What is the speed of wave propagation? Transverse waves are waves when the displacement of the oscillating points is directed perpendicular to the speed of propagation of the waves. Wave propagation speed

Let us assume that the point oscillating is located in the medium, all particles

which are interconnected. Then the energy of its vibration can be transferred to the surrounding -

pressing points, causing them to oscillate.

The phenomenon of vibration propagation in a medium is called a wave.

Let us immediately note that when oscillations propagate in a medium, i.e., in a wave, I oscillate -

moving particles do not move with a propagating oscillatory process, but oscillate around their equilibrium positions. Therefore, the main property of all waves, regardless of their nature, is the transfer of energy without transfer of mass of matter.

Longitudinal and transverse waves

If the particle vibrations are perpendicular to the direction of propagation of the vibration -

ny, then the wave is called transverse; rice. 1, here - acceleration, - displacement, - ampli -

there, is the period of oscillation.

If the particles oscillate along the same straight line along which they propagate

oscillation, then we will call the wave longitudinal; rice. 2, where is acceleration, is displacement,

Amplitude is the period of oscillation.

Elastic media and their properties

Are the waves propagating in the medium longitudinal or transverse?

– depends on the elastic properties of the medium.

If, when one layer of a medium is shifted relative to another layer, elastic forces arise, tending to return the shifted layer to an equilibrium position, then transverse waves can propagate in the medium. Such a medium is a solid body.

If elastic forces do not arise in the medium when parallel layers are shifted relative to each other, then transverse waves cannot form. For example, liquid and gas represent media in which transverse waves do not propagate. The latter does not apply to the surface of a liquid, in which transverse waves of a more complex nature can also propagate: in them particles move in closed circles.

vy trajectories.

If elastic forces arise in a medium due to compressive or tensile deformation, then longitudinal waves can propagate in the medium.

Only longitudinal waves propagate in liquids and gases.

In solids, longitudinal waves can propagate along with transverse ones -

The speed of propagation of longitudinal waves is inversely proportional to the square root of the elasticity coefficient of the medium and its density:

since approximately the Young’s modulus of the medium, then (1) can be replaced by the following:

The speed of shear waves depends on the shear modulus:

(3)

Wavelength, phase speed, wave surface, wave front

The distance over which a certain phase of oscillation propagates in one

The period of oscillation is called the wavelength; the wavelength is denoted by the letter .

In Fig. 3 graphically interprets the relationship between the displacement of particles of the medium participating in the wave - new process, and the distance of these particles, for example, particles, from the source of oscillations for some fixed moment in time. Given gra - fic is a graph of a harmonic transverse wave that propagates with speed along the directions - distribution leniya. From Fig. 3 it is clear that the wavelength is the shortest distance between points oscillating in the same phases. Although,

The given graph is similar to the harmonic graph - ical vibrations, but they are essentially different: if the wave graph determines the dependence of the displacement of all particles of the medium on the distance to the source of oscillations in

this moment

time, then the oscillation graph is the dependence of change -

of a given particle as a function of time.

The speed of propagation of a wave means its phase speed, i.e. the speed of propagation of a given phase of oscillation; for example, at the time point , Fig. 1, Fig. 3 had some kind of initial phase, that is, it left the equilibrium position; then, after a period of time, a point located at a distance from the point acquired the same initial phase. Consequently, the initial phase has spread over a distance in a time equal to the period. Hence, for the phase velocity according to -

we get the definition:

Let's imagine that the point from which the oscillations come (the center of oscillation) oscillates in a continuous medium. Vibrations spread from the center in all directions.

The geometric location of the points to which the oscillation has reached at a certain point in time is called the wave front.

It is also possible to identify in the environment the geometric location of points oscillating in one direction -

nak phases; this collection of points forms a surface of identical phases or waves -

The shape of the wave front determines the types of waves, for example, a plane wave is a wave whose front represents a plane, etc.

The directions in which vibrations propagate are called rays. In iso -

in a tropic environment, the rays are normal to the wave front; with a spherical wave front, the rays on -

adjusted according to radii.

Traveling Sine Wave Equation

Let us find out how the wave process can be analytically characterized,

rice. 3. Let us denote by the displacement of the point from the equilibrium position. The wave process will be known if we know what value it has at each moment of time for each point on the straight line along which the wave propagates.

Let the oscillations at the point in Fig. 3, occur according to the law:

(5)

here is the amplitude of oscillations; - circular frequency; - time counted from the moment the oscillations begin.

Let us take an arbitrary point in the direction lying from the origin of the coordinate -

nat at a distance. Oscillations, propagating from a point with phase velocity (4), will reach the point after a period of time

Consequently, the point will begin to oscillate a time later than the point. If the waves do not damp, then its displacement from the equilibrium position will be

(7)

where is the time counted from the moment when the point began to oscillate, which is related to time as follows: , because the point began to oscillate a period of time later; substituting this value into (7), we get

or, using (6) here, we have

This expression (8) gives the displacement as a function of time and the distance of the point from the center of oscillation; it represents the desired equation of the wave, propagating -

running along , Fig. 3.

Formula (8) is the equation of a plane wave propagating along Indeed, in this case, any plane, Fig. 4, perpendicular to the direction, will represent the top - ity of identical phases, and, therefore, all points of this plane have the same displacement at the same moment in time, defined - determined only by the distance at which the plane lies from the origin of coordinates. The wave in the opposite direction than wave (8) has the form: Expression (8) can be transformed if we use relation (4), according to to which you can enter the wave number:

where is the wavelength,

or, if instead of a circular frequency we introduce a regular frequency, also called a line -

frequency, then

Let's look at the example of a wave, Fig. 3, consequences arising from equation (8):

a) the wave process is a doubly periodic process: the argument of the cosine in (8) depends on two variables - time and coordinates; that is, the wave has a double periodicity: in space and in time;

b) for a given time, equation (8) gives the distribution of particle displacement as a function of their distance from the origin;

c) particles oscillating under the influence of a traveling wave at a given time are located along a cosine wave;

d) a given particle, characterized by a certain value, performs harmonic oscillatory motion in time:

e) the value is constant for a given point and represents the initial phase of oscillations at this point;

f) two points, characterized by distances and from the origin, have a phase difference:

from (15) it is clear that two points separated from each other at a distance equal to the wavelength, i.e. for which , have a phase difference; and they also have, for each given moment in time, the same magnitude and direction -

niyu offset; such two points are said to oscillate in the same phase;

for points located at a distance from each other , i.e., separated from each other by half a wave, the phase difference according to (15) is equal to ; such points oscillate in opposite phases - for each given moment they have displacements that are identical in absolute value, but different in sign: if one point is deflected upward, then the other is deflected downward, and vice versa.

In an elastic medium, waves of a different type than traveling waves (8) are possible, for example, spherical waves, for which the dependence of the displacement on coordinates and time has the form:

In a spherical wave, the amplitude decreases in inverse proportion to the distance from the source of vibration.

6. Wave energy

Energy of the section of the medium in which the traveling wave propagates (8):

consists of kinetic energy and potential energy. Let the volume of a section of the medium be equal to ; let's denote its mass by and the displacement speed of its particles by , then the kinetic energy

noticing that , where is the density of the medium, and finding an expression for the speed based on (8)

Let's rewrite expression (17) in the form:

(19)

The potential energy of a section of a solid body subjected to relative deformation is known to be equal to

(20)

where is the elastic modulus or Young's modulus; - change in the length of a solid body due to the action on its ends of forces equal in value to , - cross-sectional area.

Let us rewrite (20), introducing the elasticity coefficient and dividing and multiplying the right

part of it on, so

.

If the relative deformation is represented, using infinitesimals, in the form , where is the elementary difference in the displacements of particles spaced apart by ,

. (21)

Determining the expression for based on (8):

Let's write (21) in the form:

(22)

Comparing (19) and (22), we see that both kinetic energy and potential energy change in one phase, i.e., they reach maximum and minimum in phase and synchronously. In this way, the energy of a wave section differs significantly from the energy of an isolated oscillation

bathroom point where, at a maximum - kinetic energy - the potential has a minimum, and vice versa. When an individual point oscillates, the total energy reserve of the oscillation remains constant, and since the main property of all waves, regardless of their nature, is the transfer of energy without transfer of mass of matter, the total energy of the section of the medium in which the wave propagates does not remain constant.

Let's add the right-hand sides of (19) and (22), and calculate the total energy of an element of the medium with volume:

Since according to (1) the phase velocity of wave propagation in an elastic medium

then we transform (23) as follows

Thus, the energy of a wave segment is proportional to the square of the amplitude, the square of the cyclic frequency and the density of the medium.

The energy flux density vector is the Umov vector.

Let us introduce into consideration the energy density or volumetric energy density of an elastic wave

where is the volume of wave formation.

We see that energy density, like energy itself, is a variable quantity, but since the average value of the squared sine for a period is equal to , then, in accordance with (25), the average value of energy density

, (26)

with constant parameters, wave-like - vania, will be a constant value for an isotropic medium if there is no absorption in the medium. Due to the fact that energy (24) does not remain localized in a given volume, but the change - exists in the environment, we can introduce into consideration the concept of energy flow. Under the flow of energy through the top - we mean size, number - equal to the amount of energy passing through - cabbage soup through it per unit time. Let us take a surface perpendicular to the direction of the wave speed; then an amount of energy equal to the energy will flow through this surface in a time equal to the period

enclosed in a column of cross section and length , Fig. 5; this amount of energy is equal to the average energy density taken over the period and multiplied by the volume of the column, hence

(27)

We obtain the average energy flow (average power) by dividing this expression by the time during which the energy flows through the surface

(28)

or, using (26), we find

(29)

The amount of energy flowing per unit time through a unit surface area is called flux density. By this definition, applying (28), we obtain

Thus, this is a vector, the direction of which is determined by the direction of the phase velocity and coincides with the direction of wave propagation.

This vector was first introduced into wave theory by the Russian professor

N.A. Umov and is called the Umov vector.

Let's take a point source of vibrations and draw a sphere of radius with the center at the source. The wave and the energy that is associated with it will spread along radii,

i.e., perpendicular to the surface of the sphere. During a period, energy equal to will flow through the surface of the sphere, where is the flow of energy through the sphere. Flux density

we get if we divide this energy by the size of the surface of the sphere and time:

Since in the absence of absorption of oscillations in the medium and a steady wave process, the average energy flow is constant and does not depend on the radius of the test -

den sphere, then (31) shows that the average flux density is inversely proportional to the square of the distance from the point source.

Typically, the energy of vibrational motion in a medium is partially converted into internal energy.

new energy.

The total amount of energy that a wave will transfer will depend on the distance it travels from the source: the further away the wave surface is from the source, the less energy it has. Since according to (24) the energy is proportional to the square of the amplitude, the amplitude decreases as the wave propagates. Let us assume that when passing through a layer of thickness, the relative decrease in amplitude is proportional to , i.e. we write

,

where is a constant value depending on the nature of the medium.

The last equality can be rewritten

.

If the differentials of two quantities are equal to each other, then the quantities themselves differ from each other by an additive constant value, whence

The constant is determined from the initial conditions that when the value is equal to , where is the amplitude of oscillations in the wave source, should be equal to , thus:

(32)

The equation of a plane wave in a medium with absorption based on (32) will be

Let us now determine the decrease in wave energy with distance. Let us denote by - the average energy density at , and by - the average energy density at a distance , then using relations (26) and (32), we find

(34)

let us denote by and rewrite (34) as follows

The quantity is called absorption coefficient.

8. Wave equation

From the wave equation (8) we can obtain one more relation, which we will need further. Taking the second derivatives of with respect to the variables and , we obtain

whence follows

We obtained equation (36) by differentiating (8). Conversely, it can be shown that a purely periodic wave, to which the cosine wave (8) corresponds, satisfies the differential -

cial equation (36). It is called the wave equation, since it has been established that (36) also satisfies a number of other functions that describe the propagation of a wave disturbance of an arbitrary shape with a speed .

9. Huygens' principle

Each point to which the wave reaches serves as the center of secondary waves, and the envelope of these waves gives the position of the wave front at the next moment in time.

This is the essence of Huygens' principle, which is illustrated in the following figures:

Rice. 6 A small hole in an obstacle is a source of new waves	Rice. 7 Huygens construction for a plane wave
Rice. 8 Huygens’ construction for a spherical wave propagating - from the center	Huygens' principle is a geometric principle - cip. It does not touch upon the essence of the question of the amplitude, and, consequently, the intensity of waves propagating behind the barrier.

Group speed

Rayleigh was the first to show that, along with the phase velocity of waves, it makes sense

introduce the concept of another speed, called group speed. Group velocity refers to the case of propagation of waves of a complex non-cosine nature in a medium where the phase speed of propagation of cosine waves depends on their frequency.

The dependence of the phase velocity on their frequency or wavelength is called wave dispersion.

Let us imagine a wave on the surface of water in the form of a single hump or soliton, Fig. 9, spreading in a certain direction. According to the Fourier method, this is complex -

This oscillation can be decomposed into a group of purely harmonic oscillations. If all harmonic vibrations propagate over the surface of the water at the same speed -

tami, then the complex vibration they form will propagate at the same speed -

tion. But, if the speeds of individual cosine waves are different, then the phase differences between them continuously change, and the hump that appears as a result of their addition continuously changes its shape and moves at a speed that does not coincide with the phase speed of any of the component waves.

Any segment of a cosine wave, Fig. 10, can also be decomposed, according to the Fourier theorem, into an infinite number of ideal cosine waves unlimited in time. Thus, any real wave is a superposition - a group - of infinite cosine waves, and the speed of its propagation in a dispersive medium is different from the phase speed of the component waves. This speed of propagation of real waves in dispersive -

environment and is called group velocity. Only in a medium devoid of dispersion does a real wave propagate at a speed that coincides with the phase speed of those cosine waves by the addition of which it is formed.

Let us assume that a group of waves consists of two waves that differ little in length:

a) waves with wavelength , propagating at speed;

b) waves with wavelength , propagating at speed

The relative location of both waves for a certain point in time is shown in Fig. 11. a. The humps of both waves converge at the point ; the maximum of the resulting oscillations is located in one place. Let , then the second wave overtakes the first. After a certain period of time, she will overtake her by a segment; As a result, the humps of both waves will already add up at point , Fig. 11.b, i.e., the location of the maximum of the resulting complex oscillation will be shifted back by a segment equal to . Hence, the speed of propagation of the maximum of the resulting oscillations relative to the medium will be less than the speed of propagation of the first wave by an amount. This speed of propagation of the maximum of a complex oscillation is the group speed; denoting it through , we have, i.e., the more pronounced the dependence of the speed of wave propagation on their length, called dispersion.

If , That shorter waves overtake longer ones; this case is called anomalous dispersion.

Wave superposition principle

When several waves of small amplitude propagate in a medium, performing -

There is, discovered by Leonardo da Vinci, the principle of superposition: the oscillation of each particle of the medium is determined as the sum of independent oscillations that these particles would perform during the propagation of each wave separately. The principle of superposition is violated only for waves with very large amplitudes, for example, in nonlinear optics. Waves characterized by the same frequency and a constant, time-independent phase difference are called coherent; for example, for example, cosine -

nal or sine waves with the same frequency.

Interference is the addition of coherent waves, which results in a time-stable increase in oscillations at some points and a decrease in others. In this case, the oscillation energy is redistributed between neighboring regions of the medium. Interference of waves only occurs if they are coherent.

Standing waves

A special example of the result of interference between two waves is:

called standing waves, formed as a result of the superposition of two opposing flat waves with identical amplitudes.

Addition of two waves traveling in opposite directions

Let us assume that two plane waves with identical propagation amplitudes - are moving - one in a positive direction - phenomenon, fig. 12, the other - in negative - telny. If the origin of coordinates is taken at such a point - ke, in which the counterpropagating waves have the same displacement directions, i.e., they have the same phases, and choose the timing so that the initial phases of the eye - Elastic waves in elastic environment, standing waves. 2. Study the method of determining the speed of propagation... to the direction of propagation waves. Elastic transverse waves can only arise in such environments who have... Application of sound waves (1) Abstract >> Physics Mechanical vibrations, radiation and propagation of sound ( elastic) waves V environment, methods are being developed to measure the characteristics of sound... patterns of radiation, propagation and reception elastic fluctuations and waves in different environments and systems; conditionally her... Physics course answers Cheat sheet >> Physics ... elastic strength. T=2π root of m/k (s) – period, k – coefficient elasticity, m – mass of the load. No. 9. Waves V elastic environment. Length waves. Intensity waves. Speed waves Waves ...

Questions.

1. What is wavelength called?

The wavelength is the distance between two nearest points oscillating in the same phases.

2. What letter indicates wavelength?

Wavelength is denoted by the Greek letter λ (lambda).

3. How long does it take for the oscillatory process to spread over a distance equal to the wavelength?

The oscillatory process propagates over a distance equal to the wavelength λ during the period of complete oscillation T.

5. The distance between which points is equal to the length of the longitudinal wave shown in Figure 69?

The length of the longitudinal wave in Figure 69 is equal to the distance between points 1 and 2 (wave maximum) and 3 and 4 (wave minimum).

Exercises.

1. At what speed does a wave propagate in the ocean if the wavelength is 270 m and the oscillation period is 13.5 s?

2. Determine the wavelength at a frequency of 200 Hz if the wave speed is 340 m/s.

3. 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.

>>Physics: Speed ​​and wavelength

Each wave travels at a certain speed. Under wave speed understand the speed of propagation 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 the wave is determined by the properties of the medium in which the 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 is the distance over which a wave propagates in a time equal to the period of oscillation in it.

Direction of propagation of warriors

Since the speed of a 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. Thus, to find the wavelength, you need to multiply the speed of the wave by the period of oscillation in it:

By choosing the direction of wave propagation as the direction of the x axis and denoting the coordinates of the particles oscillating in the wave through y, we can construct wave chart. The graph of a sine wave (at a fixed time t) is shown in Figure 45.

The distance between adjacent crests (or troughs) on this graph coincides with the wavelength.

Formula (22.1) expresses the relationship between wavelength and its speed and period. Considering that the period of oscillation in a wave is inversely proportional to the frequency, i.e. T=1/ v, we can obtain a formula expressing the relationship between wavelength and its speed and frequency:

The resulting formula shows that the speed of the 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.

??? 1. What is meant by wave speed? 2. What is wavelength? 3. How is wavelength related to the speed and period of oscillation in the wave? 4. How is wavelength related to the speed and frequency of oscillations in the wave? 5. Which of the following wave characteristics change when the wave passes from one medium to another: a) frequency; b) period; c) speed; d) wavelength?

Experimental task . Pour water into the bath and, by rhythmically touching the water with your finger (or ruler), create waves on its surface. Using different oscillation frequencies (for example, touching the water once and twice per second), pay attention to the distance between adjacent wave crests. At what oscillation frequency is the wavelength longer?

S.V. Gromov, N.A. Rodina, Physics 8th grade

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What do you need to know and be able to do?

1. Determination of wavelength.
Wavelength is the distance between nearby points oscillating in the same phases.

THIS IS INTERESTING

Seismic waves.

Seismic waves are waves propagating in the Earth from the sources of earthquakes or some powerful explosions. Since the Earth is mostly solid, two types of waves can simultaneously arise in it - longitudinal and transverse. The speed of these waves is different: longitudinal ones travel faster than transverse ones. For example, at a depth of 500 km, the speed of transverse seismic waves is 5 km/s, and the speed of longitudinal waves is 10 km/s.

Registration and recording of vibrations earth's surface, caused by seismic waves, are carried out using instruments - seismographs. Propagating from the source of an earthquake, longitudinal waves arrive first at the seismic station, and after some time - transverse waves. Knowing the speed of propagation of seismic waves in earth's crust and the delay time of the shear wave, the distance to the center of the earthquake can be determined. To find out more precisely where it is located, they use data from several seismic stations.

Every year on globe Hundreds of thousands of earthquakes are recorded. The vast majority of them are weak, but some are observed from time to time. which violate the integrity of the soil, destroy buildings and lead to casualties.

The intensity of earthquakes is assessed on a 12-point scale.

1948 - Ashgabat - earthquake 9-12 points
1966 - Tashkent - 8 points
1988 - Spitak - several tens of thousands of people died
1976 - China - hundreds of thousands of victims

Resist devastating consequences earthquakes are only possible through the construction of earthquake-resistant buildings. But in which areas of the Earth will the next earthquake occur?

Earthquake prediction - Herculean task. Many research institutes in many countries around the world are engaged in solving this problem. The study of seismic waves inside our Earth allows us to study the deep structure of the planet. In addition, seismic exploration helps to detect areas favorable for the accumulation of oil and gas. Seismic research is carried out not only on Earth, but also on other celestial bodies.

In 1969, American astronauts placed seismic stations on the Moon. Every year they recorded from 600 to 3000 weak moonquakes. In 1976, with the help spaceship"Viking" (USA) seismograph was installed on Mars..

DO IT YOURSELF

Waves on paper.

You can perform many experiments using a sounding tube.
If, for example, you put a sheet of thick light paper on a soft substrate lying on a table, sprinkle a layer of potassium permanganate crystals on top, place a glass tube vertically in the middle of the sheet and excite vibrations in it by friction, then when sound appears, the potassium permanganate crystals will begin to move and form beautiful lines . The tube should only lightly touch the surface of the sheet. The pattern that appears on the sheet will depend on the length of the tube.

The tube excites vibrations in the paper sheet. A standing wave is formed in a sheet of paper, which is the result of the interference of two traveling waves. A circular wave arises from the end of the oscillating tube, which is reflected from the edge of the paper without changing phase. These waves are coherent and interfere, distributing potassium permanganate crystals on paper into bizarre patterns.

ABOUT THE SHOCK WAVE

In his lecture "On Ship Waves" Lord Kelvin said:
"...one discovery was actually made by a horse that daily pulled a boat along a rope between Glasgow
and Ardrossan. One day the horse rushed, and the driver, being an observant person, noticed that when the horse reached a certain speed, it became clearly easier to pull the boat
and there was no wave trace left behind her.”

The explanation for this phenomenon is that the speed of the boat and the speed of the wave that the boat excites in the river coincided.
If the horse ran even faster (the speed of the boat would become greater than the speed of the wave),
then a shock wave would appear behind the boat.
The shock wave from a supersonic aircraft occurs in exactly the same way.