Examples of longitudinal waves in nature. Mechanical waves are formed due to the inertness of the particles of the medium and the interaction between them, manifested in the existence of elastic forces

Disturbances that propagate in space, moving away from their place of origin, are called waves.

elastic waves- these are perturbations propagating in solid, liquid and gaseous media due to the action of elastic forces in them.

These environments 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. Pulling the rope tight, with a sharp lateral movement of the hand, we will create a short-term disturbance at its loose end. We will see that this perturbation will run along the rope and, having reached the wall, will be reflected back.

The initial perturbation of the medium, leading to the appearance of a wave in it, is caused by the action in it of some foreign body, which is called wave source. This may be the hand of a person who hit the rope, a pebble that fell into the water, etc. If the action of the source is of a short-term nature, then the so-called single wave. If the source of the wave makes a long oscillatory motion, then the waves in the medium begin to go 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.

Necessary condition The emergence of an elastic wave is the appearance at the moment of the occurrence of a perturbation of elastic forces that prevent this perturbation. These forces tend to bring neighboring particles of the medium closer together if they diverge, and move them away when they approach each other. Acting on particles of the medium that are more and more distant from the source, the elastic forces begin to take them out of their equilibrium position. Gradually, all particles of the medium, one after another, are involved in oscillatory motion. The propagation of these oscillations 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 a perturbation. A wave in which the particles of the medium oscillate along the direction of its propagation is called longitudinal, and the wave in which the 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 wave propagation is called longitudinal.

In an elastic longitudinal wave, perturbations are compressions and rarefactions of the medium. The compression deformation is accompanied by the appearance of elastic forces in any medium. Therefore, longitudinal waves can propagate in all media (in liquid, solid, and gaseous).

An example of the propagation of a longitudinal elastic wave is shown in the figure a and b above. The left end of a long spring suspended on threads is struck with a hand. From the impact, several turns approach each other, 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 rarefaction in this place (Figure b). With a rhythmic impact, the coils at the end of the spring will either approach or move away from each other, that is, they will oscillate around their equilibrium position. These vibrations will gradually be transmitted from coil to coil along the entire spring. Condensations and rarefaction of coils, or an elastic wave, will propagate along the spring.

transverse wave.

Waves in which vibrations occur perpendicular to the direction of their propagation are called transverse. In a transverse elastic wave, perturbations are 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. So transverse waves can only propagate in solids.

Flat wave.

plane wave A wave whose direction of propagation is the same at all points in space.

The amplitude of oscillations of particles 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 has the form:

.

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.

1. Wave - propagation of oscillations 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 the wave speed?

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

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

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

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

4. Wave speed 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 particle oscillations in the wave; longitudinal wave - a wave propagating in the direction coinciding with the direction of particle oscillations 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 shear deformation is required for the occurrence of a transverse wave, and it 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 appearance of a longitudinal wave.

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 compressional or compressional 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 propagating in an elastic medium) and seismic P-waves (created as a result of earthquakes and explosions). In longitudinal waves, the displacement of the medium is parallel to the direction of wave propagation.

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

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

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

transverse waves

Definition 2

transverse wave- a wave in which the direction of the molecules of the vibrations 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 easy to imagine as transverse waves.

Figure 3 light waves are an example of a transverse wave

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

Definition 3

2D 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 plane undamped wave equation $(\rm y=Acos)\left(\omega t-\frac(2\pi )(\lambda )\right)x+(\varphi )_0$ 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 values ​​of the $\ \ x$ coordinate, which are equal to $\frac(\lambda)(4)$; $\frac(\lambda)(2)$; $\frac(0.75)(\lambda)$.

Figure 4

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

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

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

Longitudinal wave- this is a wave, during the propagation of which the displacement of the particles of the medium occurs in the direction of the wave propagation (Fig. 1, a).

The reason for the appearance of a longitudinal wave is the compression / extension deformation, i.e. the resistance of a medium to a change 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 displacement of the particles of the medium occurs in the direction perpendicular to the propagation of the wave (Fig. 1b).

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

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

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

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

Question 30

Wavelength.

Each wave propagates at a certain speed. Under wave speed understand the propagation speed 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 a wave is determined by the properties of the medium in which this wave propagates. When a wave passes from one medium to another, its speed changes.

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

Since the speed of the 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. In this way, 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 oscillations in the wave; λ ( Greek letter lambda is the wavelength.

By choosing the direction of wave propagation beyond 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 (for a fixed time t) is shown in Figure 45. The distance between adjacent crests (or troughs) on this graph coincides with the wavelength λ.

Formula (22.1) expresses the relationship of the wavelength with its speed and period. Taking into account that the period of oscillations 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 a 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, the analytical expression of the function of two variables S = f(t, x) , imagine that, at some point in space, harmonic oscillations arise with a circular frequency w and the initial phase, for simplification equal to zero (see Fig. 8). Point offset M: S m = A sin w t, where A- amplitude. Since the particles of the medium filling the space are interconnected, the oscillations from the 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 as.

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

We fix one end of the cord, pull it slightly and move the free end of the cord up and then down (we will bring it into oscillation). We will see that a wave will “run” along the cord (Fig. 84). The parts of the cord have inertia, so they will move relative to the equilibrium position not simultaneously, but with some delay. Gradually, all sections of the cord will come into oscillation. An oscillation will spread through 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, while the particle oscillates in the vertical direction.

Waves, the direction of propagation of which is perpendicular to the direction of oscillation of the particles of the medium, are called transverse.

Transverse waves are an alternation humps and hollows.

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

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

Let us fasten one end of a long spring suspended by threads and hit its other end. We will see how the condensation of coils that has arisen at the end of the spring “runs” along it (Fig. 85). There is a displacement condensations 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 the 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 oscillations; the oscillation frequency of all particles is the same and equal to the frequency of the oscillation source;

- the oscillation of each particle occurs with a delay, which is due to its inertia; this delay is the greater, the farther the particle is from the source of oscillations.

An important property wave motion is that matter is not transported along with the wave. This is easy to verify. If you throw pieces of cork on the surface of the water and create a wave motion, you can see that the waves “run” along the surface of the water. Pieces of cork will rise up on the crest of the wave and fall down on the trough.

3. 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 both in solids and in liquid and gaseous bodies, since in all these bodies, when their volume changes, elastic forces arise.

The propagation of transverse waves is associated mainly with a change in the shape of the body. In gases and liquids, when their shape changes, elastic forces do not arise, therefore 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 an earthquake. The seismic station first receives longitudinal waves, and then transverse ones, since the speed of the latter is less. 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 period of oscillation.

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

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

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

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

As observations show, at the same frequency, the speed of the wave, and, accordingly, the wavelength depends 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 located closer to each other than in gases and interact more strongly.

Table 15

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 others, and, accordingly, they have less inertia.

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

Questions for self-examination

1. What is a mechanical wave?

2. What is a transverse wave? longitudinal?

3. What are the features of wave motion?

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

5. What is the wavelength?

6. How is the speed of a wave related to the wavelength and the period of oscillation? With wavelength and frequency?

7. What determines the speed of a wave at a constant frequency of oscillation?

Task 27

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

2 * . Does wave motion transfer energy? Explain the answer.

3. What is the distance between the 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 motion in a 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, at an oscillation frequency of 400 Hz, the wavelength is 11.8 m?

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

7. Determine the frequency of the vibrator, which creates a wave length of 15 m in water at 25 ° C.