Longitudinal and transverse waves.

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 oscillation; the vibration frequency of all particles is the same and equal to the frequency of the vibration source;

- the oscillation 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 mainly associated 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 Oh.

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 the 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.

If oscillatory motion is excited at any point in the medium, then it spreads from one point to another as a result of the interaction of particles of the substance. The process of vibration propagation is called a wave.

When considering mechanical waves, we will not pay attention to internal structure environment. In this case, we consider the substance to be a continuous medium that changes from one point to another.

Particle ( material point), we will call a small element of the volume of the medium, the dimensions of which are much larger than the distances between the molecules.

Mechanical waves propagate only in media that have elastic properties. The elastic forces in such substances under small deformations are proportional to the magnitude of the deformation.

The main property of the wave process is that the wave, while transferring energy and oscillatory motion, does not transfer mass.

Waves are longitudinal and transverse.

Longitudinal waves

I call a wave longitudinal if the particles of the medium oscillate in the direction of propagation of the wave.

Longitudinal waves propagate in a substance in which elastic forces arise during tensile and compressive deformation in a substance in any state of aggregation.

When a longitudinal wave propagates in a medium, alternations of condensations and rarefactions of particles appear, moving in the direction of wave propagation at a speed of $(\rm v)$. The displacement of particles in this wave occurs along a line that connects their centers, that is, it causes a change in volume. Throughout the existence of the wave, the elements of the medium oscillate at their equilibrium positions, while different particles oscillate with a phase shift. In solids, the speed of propagation of longitudinal waves is greater than the speed of transverse waves.

Waves in liquids and gases are always longitudinal. In a solid, the type of wave depends on the method of its excitation. Waves on the free surface of a liquid are mixed; they are both longitudinal and transverse. The trajectory of a water particle on the surface during a wave process is an ellipse or an even more complex figure.

Acoustic waves (example of longitudinal waves)

Sound (or acoustic) waves are longitudinal waves. Sound waves in liquids and gases are pressure fluctuations propagating through a medium. Longitudinal waves with frequencies from 17 to 20~000 Hz are called sound waves.

Acoustic vibrations with a frequency below the limit of audibility are called infrasound. Acoustic vibrations with a frequency above 20~000 Hz are called ultrasound.

Acoustic waves cannot propagate in a vacuum because elastic waves are able to spread only in the environment where there is a connection between individual particles of the substance. The speed of sound in air is on average 330 m/s.

The propagation of longitudinal sound waves in an elastic medium is associated with volumetric deformation. In this process, the pressure at each point in the medium changes continuously. This pressure is equal to the sum of the equilibrium pressure of the medium and the additional pressure (sound pressure) that appears as a result of deformation of the medium.

Compression and extension of a spring (example of longitudinal waves)

Let us assume that an elastic spring is suspended horizontally by threads. One end of the spring is struck so that the deformation force is directed along the axis of the spring. The impact brings several coils of the spring closer together, and an elastic force arises. Under the influence of elastic force, the coils diverge. Moving by inertia, the coils of the spring pass the equilibrium position, and a vacuum is formed. For some time, the coils of the spring at the end at the point of impact will oscillate around their equilibrium position. These vibrations are transmitted over time from coil to coil throughout the spring. As a result, the condensation and rarefaction of the coils spread, and a longitudinal elastic wave propagates.

Similarly, a longitudinal wave propagates along a metal rod if its end is struck with a force directed along its axis.

Transverse waves

A wave is called a transverse wave if the vibrations of the particles of the medium occur in directions perpendicular to the direction of propagation of the wave.

Mechanical waves can be transverse only in a medium in which shear deformations are possible (the medium has elasticity of shape). Transverse mechanical waves arise in solids.

Wave propagating along a string (an example of a transverse wave)

Let a one-dimensional transverse wave propagate along the X axis, from the wave source located at the origin of coordinates - point O. An example of such a wave is a wave that propagates in an elastic infinite string, one of the ends of which is forced to oscillatory movements. The equation of such a one-dimensional wave is:

\\ )\left(1\right),\]

$k$ -wavenumber$;;\ \lambda$ - wavelength; $v$ is the phase velocity of the wave; $A$ - amplitude; $\omega$ - cyclic oscillation frequency; $\varphi $ - initial phase; the quantity $\left[\omega t-kx+\varphi \right]$ is called the phase of the wave at an arbitrary point.

Examples of problems with solutions

Example 1

Exercise. What is the length of the transverse wave if it propagates along an elastic string with a speed of $v=10\ \frac(m)(s)$, while the period of oscillation of the string is $T=1\ c$?

Solution. Let's make a drawing.

The wavelength is the distance that the wave travels in one period (Fig. 1), therefore, it can be found using the formula:

\[\lambda =Tv\ \left(1.1\right).\]

Let's calculate the wavelength:

\[\lambda =10\cdot 1=10\ (m)\]

Answer.$\lambda =10$ m

Example 2

Exercise. Sound vibrations with frequency $\nu $ and amplitude $A$ propagate in an elastic medium. What is maximum speed movement of particles in the environment?

Solution. Let's write the equation of a one-dimensional wave:

\\ )\left(2.1\right),\]

The speed of movement of particles of the medium is equal to:

\[\frac(ds)(dt)=-A\omega (\sin \left[\omega t-kx+\varphi \right]\ )\ \left(2.2\right).\]

The maximum value of expression (2.2), taking into account the range of values ​​of the sine function:

\[(\left(\frac(ds)(dt)\right))_(max)=\left|A\omega \right|\left(2.3\right).\]

We find the cyclic frequency as:

\[\omega =2\pi \nu \ \left(2.4\right).\]

Finally, the maximum value of the speed of movement of particles of the medium in our longitudinal (sound) wave is equal to:

\[(\left(\frac(ds)(dt)\right))_(max)=2\pi A\nu .\]

Answer.$(\left(\frac(ds)(dt)\right))_(max)=2\pi A\nu$

1. Wave - propagation of vibrations from point to point from particle to particle. For a wave to occur in a medium, deformation is necessary, since without it there will be no elastic force.

2. What is wave speed?

2. Wave speed - the speed of propagation of vibrations in space.

3. How are speed, wavelength and frequency of oscillations of particles in a wave related to each other?

3. The speed of the wave is equal to the product of the wavelength and the oscillation frequency of the particles in the wave.

4. How are speed, wavelength and period of oscillation of particles in a wave related to each other?

4. The speed of the wave is equal to the wavelength divided by the period of oscillation in the wave.

5. What wave is called longitudinal? Transverse?

5. Transverse wave - a wave propagating in a direction perpendicular to the direction of oscillation of particles in the wave; longitudinal wave - a wave propagating in a direction coinciding with the direction of oscillation of particles in the wave.

6. In what media can transverse waves arise and propagate? Longitudinal waves?

6. Transverse waves can arise and propagate only in solid media, since the occurrence of a transverse wave requires shear deformation, and this is possible only in solids. Longitudinal waves can arise and propagate in any medium (solid, liquid, gaseous), since compression or tension deformation is necessary for the occurrence of a longitudinal wave.

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 vibrational 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 only propagate 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 the 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.