Physics of longitudinal and transverse waves. Longitudinal mechanical waves can propagate in any media - solid, liquid and gaseous

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 the 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 shear 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.

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 wave propagation (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 brought into vertical oscillatory motion.

Let us consider in more detail the process of formation of transverse waves. Let us take as a model of a real string a chain of balls ( material points) connected to each other by elastic forces (Fig. 15.4, a). Figure 15.4 depicts the 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 (t0 = 0) all points are in a state of equilibrium (Fig. 15.4, a). Then we cause a perturbation by deviating point 1 from the equilibrium position by the value 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 propagate 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 down, 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 t5 = T The 1st point, having made a complete oscillation, passes through the equilibrium position, and the oscillatory movement will spread to the 13th point (Fig. 15.4, e). All points from the 1st to the 13th are located so that they form a complete wave consisting of hollows 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, one can notice how successive condensations and rarefaction of its coils will spread along the spring, running one after another. In Figure 15.6, the dots show the position of the coils of the spring at rest, and then the positions of the coils of the spring at successive intervals of time equal to a quarter of the period.

Thus, the longitudinal wave in the case under consideration is an alternating cluster (Sg) and rarefaction (Once) spring coils.

The type of wave depends on the type of deformation of the medium. Longitudinal waves are due to compressive - tensile deformation, transverse waves - to shear deformation. Therefore, in gases and liquids, in which elastic forces occur only during compression, the propagation of transverse waves is impossible. In solids, elastic forces arise both during longevity (tension) and shear, therefore, 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 transmitted from one point of the medium to another. An important difference between elastic waves in a medium and any other ordered motion of its particles is that the propagation of waves is not associated with the transfer of matter in the medium.

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

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

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

Wave I intensity shows how much energy, on average, is transferred by a wave per unit of time through a unit area of ​​the surface perpendicular to the direction of wave propagation \\

The SI unit of wave intensity is the watt per square meter J / (m 2 \ (\ cdot \) c) \u003d W / m 2

The energy and intensity of the 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: Proc. allowance for institutions providing general. environments, education / L. A. Aksenovich, N. N. Rakina, K. S. Farino; Ed. K. S. Farino. - Mn.: Adukatsia i vykhavanne, 2004. - S. 425-428.

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

Considering mechanical waves, we will not pay attention to the internal structure of the medium. In this case, we consider the substance as a continuous medium, which changes from one point to another.

A particle (material point) is a small volume element of the medium, the dimensions of which are much larger than the distances between molecules.

Mechanical waves propagate only in media that have elastic properties. The elastic forces in such substances at 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, in the event that the particles of the medium oscillate in the direction of wave propagation.

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

During the propagation of a longitudinal wave in a medium, alternations of concentrations and rarefaction of particles appear, moving in the direction of wave propagation with a speed of $(\rm v)$. The shift 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 perform oscillations at their equilibrium positions, while different particles oscillate out of phase. In solids, the propagation velocity of longitudinal waves is greater than the velocity of transverse waves.

Waves in liquids and gases are always longitudinal. In a solid body, the type of wave depends on the method of its excitation. The waves on the free surface of the 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 in a medium. Longitudinal waves having frequencies from 17 to 20~000 Hz are called sound waves.

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

Acoustic waves cannot propagate in a vacuum, since elastic waves can only propagate in a medium where there is a connection between individual particles of matter. 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 of the medium changes continuously. This pressure is equal to the sum of the equilibrium pressure of the medium and the additional pressure ( sound pressure), which 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 on threads. One end of the spring is struck so that the deformation force is directed along the axis of the spring. From the impact, several coils of the spring come together, and an elastic force arises. Under the influence of the elastic force, the coils diverge. Moving by inertia, the coils of the spring pass the equilibrium position, a rarefaction is formed. For some time, the coils of the spring at the end at the point of impact will oscillate about their equilibrium position. These vibrations over time are transmitted from coil to coil throughout the spring. As a result, the condensation and rarefaction of the coils propagate, and a longitudinal elastic wave propagates.

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

transverse waves

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

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

A wave propagating along a string (example of a transverse wave)

Let a one-dimensional transverse wave propagate along the X axis, from the wave source located at the origin - 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 oscillate. The equation for such a one-dimensional wave is:

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

$k$ - wavenumber$;;\ \lambda $ - wavelength; $v$ - wave phase velocity; $A$ - 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 a solution

Example 1

Exercise. What is the length of a transverse wave if it propagates along an elastic string with a speed $v=10\ \frac(m)(s)$, while the period of the string's oscillations 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 by 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 the maximum speed of particles in the medium?

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

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

The speed of movement of the 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), given the range 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.$(\left(\frac(ds)(dt)\right))_(max)=2\pi A\nu$

Let the oscillating body be in a medium, all particles of which are interconnected. The particles of the medium in contact with it will begin to oscillate, 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 have appeared in some place of the 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 involved by the wave in translational motion, but perform oscillatory motions around their equilibrium positions, only elastic deformation is transmitted from one part of the medium to another.

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

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

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

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

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 shows 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 (t0 = 0) all points are in equilibrium. Then we cause a perturbation by deviating point 1 from the equilibrium position by the value 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 period of oscillation ( t 2 = T 4 ) 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 propagated to the 7th point, etc.

By the time t5 = T The 1st point, having made 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 hollows and comb.

Demonstration of shear wave propagation

The type of wave depends on the type of deformation of the medium. Longitudinal waves are due to compressive - tensile deformation, transverse waves - to 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, therefore, 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 motion of its particles is that the propagation of waves is not associated with the transfer of matter in the medium.