Examples of longitudinal and transverse waves. A

There are longitudinal and transverse waves. The wave is called transverse, if the particles of the medium oscillate in a direction perpendicular to the direction of propagation of the wave (Fig. 15.3). A transverse wave propagates, for example, along a stretched horizontal rubber cord, one of the ends of which is fixed and the other is set in a vertical oscillatory motion.

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 (Fig. 15.4, a). Figure 15.4 depicts the process of shear wave propagation 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 (Fig. 15.4, a). 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 period, the oscillations \(\Bigr(t_2 = \frac(T)(4) \Bigl)\) 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 oscillation amplitude A ( Fig. 15.4, b). After half a period, the 1st point, moving downward, will return to the equilibrium position, the 4th deviated from the equilibrium position by a distance equal to the amplitude of oscillations A (Fig. 15.4, c), the wave 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 (Fig. 15.4, d). All points from the 1st to the 13th are located so that they form a complete wave consisting of depressions And hump.

The wave is called longitudinal, if the particles of the medium oscillate in the direction of wave propagation (Fig. 15.5).

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 Figure 15.6, the dots show the position of the spring coils at rest, and then the positions of the spring coils at successive intervals equal to a quarter of the period.

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

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 tension (tension) and shear, so the propagation of both longitudinal and transverse waves is possible in them.

As Figures 15.4 and 15.6 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 vibrating particles and from potential energy elastic deformation of the medium.

Consider, for example, a longitudinal wave in an elastic spring. At a fixed moment in time, kinetic energy is distributed unevenly over the spring, since some coils of the spring are at rest at this moment, while others, on the contrary, move with maximum speed. The same is true for potential energy, since at this moment some elements of the spring are not deformed, while others are deformed to the maximum. Therefore, when considering wave energy, a characteristic is introduced such as the density \(\omega\) of kinetic and potential energies (\(\omega=\frac(W)(V) \) - energy per unit volume). The wave energy density at each point of the medium does not remain constant, but changes periodically as the wave passes: the energy spreads along with the wave.

Any source of waves has energy W, which the wave transmits to the particles of the medium during its propagation.

Wave I intensity shows the average energy a wave transfers per unit time through a unit surface area perpendicular to the direction of propagation of the wave\

The SI unit of wave intensity is watt per square meter J/(m 2 \(\cdot\) c) = W/m 2

The energy and intensity of a wave are directly proportional to the square of its amplitude \(~I \sim A^2\).

Literature

Aksenovich L. A. Physics in high school: Theory. Tasks. Tests: Textbook. allowance for institutions providing general education. environment, education / L. A. Aksenovich, N. N. Rakina, K. S. Farino; Ed. K. S. Farino. - Mn.: Adukatsiya i vyhavanne, 2004. - P. 425-428.

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 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.

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. The 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.

Longitudinal waves

Definition 1

A wave in which oscillations occur in the direction of its propagation. An example of a longitudinal wave is a sound wave.

Figure 1. Longitudinal wave

Mechanical longitudinal waves are also called compression waves or compression waves because they produce compression as they move through a medium. Transverse mechanical waves are also called "T-waves" or "shear waves".

Longitudinal waves include acoustic waves (the speed of particles traveling in an elastic medium) and seismic P waves (created by earthquakes and explosions). In longitudinal waves, the displacement of the medium is parallel to the direction of propagation of the wave.

Sound waves

In the case of longitudinal harmonic sound waves, the frequency and wavelength can be described by the formula:

$y_0-$ oscillation amplitude;\textit()

$\omega -$ wave angular frequency;

$c-$ wave speed.

The usual frequency of the $\left((\rm f)\right)$wave is given by

The speed of sound propagation depends on the type, temperature and composition of the medium through which it travels.

In an elastic medium, a harmonic longitudinal wave travels in the positive direction along the axis.

Transverse waves

Definition 2

Transverse wave- a wave in which the direction of the molecules of vibration of the medium is perpendicular to the direction of propagation. An example of transverse waves is an electromagnetic wave.

Figure 2. Longitudinal and transverse waves

Ripples in a pond and waves on a string are easily represented as transverse waves.

Figure 3. Light waves are an example of a transverse wave

Transverse waves are waves that oscillate perpendicular to the direction of propagation. There are two independent directions in which wave movements can occur.

Definition 3

Two-dimensional shear waves exhibit a phenomenon called polarization.

Electromagnetic waves behave in the same way, although it's a little harder to see. Electromagnetic waves are also two-dimensional transverse waves.

Example 1

Prove that the equation of a plane undamped wave is $(\rm y=Acos)\left(\omega t-\frac(2\pi )(\lambda )\right)x+(\varphi )_0$ for the wave shown in the figure , can be written as $(\rm y=Asin)\left(\frac(2\pi )(\lambda )\right)x$. Verify this by substituting the coordinate values ​​$\ \ x$ that are $\frac(\lambda)(4)$; $\frac(\lambda)(2)$; $\frac(0.75)(\lambda)$.

Figure 4.

The equation $y\left(x\right)$ for a plane undamped wave does not depend on $t$, which means that the moment of time $t$ can be chosen arbitrarily. Let us choose the moment of time $t$ such that

\[\omega t=\frac(3)(2)\pi -(\varphi )_0\] \

Let's substitute this value into the equation:

\ \[=Acos\left(2\pi -\frac(\pi )(2)-\left(\frac(2\pi )(\lambda )\right)x\right)=Acos\left(2\ pi -\left(\left(\frac(2\pi )(\lambda )\right)x+\frac(\pi )(2)\right)\right)=\] \[=Acos\left(\left (\frac(2\pi )(\lambda )\right)x+\frac(\pi )(2)\right)=Asin\left(\frac(2\pi )(\lambda )\right)x\] \ \ \[(\mathbf x)(\mathbf =)\frac((\mathbf 3))((\mathbf 4))(\mathbf \lambda )(\mathbf =)(\mathbf 18),(\mathbf 75)(\mathbf \ cm,\ \ \ )(\mathbf y)(\mathbf =\ )(\mathbf 0),(\mathbf 2)(\cdot)(\mathbf sin)\frac((\mathbf 3 ))((\mathbf 2))(\mathbf \pi )(\mathbf =-)(\mathbf 0),(\mathbf 2)\]

Answer: $Asin\left(\frac(2\pi )(\lambda )\right)x$

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.

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 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 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 are caused 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 during both compression (tension) and shear, so 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 energy of a wave in an elastic medium consists of the kinetic energy of oscillating particles and the potential energy of elastic deformation of the medium.