Open Library - an open library of educational information. Propagation of vibrations in an elastic medium. Longitudinal and transverse waves

10.10.2019

Mechanical vibrations propagating in an elastic medium (solid, liquid or gaseous) are called mechanical or elastic waves.

The process of propagation of vibrations in a continuous medium is called a wave process or wave. Particles of the medium in which the wave propagates are not drawn into translational motion by the wave. They only oscillate around their equilibrium positions. Together with the wave, only the state of oscillatory motion and its energy are transferred from particle to particle of the medium. That's why the main property of all waves, regardless of their nature, is the transfer of energy without transfer of matter.

Depending on the direction of particle vibrations relative to

to the direction in which the wave propagates, there are pro-

lobar And transverse waves.

An elastic wave is called longitudinal, if the vibrations of the particles of the medium occur in the direction of wave propagation. Longitudinal waves are associated with volumetric tensile-compression deformation of the medium, therefore they can propagate both in solids and

in liquids and gaseous media.

x subject to shear deformation. Only solid bodies have this property.

λ In Fig. 6.1.1 presents the harmonic

dependence of the displacement of all particles of the medium on the distance to the source of oscillations at a given time. The distance between nearby particles vibrating in the same phase is called wavelength. The wavelength is also equal to the distance over which a certain phase of oscillation extends during the oscillation period

Not only particles located along the 0 axis oscillate X, but a collection of particles contained in a certain volume. The geometric location of the points to which the oscillations reach at the moment of time t, called wave front. The wave front is the surface that separates the part of space already involved in the wave process from the area in which oscillations have not yet arisen. The geometric location of points oscillating in the same phase is called wave surface. The wave surface can be drawn through any point in space covered by the wave process. Wave surfaces can be of any shape. In the simplest cases, they have the shape of a plane or sphere. Accordingly, the wave in these cases is called flat or spherical. In a plane wave, the wave surfaces are a set of planes parallel to each other, and in a spherical wave they are a set of concentric spheres.

Plane wave equation

The plane wave equation is an expression that gives the displacement of an oscillating particle as a function of its coordinates x, y, z and time t

S=S(x,y,z,t).

(6.2.1)

This function must be periodic both with respect to time t, and relative to the coordinates x, y, z. Periodicity in time follows from the fact that the displacement S describes the vibrations of a particle with coordinates x, y, z, and periodicity in coordinates follows from the fact that points spaced from each other at a distance equal to the wavelength vibrate in the same way.

Let us assume that the oscillations are harmonic in nature, and the axis 0 X coincides with the direction of wave propagation. Then the wave surfaces will be perpendicular to the 0 axis X and since everything

points of the wave surface oscillate equally, displacement S will depend only on the coordinate X and time t

Let us find the type of oscillation of points in the plane corresponding to an arbitrary value X. In order to travel the path from the plane X= 0 to plane X, the wave requires time τ = x/v. Consequently, vibrations of particles lying in the plane X, will lag in time by τ from particle oscillations in the plane X= 0 and described by the equation

S(x;t)=A cosω( t− τ)+ϕ	= A cos	ω t −	x	+ϕ	. (6.2.4)


				υ

Where A− wave amplitude; ϕ 0 – initial phase of the wave (determined by the choice of reference points X And t).

Let us fix some phase value ω( t − xυ) +ϕ 0 = const.

This expression defines the relationship between time t and that place X, in which the phase has a fixed value. Differentiating this expression, we get

Let us give the plane wave equation a symmetrical relative

strictly X And t view. To do this, we introduce the quantity k= 2 λ π, which is called

yes wave number, which can be represented in the form

We assumed that the amplitude of oscillations does not depend on X. For a plane wave, this is observed in the case when the wave energy is not absorbed by the medium. When propagating in an energy-absorbing medium, the intensity of the wave gradually decreases with distance from the source of oscillations, i.e., wave attenuation is observed. In a homogeneous medium, such attenuation occurs exponentially

law A = A 0 e −β x. Then the plane wave equation for the absorbing medium has the form

Where r r – radius vector, wave points; k = k ⋅n r − wave vector; n r is the unit vector of the normal to the wave surface.

Wave vector− is a vector equal in magnitude to the wave number k and having the direction of the normal to the wave surface on-

called.
Let's move from the radius vector of a point to its coordinates x, y, z
r	r	(6.3.2)
k	⋅r=k x x+k y y+k z z.
Then equation (6.3.1) will take the form
S(x,y,z;t)=A cos(ω t−k x x−k y y−k z z+ϕ 0).	(6.3.3)

Let us establish the form of the wave equation. To do this, we find the second partial derivatives with respect to coordinates and time, expression (6.3.3)

∂ 2 S

∂t

= −ω A cos

(ω t − k ⋅ r

+ϕ 0) = −ω S;

∂ 2 S

∂x

= − k x A cos(ω t − k

⋅r

+ϕ 0) = − k x S

(6.3.4)

∂ 2 S

∂y

= − k y A cos

(ω t − k ⋅ r

+ϕ 0) = − k y S;

∂ 2 S

∂z

= − k z A cos(ω t − k

⋅r

+ϕ 0) = − k z S

Adding derivatives with respect to coordinates, and taking into account the derivative

in time, we get

∂

S 2

+ ∂

S 2

= − (k x 2 + k y 2 + k z 2)S

= − k 2 S =

S 2 .

(6.3.5)

∂t

∂x

∂y

∂z

We will make a replacement

ω 2

and we get the wave equation

∂ 2 S

1 ∂ 2 S

(6.3.6)

∂x 2

∂y 2

∂z 2

υ 2 ∂ t 2

where =

∂ 2

− Laplace operator.

∂x 2

∂y 2

∂z 2

We present to your attention a video lesson on the topic “Propagation of vibrations in an elastic medium. Longitudinal and transverse waves." In this lesson we will study issues related to the propagation of vibrations in an elastic medium. You will learn what a wave is, how it appears, and how it is characterized. Let's study the properties and differences between longitudinal and transverse waves.

We move on to studying issues related to waves. Let's talk about what a wave is, how it appears and how it is characterized. It turns out that, in addition to simply an oscillatory process in a narrow region of space, it is also possible for these oscillations to propagate in a medium; it is precisely this propagation that is wave motion.

Let's move on to discuss this distribution. To discuss the possibility of the existence of oscillations in a medium, we must decide what a dense medium is. A dense medium is a medium that consists of a large number of particles whose interaction is very close to elastic. Let's imagine the following thought experiment.

Rice. 1. Thought experiment

Let us place a ball in an elastic medium. The ball will shrink, decrease in size, and then expand like a heartbeat. What will be observed in this case? In this case, the particles that are adjacent to this ball will repeat its movement, i.e. moving away, approaching - thereby they will oscillate. Since these particles interact with other particles more distant from the ball, they will also oscillate, but with some delay. Particles that come close to this ball vibrate. They will be transmitted to other particles, more distant. Thus, the vibration will spread in all directions. Please note that in this case the vibration state will propagate. We call this propagation of a state of oscillation a wave. It can be said that the process of propagation of vibrations in an elastic medium over time is called a mechanical wave.

Please note: when we talk about the process of occurrence of such oscillations, we must say that they are possible only if there is interaction between particles. In other words, a wave can only exist when there is an external disturbing force and forces that resist the action of the disturbance force. In this case, these are elastic forces. The propagation process in this case will be related to the density and strength of interaction between the particles of a given medium.

Let's note one more thing. The wave does not transport matter. After all, particles oscillate near the equilibrium position. But at the same time, the wave transfers energy. This fact can be illustrated by tsunami waves. Matter is not carried by the wave, but the wave carries such energy that it brings great disasters.

Let's talk about wave types. There are two types - longitudinal and transverse waves. What's happened longitudinal waves? These waves can exist in all media. And the example with a pulsating ball inside a dense medium is just an example of the formation of a longitudinal wave. Such a wave is a propagation in space over time. This alternation of compaction and rarefaction is a longitudinal wave. I repeat once again that such a wave can exist in all media - liquid, solid, gaseous. A longitudinal wave is a wave whose propagation causes particles of the medium to oscillate along the direction of propagation of the wave.

Rice. 2. Longitudinal wave

As for the transverse wave, then transverse wave can exist only in solids and on the surface of liquids. A transverse wave is a wave whose propagation causes particles of the medium to oscillate perpendicular to the direction of propagation of the wave.

Rice. 3. Transverse wave

The speed of propagation of longitudinal and transverse waves is different, but this is the topic of the following lessons.

List of additional literature:

Are you familiar with the concept of a wave? // Quantum. - 1985. - No. 6. — P. 32-33. Physics: Mechanics. 10th grade: Textbook. for in-depth study of physics / M.M. Balashov, A.I. Gomonova, A.B. Dolitsky and others; Ed. G.Ya. Myakisheva. - M.: Bustard, 2002. Elementary physics textbook. Ed. G.S. Landsberg. T. 3. - M., 1974.

Let's start with the definition of an elastic medium. As one can conclude from the name, an elastic medium is a medium in which elastic forces act. With regard to our goals, we will add that with any disturbance of this environment (not an emotional violent reaction, but a deviation of the parameters of the environment in some place from equilibrium), forces arise in it, striving to return our environment to its original equilibrium state. In this case, we will consider extended media. We will clarify how extensive this is in the future, but for now we will assume that this is enough. For example, imagine a long spring attached at both ends. If several turns of the spring are compressed in some place, the compressed turns will tend to expand, and the adjacent turns that are stretched will tend to compress. Thus, our elastic medium - the spring - will try to return to its original calm (undisturbed) state.

Gases, liquids, and solids are elastic media. An important thing in the previous example is the fact that the compressed section of the spring acts on neighboring sections, or, in scientific terms, transmits a disturbance. In a similar way, in gas, creating in some place, for example, an area of low pressure, neighboring areas, trying to equalize the pressure, will transmit the disturbance to their neighbors, who, in turn, to their own, and so on.

A few words about physical quantities. In thermodynamics, as a rule, the state of a body is determined by parameters common to the entire body, gas pressure, its temperature and density. Now we will be interested in the local distribution of these quantities.

If an oscillating body (string, membrane, etc.) is in an elastic medium (gas, as we already know, is an elastic medium), then it sets the particles of the medium in contact with it into oscillatory motion. As a result, periodic deformations (for example, compression and discharge) occur in the elements of the environment adjacent to the body. With these deformations, elastic forces appear in the medium, tending to return the elements of the medium to their original states of equilibrium; Due to the interaction of neighboring elements of the medium, elastic deformations will be transmitted from one part of the medium to others, more distant from the oscillating body.

Thus, periodic deformations caused in some place of an elastic medium will propagate in the medium at a certain speed, depending on its physical properties. In this case, the particles of the medium perform oscillatory movements around equilibrium positions; Only the state of deformation is transmitted from one part of the medium to another.

When a fish “bites” (pulls the hook), circles scatter across the surface of the water from the float. Together with the float, the water particles in contact with it move, which involve other particles closest to them in movement, and so on.

The same phenomenon occurs with particles of a stretched rubber cord if one end of it is vibrated (Fig. 1.1).

The propagation of oscillations in a medium is called wave motion. Let us consider in more detail how a wave arises on a cord. If we fix the positions of the cord every 1/4 T (T is the period with which the hand oscillates in Fig. 1.1) after the start of oscillation of its first point, you will get the picture shown in Fig. 1.2, b-d. Position a corresponds to the beginning of oscillations of the first point of the cord. Its ten points are marked with numbers, and the dotted lines show where the same points of the cord are located at different points in time.

1/4 T after the start of oscillation, point 1 occupies the highest position, and point 2 is just beginning its movement. Since each subsequent point of the cord begins its movement later than the previous one, then in the interval 1-2 points are located, as shown in Fig. 1.2, b. After another 1/4 T, point 1 will take the equilibrium position and move downward, and point 2 will take the upper position (position c). Point 3 at this moment is just beginning to move.

Over the entire period, the oscillations propagate to point 5 of the cord (position d). At the end of period T, point 1, moving upward, will begin its second oscillation. At the same time, point 5 will begin to move upward, making its first oscillation. In the future, these points will have the same oscillation phases. The combination of cord points in the interval 1-5 forms a wave. When point 1 completes the second oscillation, another 5-10 points on the cord will be involved in the movement, i.e. a second wave will form.

If you trace the position of points that have the same phase, you will see that the phase seems to move from point to point and moves to the right. Indeed, if in position b point 1 has phase 1/4, then in position c point 2 has the same phase, etc.

Waves in which the phase moves at a certain speed are called traveling. When observing waves, it is the phase propagation that is visible, such as the movement of the wave crest. Note that all points of the medium in the wave oscillate around their equilibrium position and do not move with the phase.

The process of propagation of oscillatory motion in a medium is called a wave process or simply a wave.

Depending on the nature of the elastic deformations that arise, waves are distinguished longitudinal And transverse. In longitudinal waves, particles of the medium oscillate along a line coinciding with the direction of propagation of the oscillations. In transverse waves, particles of the medium oscillate perpendicular to the direction of propagation of the wave. In Fig. Figure 1.3 shows the location of particles of the medium (conventionally depicted as dashes) in longitudinal (a) and transverse (b) waves.

Liquid and gaseous media do not have shear elasticity and therefore only longitudinal waves are excited in them, propagating in the form of alternating compression and rarefaction of the medium. The waves excited on the surface of the hearth are transverse: they owe their existence to gravity. In solids, both longitudinal and transverse waves can be generated; A particular type of transverse will is torsional, excited in elastic rods to which torsional vibrations are applied.

Let us assume that a point source of a wave began to excite oscillations in the medium at the moment of time t= 0; after time has passed t this vibration will spread in different directions at a distance r i =c i t, Where with i- wave speed in a given direction.

The surface to which the oscillation reaches at some point in time is called the wave front.

It is clear that the wave front (wave front) moves with time in space.

The shape of the wave front is determined by the configuration of the oscillation source and the properties of the medium. In homogeneous media, the speed of wave propagation is the same everywhere. The environment is called isotropic, if this speed is the same in all directions. The wave front from a point source of oscillations in a homogeneous and isotropic medium has the shape of a sphere; such waves are called spherical.

In a non-uniform and non-isotropic ( anisotropic) environment, as well as from non-point sources of oscillations, the wave front has a complex shape. If the wave front is a plane and this shape is maintained as vibrations propagate in the medium, then the wave is called flat. Small sections of the wave front of a complex shape can be considered a plane wave (if we only consider the short distances traveled by this wave).

When describing wave processes, surfaces are identified in which all particles vibrate in the same phase; these “surfaces of the same phase” are called wave or phase.

It is clear that the wave front represents the front wave surface, i.e. the most distant from the source creating the waves, and the wave surfaces can also be spherical, flat, or have a complex shape, depending on the configuration of the source of oscillations and the properties of the medium. In Fig. 1.4 conventionally shows: I - a spherical wave from a point source, II - a wave from a vibrating plate, III - an elliptical wave from a point source in an anisotropic medium in which the wave propagation speed With changes smoothly as the angle α increases, reaching a maximum along the AA direction and a minimum along BB.

Rice. 1. Thought experiment

Rice. 2. Longitudinal wave

Rice. 3. Transverse wave

The speed of propagation of longitudinal and transverse waves is different, but this is the topic of the following lessons.

List of additional literature: