, since the speed of the latter is lower. If the velocities of transverse and longitudinal waves are known and the time interval between their arrival is measured, then the distance from the center of the earthquake to the station can be determined.

4. You are already familiar with the concept of wavelength. Let's remember him.

The wavelength is the distance over which the wave propagates in a time equal to the oscillation period.

We can also say that the wavelength is the distance between the two nearest humps or troughs of the transverse wave (Fig. 86, A) or the distance between the two nearest condensations or rarefactions of the longitudinal wave (Fig. 86, b).

The wavelength is designated by the letter l and is measured in meters(m).

5. Knowing the wavelength, you can determine its speed.

The wave speed is taken to be the speed of movement of a crest or trough in a transverse wave, thickening or rarefaction in a longitudinal wave .

v = .

As observations show, at the same frequency, the wave speed, and accordingly the wavelength, depend on the medium in which they propagate. Table 15 shows the speed of sound in different environments at different temperatures. The table shows that in solids the speed of sound is greater than in liquids and gases, and in liquids it is greater than in gases. This is due to the fact that the molecules in liquids and solids are arranged closer friend to each other than in gases, and interact more strongly.

Table 15

Wednesday	Temperature,° WITH	Speed, m/s
Carbon dioxide	0	259
Air	0	332
Air	10	338
Air	30	349
Helium	0	965
Hydrogen	0	128
Kerosene	15	1330
Water	25	1497
Copper	20	4700
Steel	20	50006100
Glass	20	5500

The relatively high speed of sound in helium and hydrogen is explained by the fact that the mass of the molecules of these gases is less than that of others, and accordingly they have less inertia.

The speed of waves also depends on temperature. In particular, the higher the air temperature, the higher the speed of sound. The reason for this is that as the temperature increases, the mobility of the particles increases.

Self-test questions

1. What is called a mechanical wave?

2. What wave is called transverse? longitudinal?

3. What are the features of wave motion?

4. In which media do longitudinal waves propagate, and in which do transverse waves propagate? Why?

5. What is wavelength called?

6. How is wave speed related to wavelength and oscillation period? With wavelength and vibration frequency?

7. What does the speed of a wave depend on at a constant oscillation frequency?

Task 27

1. The transverse wave moves to the left (Fig. 87). Determine the direction of particle motion A in this wave.

2 * . Does energy transfer occur during wave motion? Explain your answer.

3. What is the distance between points A And B; A And C; A And D; A And E; A And F; B And F transverse wave (Fig. 88)?

4. Figure 89 shows the instantaneous position of the particles of the medium and the direction of their movement in the transverse wave. Draw the position of these particles and indicate the direction of their movement at intervals equal to T/4, T/2, 3T/4 and T.

5. What is the speed of sound in copper if the wavelength is 11.8 m at an oscillation frequency of 400 Hz?

6. A boat rocks on waves traveling at a speed of 1.5 m/s. The distance between the two nearest wave crests is 6 m. Determine the period of oscillation of the boat.

7. Determine the frequency of a vibrator that creates waves 15 m long in water at 25 °C.

Disturbances propagating in space, moving away from the place of their origin, are called waves.

Elastic waves- these are disturbances that propagate in solid, liquid and gaseous media due to the action of elastic forces in them.

These environments themselves are called elastic. Perturbation of an elastic medium is any deviation of the particles of this medium from their equilibrium position.

Take, for example, a long rope (or rubber tube) and attach one of its ends to the wall. Having pulled the rope tightly, with a sharp lateral movement of the hand we will create a short-term disturbance at its loose end. We will see that this disturbance will run along the rope and, reaching the wall, will be reflected back.

The initial disturbance of the medium, leading to the appearance of a wave in it, is caused by the action of some foreign body which is called wave source. This could be the hand of a person hitting the rope, a pebble falling into the water, etc. If the action of the source is short-term, then a so-called single wave. If the source of the wave performs a long oscillatory motion, then the waves in the medium begin to move one after another. A similar picture can be seen by placing a vibrating plate with a tip lowered into the water over a bath of water.

A necessary condition the occurrence of an elastic wave is the appearance at the moment of the disturbance of elastic forces that interfere with this disturbance. These forces tend to bring neighboring particles of the medium closer together when they move apart, and move them away when they come closer. Acting on particles of the medium that are increasingly distant from the source, elastic forces begin to remove them from their equilibrium position. Gradually, all particles of the medium, one after another, are involved in oscillatory motion. The propagation of these vibrations manifests itself in the form of a wave.

In any elastic medium, two types of motion simultaneously exist: oscillations of the particles of the medium and the propagation of disturbances. A wave in which particles of the medium oscillate along the direction of its propagation is called longitudinal, and a wave in which particles of the medium oscillate across the direction of its propagation is called transverse.

Longitudinal wave.

A wave in which oscillations occur along the direction of propagation of the wave is called longitudinal.

In an elastic longitudinal wave, disturbances represent compression and rarefaction of the medium. Compressive deformation is accompanied by the appearance of elastic forces in any medium. Therefore, longitudinal waves can propagate in all media (liquid, solid, and gaseous).

An example of the propagation of a longitudinal elastic wave is shown in the figure A And b higher. The left end of a long spring suspended by threads is struck with the hand. The impact brings several turns closer together, and an elastic force arises, under the influence of which these turns begin to diverge. Continuing to move by inertia, they will continue to diverge, bypassing the equilibrium position and forming a vacuum in this place (Figure b). With rhythmic action, the coils at the end of the spring will either approach or move away from each other, i.e., oscillate around their equilibrium position. These vibrations will gradually be transmitted from coil to coil along the entire spring. Condensations and rarefaction of turns, or an elastic wave, will spread along the spring.

Transverse wave.

Waves in which oscillations occur perpendicular to the direction of their propagation are called transverse. In a transverse elastic wave, disturbances represent displacements (shifts) of some layers of the medium relative to others.

Shear deformation leads to the appearance of elastic forces only in solids: the shift of layers in gases and liquids is not accompanied by the appearance of elastic forces. Therefore, transverse waves can propagate only in solids.

Plane wave.

Plane wave is a wave whose direction of propagation is the same at all points in space.

The amplitude of particle oscillations in a spherical wave necessarily decreases with distance from the source. The energy emitted by the source is evenly distributed over the surface of the sphere, the radius of which continuously increases as the wave propagates. The spherical wave equation is:

.

Unlike a plane wave, where s m = A- the amplitude of the wave is a constant value; in a spherical wave it decreases with distance from the center of the wave.