How longitudinal and transverse waves propagate. Elastic waves (mechanical waves)

Longitudinal waves

Definition 1

A wave in which oscillations occur in the direction of its propagation. Example longitudinal wave can serve as 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(speed of particles propagating in 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 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 can easily be represented as transverse waves.

Figure 3. Light waves are an example shear 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: $Asin\left(\frac(2\pi )(\lambda )\right)x$

If oscillatory motion excited at any point in the medium, then it propagates 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 continuous medium, which changes from one point to another.

A particle (material point) is 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. 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 perform oscillations 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 free surface fluids are mixed, they are both longitudinal and transverse. The trajectory of a water particle on the surface at 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, since elastic waves can propagate only in a medium where there is a connection between individual particles substances. 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 endless string, one of the ends of which is forced to perform oscillatory movements. The equation of such a one-dimensional wave is:

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

$k$ -wavenumber$;;\ \lambda$ - wavelength; $v$ - phase speed waves; $A$ - 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 the maximum speed of movement of particles in the medium?

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

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. When deformations occur in the environment, elastic forces, which strive 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 the wave by the wave. forward motion, 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.

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'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 to the maximum distance, equal to amplitude oscillations A. After half the period, the 1st point, moving downwards, 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 spread 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 1st to 13th are located so that they form full 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. Important difference elastic waves in a medium from any other ordered movement of its particles is that the propagation of waves is not associated with the transfer of matter in the medium.

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 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 (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.

IN starting moment 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 downwards, 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 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 both during tension (tension) and shear, so 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 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.

Consider, for example, a longitudinal wave in an elastic spring. At a fixed point in time kinetic energy 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 how much energy on average 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. 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 the direction perpendicular to the direction vibrations of particles in a 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 shear deformation is required for the occurrence of a transverse wave, 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.