Mechanical waves speed. Mechanical and sound waves. Key points
A mechanical or elastic wave is the process of propagation of oscillations in an elastic medium. For example, air begins to oscillate around a vibrating string or speaker cone - the string or speaker has become sources of a sound wave.
For the occurrence of a mechanical wave, two conditions must be met - the presence of a wave source (it can be any oscillating body) and an elastic medium (gas, liquid, solid).
Find out the cause of the wave. Why do the particles of the medium surrounding any oscillating body also come into oscillatory motion?
The simplest model of a one-dimensional elastic medium is a chain of balls connected by springs. Balls are models of molecules, the springs connecting them model the forces of interaction between molecules.
Suppose the first ball oscillates with a frequency ω. Spring 1-2 is deformed, an elastic force arises in it, which changes with frequency ω. Under the action of an external periodically changing force, the second ball begins to perform forced oscillations. Since forced oscillations always occur at the frequency of the external driving force, the oscillation frequency of the second ball will coincide with the oscillation frequency of the first. However, the forced oscillations of the second ball will occur with some phase delay relative to the external driving force. In other words, the second ball will begin to oscillate somewhat later than the first ball.
The vibrations of the second ball will cause a periodically changing deformation of the spring 2-3, which will make the third ball oscillate, and so on. Thus, all the balls in the chain will alternately be involved in an oscillatory motion with the oscillation frequency of the first ball.
Obviously, the cause of wave propagation in an elastic medium is the presence of interaction between molecules. The oscillation frequency of all particles in the wave is the same and coincides with the oscillation frequency of the wave source.
According to the nature of particle oscillations in a wave, waves are divided into transverse, longitudinal and surface waves.
IN longitudinal wave particles oscillate along the direction of wave propagation.
The propagation of a longitudinal wave is associated with the occurrence of tensile-compressive deformation in the medium. In the stretched areas of the medium, a decrease in the density of the substance is observed - rarefaction. In compressed areas of the medium, on the contrary, there is an increase in the density of the substance - the so-called thickening. For this reason, a longitudinal wave is a movement in space of areas of condensation and rarefaction.
Tensile-compressive deformation can occur in any elastic medium, so longitudinal waves can propagate in gases, liquids and solids. An example of a longitudinal wave is sound.
IN shear wave particles oscillate perpendicular to the direction of wave propagation.
The propagation of a transverse wave is associated with the occurrence of shear deformation in the medium. This kind of deformation can only exist in solids, so transverse waves can only propagate in solids. An example of a shear wave is the seismic S-wave.
surface waves occur at the interface between two media. Oscillating particles of the medium have both transverse, perpendicular to the surface, and longitudinal components of the displacement vector. During their oscillations, the particles of the medium describe elliptical trajectories in a plane perpendicular to the surface and passing through the direction of wave propagation. An example of surface waves are waves on the water surface and seismic L - waves.
The wave front is the locus of points reached by the wave process. The shape of the wave front can be different. The most common are plane, spherical and cylindrical waves.
Note that the wavefront is always located perpendicular direction of the wave! All points of the wavefront will begin to oscillate in one phase.
To characterize the wave process, the following quantities are introduced:
1. Wave frequencyν is the oscillation frequency of all the particles in the wave.
2. Wave amplitude A is the oscillation amplitude of the particles in the wave.
3. Wave speedυ is the distance over which the wave process (perturbation) propagates per unit time.
Please note that the speed of the wave and the speed of oscillation of the particles in the wave are different concepts! The speed of a wave depends on two factors: the type of wave and the medium in which the wave propagates.
The general pattern is as follows: the speed of a longitudinal wave in a solid is greater than in liquids, and the speed in liquids, in turn, is greater than the speed of a wave in gases.
It is not difficult to understand the physical reason for this regularity. The cause of wave propagation is the interaction of molecules. Naturally, the perturbation propagates faster in the medium where the interaction of molecules is stronger.
In the same medium, the regularity is different - the speed of the longitudinal wave is greater than the speed of the transverse wave.
For example, the speed of a longitudinal wave in a solid, where E is the elastic modulus (Young's modulus) of the substance, ρ is the density of the substance.
Shear wave velocity in a solid, where N is the shear modulus. Since for all substances , then . One of the methods for determining the distance to the source of an earthquake is based on the difference in the velocities of longitudinal and transverse seismic waves.
The speed of a transverse wave in a stretched cord or string is determined by the tension force F and the mass per unit length μ:
4. Wavelengthλ is the minimum distance between points that oscillate equally.
For waves traveling on the surface of water, the wavelength is easily defined as the distance between two adjacent humps or adjacent depressions.
For a longitudinal wave, the wavelength can be found as the distance between two adjacent concentrations or rarefactions.
5. In the process of wave propagation, sections of the medium are involved in an oscillatory process. An oscillating medium, firstly, moves, therefore, it has kinetic energy. Secondly, the medium through which the wave runs is deformed, therefore, it has potential energy. It is easy to see that wave propagation is associated with the transfer of energy to unexcited parts of the medium. To characterize the energy transfer process, we introduce wave intensity I.
In the 7th grade physics course, you studied mechanical vibrations. It often happens that, having arisen in one place, vibrations propagate to neighboring regions of space. Recall, for example, the propagation of vibrations from a pebble thrown into the water or the vibrations of the earth's crust propagating from the epicenter of an earthquake. In such cases, they speak of wave motion - waves (Fig. 17.1). In this section, you will learn about the features of wave motion.
Create mechanical waves
Take a fairly long rope, one end of which we attach to vertical surface, and we will move the second one up and down (oscillate). Vibrations from the hand will spread along the rope, gradually involving more and more distant points in the oscillatory movement - a mechanical wave will run along the rope (Fig. 17.2).
A mechanical wave is the propagation of oscillations in an elastic medium*.
Now we fix a long soft spring horizontally and apply a series of successive blows to its free end - a wave will run in the spring, consisting of condensations and rarefaction of the coils of the spring (Fig. 17.3).
The waves described above can be seen, however, most mechanical waves are invisible, for example sound waves(Fig. 17.4).
At first glance, all mechanical waves are completely different, but the reasons for their occurrence and propagation are the same.
We find out how and why a mechanical wave propagates in a medium
Any mechanical wave is created by an oscillating body - the source of the wave. Performing an oscillatory motion, the wave source deforms the layers of the medium closest to it (compresses and stretches them or displaces them). As a result, elastic forces arise that act on neighboring layers of the medium and force them to carry out forced oscillations. These layers, in turn, deform the next layers and cause them to oscillate. Gradually, one by one, all layers of the medium are involved in oscillatory motion - a mechanical wave propagates in the medium.
Rice. 17.6. In a longitudinal wave, the layers of the medium oscillate along the direction of wave propagation
Distinguish between transverse and longitudinal mechanical waves
Let's compare wave propagation along a rope (see Fig. 17.2) and in a spring (see Fig. 17.3).
Separate parts of the rope move (oscillate) perpendicular to the direction of wave propagation (in Fig. 17.2, the wave propagates from right to left, and parts of the rope move up and down). Such waves are called transverse (Fig. 17.5). During the propagation of transverse waves, some layers of the medium are displaced relative to others. Displacement deformation is accompanied by the appearance of elastic forces only in solids, so transverse waves cannot propagate in liquids and gases. So, transverse waves propagate only in solids.
When a wave propagates in a spring, the coils of the spring move (oscillate) along the direction of wave propagation. Such waves are called longitudinal (Fig. 17.6). When a longitudinal wave propagates, compressive and tensile deformations occur in the medium (along the direction of wave propagation, the density of the medium either increases or decreases). Such deformations in any medium are accompanied by the appearance of elastic forces. Therefore, longitudinal waves propagate in solids, and in liquids, and in gases.
Waves on the surface of a liquid are neither longitudinal nor transverse. They have a complex longitudinal-transverse character, while the liquid particles move along ellipses. This is easy to verify if you throw a light chip into the sea and watch its movement on the surface of the water.
Finding out the basic properties of waves
1. Oscillatory motion from one point of the medium to another is not transmitted instantly, but with some delay, so the waves propagate in the medium with a finite speed.
2. The source of mechanical waves is an oscillating body. When a wave propagates, the vibrations of parts of the medium are forced, so the frequency of vibrations of each part of the medium is equal to the frequency of vibrations of the wave source.
3. Mechanical waves cannot propagate in a vacuum.
4. Wave motion is not accompanied by the transfer of matter - parts of the medium only oscillate about the equilibrium positions.
5. With the arrival of the wave, parts of the medium begin to move (acquire kinetic energy). This means that when the wave propagates, energy is transferred.
Transfer of energy without transfer of matter - the most important property any wave.
Remember the propagation of waves on the surface of the water (Fig. 17.7). What observations confirm the basic properties of wave motion?
We recall the physical quantities characterizing the oscillations
A wave is the propagation of oscillations, so the physical quantities that characterize oscillations (frequency, period, amplitude) also characterize the wave. So, let's remember the material of the 7th grade:
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Physical quantities characterizing oscillations |
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Oscillation frequency ν |
Oscillation period T |
Oscillation amplitude A |
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Define |
number of oscillations per unit of time |
time of one oscillation |
the maximum distance a point deviates from its equilibrium position |
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Formula to determine |
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 |
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N is the number of oscillations per time interval t |
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Unit in SI |
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second (s) |
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Note! When a mechanical wave propagates, all parts of the medium in which the wave propagates oscillate with the same frequency (ν), which is equal to the oscillation frequency of the wave source, so the period
oscillations (T) for all points of the medium is also the same, because
But the amplitude of oscillations gradually decreases with distance from the source of the wave.
We find out the length and speed of propagation of the wave
Remember the propagation of a wave along a rope. Let the end of the rope carry out one complete oscillation, that is, the propagation time of the wave is equal to one period (t = T). During this time, the wave propagated over a certain distance λ (Fig. 17.8, a). This distance is called the wavelength.
The wavelength λ is the distance over which the wave propagates in a time equal to the period T:
where v is the speed of wave propagation. The unit of wavelength in SI is the meter:
It is easy to see that the points of the rope, located at a distance of one wavelength from each other, oscillate synchronously - they have the same phase of oscillation (Fig. 17.8, b, c). For example, points A and B of the rope move up at the same time, reach the crest of a wave at the same time, then start moving down at the same time, and so on.
Rice. 17.8. The wavelength is equal to the distance that the wave propagates during one oscillation (this is also the distance between the two nearest crests or the two nearest troughs)
Using the formula λ = vT, we can determine the propagation velocity
we obtain the formula for the relationship between the length, frequency and speed of wave propagation - the wave formula:
If a wave passes from one medium to another, its propagation speed changes, but the frequency remains the same, since the frequency is determined by the source of the wave. Thus, according to the formula v = λν, when a wave passes from one medium to another, the wavelength changes.
Wave formula
Learning to solve problems
Task. The transverse wave propagates along the cord at a speed of 3 m/s. On fig. 1 shows the position of the cord at some point in time and the direction of wave propagation. Assuming that the side of the cage is 15 cm, determine:
1) amplitude, period, frequency and wavelength;
Analysis of a physical problem, solution
The wave is transverse, so the points of the cord oscillate perpendicular to the direction of wave propagation (they move up and down relative to some equilibrium positions).
1) From fig. 1 we see that the maximum deviation from the equilibrium position (amplitude A of the wave) is equal to 2 cells. So A \u003d 2 15 cm \u003d 30 cm.
The distance between the crest and trough is 60 cm (4 cells), respectively, the distance between the two nearest crests (wavelength) is twice as large. So, λ = 2 60 cm = 120 cm = 1.2m.
We find the frequency ν and the period T of the wave using the wave formula:
2) To find out the direction of movement of the points of the cord, we perform an additional construction. Let the wave move over a small distance over a short time interval Δt. Since the wave shifts to the right, and its shape does not change with time, the pinch points will take the position shown in Fig. 2 dotted.
The wave is transverse, that is, the points of the cord move perpendicular to the direction of wave propagation. From fig. 2 we see that point K after a time interval Δt will be below its initial position, therefore, its speed is directed downwards; point B will move higher, therefore, the speed of its movement is directed upwards; point C will move lower, therefore, the speed of its movement is directed downward.
Answer: A = 30 cm; T = 0.4 s; ν = 2.5 Hz; λ = 1.2 m; K and C - down, B - up.
Summing up
The propagation of oscillations in an elastic medium is called a mechanical wave. A mechanical wave in which parts of the medium oscillate perpendicular to the direction of wave propagation is called transverse; a wave in which parts of the medium oscillate along the direction of wave propagation is called longitudinal.
The wave propagates in space not instantly, but with a certain speed. When a wave propagates, energy is transferred without transfer of matter. The distance over which the wave propagates in a time equal to the period is called the wavelength - this is the distance between the two nearest points that oscillate synchronously (have the same phase of oscillation). The length λ, frequency ν and velocity v of wave propagation are related by the wave formula: v = λν.
Control questions
1. Define a mechanical wave. 2. Describe the mechanism of formation and propagation of a mechanical wave. 3. Name the main properties of wave motion. 4. What waves are called longitudinal? transverse? In what environments do they spread? 5. What is the wavelength? How is it defined? 6. How are the length, frequency and speed of wave propagation related?
Exercise number 17
1. Determine the length of each wave in fig. 1.
2. In the ocean, the wavelength reaches 270 m, and its period is 13.5 s. Determine the propagation speed of such a wave.
3. Do the speed of wave propagation and the speed of movement of the points of the medium in which the wave propagates coincide?
4. Why does a mechanical wave not propagate in a vacuum?
5. As a result of the explosion produced by geologists, in earth's crust the wave propagated at a speed of 4.5 km/s. Reflected from the deep layers of the Earth, the wave was recorded on the Earth's surface 20 s after the explosion. At what depth does the rock lie, the density of which differs sharply from the density of the earth's crust?
6. In fig. 2 shows two ropes along which transverse wave. Each rope shows the direction of oscillation of one of its points. Determine the directions of wave propagation.
7. In fig. 3 shows the position of two filaments along which the wave propagates, showing the direction of propagation of each wave. For each case a and b determine: 1) amplitude, period, wavelength; 2) the direction in which points A, B and C of the cord are moving at a given time; 3) the number of oscillations that any point of the cord makes in 30 s. Consider that the side of the cage is 20 cm.
8. A man standing on the seashore determined that the distance between adjacent wave crests is 15 m. In addition, he calculated that 16 wave crests reach the shore in 75 seconds. Determine the speed of wave propagation.
This is textbook material.
The existence of a wave requires a source of oscillation and a material medium or field in which this wave propagates. Waves are of the most diverse nature, but they obey similar patterns.
By physical nature distinguish:
According to the orientation of disturbances distinguish:
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Longitudinal waves -
The displacement of particles occurs along the direction of propagation; it is necessary to have an elastic force in the medium during compression; can be distributed in any environment. Examples: sound waves  |
Transverse waves -
The displacement of particles occurs across the direction of propagation; can propagate only in elastic media; it is necessary to have a shear elastic force in the medium; can propagate only in solid media (and at the boundary of two media). Examples: elastic waves in a string, waves on water
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According to the nature of the dependence on time distinguish:
elastic waves - mechanical displacements (deformations) propagating in an elastic medium. The elastic wave is called harmonic(sinusoidal) if the vibrations of the medium corresponding to it are harmonic.
running waves - Waves that carry energy in space.
According to the shape of the wave surface : plane, spherical, cylindrical wave.
wave front- the locus of points, to which the oscillations have reached a given point in time.
wave surface- locus of points oscillating in one phase.
Wave characteristics
Wavelength λ - the distance over which the wave propagates in a time equal to the period of oscillation
Wave amplitude A - amplitude of oscillations of particles in a wave
Wave speed v - speed of propagation of perturbations in the medium
Wave period T - oscillation period
Wave frequency ν - the reciprocal of the period
Traveling wave equation
During the propagation of a traveling wave, the disturbances of the medium reach the next points in space, while the wave transfers energy and momentum, but does not transfer matter (the particles of the medium continue to oscillate in the same place in space).
Where v- speed , φ 0 - initial phase , ω – cyclic frequency , A– amplitude
Properties of mechanical waves
1. wave reflection – mechanical waves of any origin have the ability to be reflected from the interface between two media. If a mechanical wave propagating in a medium encounters an obstacle in its path, it can dramatically change the nature of its behavior. For example, at the interface between two media with different mechanical properties the wave is partially reflected, and partially penetrates into the second medium.
2. Refraction of waves – during the propagation of mechanical waves, one can also observe the phenomenon of refraction: a change in the direction of propagation of mechanical waves during the transition from one medium to another.
3. Wave diffraction – deviation of waves from rectilinear propagation, that is, their bending around obstacles.
4. Wave interference – addition of two waves. In a space where several waves propagate, their interference leads to the appearance of regions with the minimum and maximum values of the oscillation amplitude
Interference and diffraction of mechanical waves.
A wave running along a rubber band or string is reflected from a fixed end; this creates a wave traveling in the opposite direction.
When waves are superimposed, the phenomenon of interference can be observed. The phenomenon of interference occurs when coherent waves are superimposed.
coherent calledwaveshaving the same frequencies, a constant phase difference, and the oscillations occur in the same plane.
interference called the time-constant phenomenon of mutual amplification and attenuation of oscillations at different points of the medium as a result of the superposition of coherent waves.
The result of the superposition of waves depends on the phases in which the oscillations are superimposed on each other.
If waves from sources A and B arrive at point C in the same phases, then the oscillations will increase; if it is in opposite phases, then there is a weakening of the oscillations. As a result, a stable pattern of alternating regions of enhanced and weakened oscillations is formed in space.
Maximum and minimum conditions
If the oscillations of points A and B coincide in phase and have equal amplitudes, then it is obvious that the resulting displacement at point C depends on the difference between the paths of the two waves.
Maximum conditions
If the difference between the paths of these waves is equal to an integer number of waves (i.e., an even number of half-waves) Δd = kλ , Where k= 0, 1, 2, ..., then an interference maximum is formed at the point of superposition of these waves.
Maximum condition
: 
A = 2x0.
Minimum condition
If the path difference of these waves is equal to an odd number of half-waves, then this means that the waves from points A and B will come to point C in antiphase and cancel each other out.
Minimum condition:
The amplitude of the resulting oscillation A = 0.
If Δd is not equal to an integer number of half-waves, then 0< А < 2х 0 .
Diffraction of waves.
The phenomenon of deviation from rectilinear propagation and rounding of obstacles by waves is calleddiffraction.
The relationship between the wavelength (λ) and the size of the obstacle (L) determines the behavior of the wave. Diffraction is most pronounced if the incident wavelength more sizes obstacles. Experiments show that diffraction always exists, but becomes noticeable under the condition d<<λ , where d is the size of the obstacle.
Diffraction is a common property of waves of any nature, which always occurs, but the conditions for its observation are different.
A wave on the water surface propagates towards a sufficiently large obstacle, behind which a shadow is formed, i.e. no wave process is observed. This property is used in the construction of breakwaters in ports. If the size of the obstacle is comparable to the wavelength, then there will be a wave behind the obstacle. Behind him, the wave propagates as if there was no obstacle at all, i.e. wave diffraction is observed.
Examples of the manifestation of diffraction . Hearing a loud conversation around the corner of the house, sounds in the forest, waves on the surface of the water.
standing waves
standing waves are formed by adding the direct and reflected waves if they have the same frequency and amplitude.
In a string fixed at both ends, complex vibrations arise, which can be considered as the result of superposition ( superpositions) two waves propagating in opposite directions and experiencing reflections and re-reflections at the ends. Vibrations of strings fixed at both ends create the sounds of all stringed musical instruments. A very similar phenomenon occurs with the sound of wind instruments, including organ pipes.
string vibrations. In a stretched string fixed at both ends, when transverse vibrations are excited, standing waves , and knots should be located in the places where the string is fixed. Therefore, the string is excited with noticeable intensity only such vibrations, half of the wavelength of which fits on the length of the string an integer number of times.
This implies the condition 
Wavelengths correspond to frequencies 
n = 1, 2, 3...Frequencies vn called natural frequencies strings.
Harmonic vibrations with frequencies vn called own or normal vibrations . They are also called harmonics. In general, the vibration of a string is a superposition of different harmonics.
Standing wave equation :
At points where the coordinates satisfy the condition 
(n= 1, 2, 3, ...), the total amplitude is equal to the maximum value - this antinodes
standing wave. Antinode coordinates
: 
At points whose coordinates satisfy the condition
(n= 0, 1, 2,…), the total oscillation amplitude is equal to zero – This nodes standing wave. Node coordinates: 
The formation of standing waves is observed when the traveling and reflected waves interfere. At the boundary where the wave is reflected, an antinode is obtained if the medium from which the reflection occurs is less dense (a), and a knot is obtained if it is more dense (b).
If we consider traveling wave , then in the direction of its propagation energy is transferred oscillatory movement. When same there is no standing wave of energy transfer , because incident and reflected waves of the same amplitude carry the same energy in opposite directions.
Standing waves arise, for example, in a string stretched at both ends when transverse vibrations are excited in it. Moreover, in the places of fixings, there are nodes of a standing wave.
If a standing wave is established in an air column that is open at one end (sound wave), then an antinode is formed at the open end, and a knot is formed at the opposite end.
§ 1.7. mechanical waves
The vibrations of a substance or field propagating in space are called a wave. Fluctuations of matter generate elastic waves (a special case is sound).
mechanical wave is the propagation of oscillations of the particles of the medium over time.
Waves in a continuous medium propagate due to the interaction between particles. If any particle comes into oscillatory motion, then, due to the elastic connection, this motion is transferred to neighboring particles, and the wave propagates. In this case, the oscillating particles themselves do not move with the wave, but hesitate around their equilibrium positions.
Longitudinal waves are waves in which the direction of particle oscillations x coincides with the direction of wave propagation 
. Longitudinal waves propagate in gases, liquids and solids.
P 
opera waves- these are waves in which the direction of particle oscillations is perpendicular to the direction of wave propagation 
. Transverse waves propagate only in solid media.
Waves have two periodicity - in time and space. Periodicity in time means that each particle of the medium oscillates around its equilibrium position, and this movement is repeated with an oscillation period T. Periodicity in space means that the oscillatory motion of the particles of the medium is repeated at certain distances between them.
The periodicity of the wave process in space is characterized by a quantity called the wavelength and denoted
.
The wavelength is the distance over which a wave propagates in a medium during one period of particle oscillation. 
.
From here 
, Where 
- particle oscillation period, 
- oscillation frequency, 
- speed of wave propagation, depending on the properties of the medium.
TO 
how to write the wave equation? Let a piece of cord located at point O (the source of the wave) oscillate according to the cosine law
Let some point B be at a distance x from the source (point O). It takes time for a wave propagating with a speed v to reach it. 
. This means that at point B, oscillations will begin later on 
. That is. After substituting into this equation the expressions for 
and a number of mathematical transformations, we get
,
. Let's introduce the notation: 
. Then. Due to the arbitrariness of the choice of point B, this equation will be the required plane wave equation 
.
The expression under the cosine sign is called the phase of the wave 
.
E 
If two points are at different distances from the source of the wave, then their phases will be different. For example, the phases of points B and C, located at distances 
And 
from the source of the wave, will be respectively equal to
The phase difference of the oscillations occurring at point B and at point C will be denoted 
and it will be equal
In such cases, it is said that between the oscillations occurring at points B and C there is a phase shift Δφ. It is said that oscillations at points B and C occur in phase if 
. If 
, then the oscillations at points B and C occur in antiphase. In all other cases, there is simply a phase shift.
The concept of "wavelength" can be defined in another way:
Therefore, k is called the wave number.
We have introduced the notation 
and showed that 
. Then
.
Wavelength is the path traveled by a wave in one period of oscillation.
Let us define two important concepts in the wave theory.
wave surface is the locus of points in the medium that oscillate in the same phase. The wave surface can be drawn through any point of the medium, therefore, there are an infinite number of them.
Wave surfaces can be of any shape, and in the simplest case they are a set of planes (if the wave source is an infinite plane) parallel to each other, or a set of concentric spheres (if the wave source is a point).
wave front(wave front) - the locus of points to which fluctuations reach by the moment of time 
. The wave front separates the part of space involved in the wave process from the area where oscillations have not yet arisen. Therefore, the wave front is one of the wave surfaces. It separates two areas: 1 - which the wave reached by the time t, 2 - did not reach.
There is only one wave front at any moment of time, and it moves all the time, while the wave surfaces remain stationary (they pass through the equilibrium positions of particles oscillating in the same phase).
plane wave- this is a wave in which the wave surfaces (and the wave front) are parallel planes.
spherical wave is a wave whose wave surfaces are concentric spheres. Spherical wave equation: 
.
Each point of the medium reached by two or more waves will take part in the oscillations caused by each wave separately. What will be the resulting vibration? It depends on a number of factors, in particular, on the properties of the medium. If the properties of the medium do not change due to the process of wave propagation, then the medium is called linear. Experience shows that waves propagate independently of each other in a linear medium. We will consider waves only in linear media. And what will be the fluctuation of the point, which reached two waves at the same time? To answer this question, it is necessary to understand how to find the amplitude and phase of the oscillation caused by this double action. To determine the amplitude and phase of the resulting oscillation, it is necessary to find the displacements caused by each wave, and then add them. How? Geometrically!
The principle of superposition (overlay) of waves: when several waves propagate in a linear medium, each of them propagates as if there were no other waves, and the resulting displacement of a particle of the medium at any time is equal to the geometric sum of the displacements that the particles receive, participating in each of the components of the wave processes.
An important concept of wave theory is the concept coherence - coordinated flow in time and space of several oscillatory or wave processes. If the phase difference of the waves arriving at the observation point does not depend on time, then such waves are called coherent. Obviously, only waves having the same frequency can be coherent.
R 
Let's consider what will be the result of adding two coherent waves coming to some point in space (observation point) B. In order to simplify mathematical calculations, we will assume that the waves emitted by sources S 1 and S 2 have the same amplitude and initial phases equal to zero. At the point of observation (at point B), the waves coming from the sources S 1 and S 2 will cause oscillations of the particles of the medium: 
And 
. The resulting fluctuation at point B is found as a sum.
Usually, the amplitude and phase of the resulting oscillation that occurs at the point of observation is found using the method of vector diagrams, representing each oscillation as a vector rotating with an angular velocity ω. The length of the vector is equal to the amplitude of the oscillation. Initially, this vector forms an angle with the chosen direction equal to the initial phase of oscillations. Then the amplitude of the resulting oscillation is determined by the formula.
For our case of adding two oscillations with amplitudes 
,
and phases 
,
.
Therefore, the amplitude of the oscillations that occur at point B depends on what is the path difference 
traversed by each wave separately from the source to the observation point ( 
is the path difference between the waves arriving at the observation point). Interference minima or maxima can be observed at those points for which 
. And this is the equation of a hyperbola with foci at the points S 1 and S 2 .
At those points in space for which 
, the amplitude of the resulting oscillations will be maximum and equal to 
. Because 
, then the oscillation amplitude will be maximum at those points for which.
at those points in space for which 
, the amplitude of the resulting oscillations will be minimal and equal to 
.oscillation amplitude will be minimal at those points for which .
The phenomenon of energy redistribution resulting from the addition of a finite number of coherent waves is called interference.
The phenomenon of waves bending around obstacles is called diffraction.
Sometimes diffraction is called any deviation of wave propagation near obstacles from the laws of geometric optics (if the dimensions of the obstacles are commensurate with the wavelength).
B 
Due to diffraction, waves can enter the region of a geometric shadow, go around obstacles, penetrate through small holes in screens, etc. How to explain the hit of waves in the area of geometric shadow? The phenomenon of diffraction can be explained using the Huygens principle: each point that a wave reaches is a source of secondary waves (in a homogeneous spherical medium), and the envelope of these waves sets the position of the wave front at the next moment in time.
Insert from light interference to see what might come in handy
wave called the process of propagation of vibrations in space.
wave surface is the locus of points at which oscillations occur in the same phase.
wave front called the locus of points to which the wave reaches a certain point in time t. The wave front separates the part of space involved in the wave process from the area where oscillations have not yet arisen.
For a point source, the wave front is a spherical surface centered at the source location S. 1, 2,
3
- wave surfaces; 1
- wave front. The equation of a spherical wave propagating along the beam emanating from the source: . Here 
- wave propagation speed, 
- wavelength; A- oscillation amplitude; 
- circular (cyclic) oscillation frequency; 
- displacement from the equilibrium position of a point located at a distance r from a point source at time t.
plane wave is a wave with a flat wave front. The equation of a plane wave propagating along the positive direction of the axis y:
, Where x- displacement from the equilibrium position of a point located at a distance y from the source at time t.
Experience shows that oscillations excited at any point of an elastic medium are transmitted over time to its other parts. So from a stone thrown into the calm water of the lake, waves diverge in circles, which eventually reach the shore. The vibrations of the heart, located inside the chest, can be felt on the wrist, which is used to determine the pulse. The above examples are related to the propagation of mechanical waves.
- mechanical wave called the process of propagation of oscillations in an elastic medium, which is accompanied by the transfer of energy from one point of the medium to another. Note that mechanical waves cannot propagate in a vacuum.
The source of a mechanical wave is an oscillating body. If the source oscillates sinusoidally, then the wave in the elastic medium will also have the form of a sinusoid. Oscillations caused in any place of an elastic medium propagate in the medium at a certain speed, depending on the density and elastic properties of the medium.
We emphasize that when the wave propagates no transfer of matter, i.e., particles only oscillate near equilibrium positions. The average displacement of particles relative to the equilibrium position over a long period of time is zero.
Main characteristics of the wave
Consider the main characteristics of the wave.
- "Wave front"- this is an imaginary surface to which the wave disturbance has reached at a given moment of time.
- A line drawn perpendicular to the wave front in the direction of wave propagation is called beam.
The beam indicates the direction of wave propagation.
Depending on the shape of the wave front, waves are plane, spherical, etc.
IN plane wave wave surfaces are planes perpendicular to the direction of wave propagation. Plane waves can be obtained on the surface of water in a flat bath using oscillations of a flat rod (Fig. 1).
IN spherical wave wave surfaces are concentric spheres. A spherical wave can be created by a ball pulsating in a homogeneous elastic medium. Such a wave propagates with the same speed in all directions. The rays are the radii of the spheres (Fig. 2).
The main characteristics of the wave:
- amplitude (A) is the modulus of maximum displacement of points of the medium from equilibrium positions during vibrations;
- period (T) is the time of complete oscillation (the period of oscillation of the points of the medium is equal to the period of oscillation of the wave source)
\(T=\dfrac(t)(N),\)
Where t- the period of time during which N fluctuations;
- frequency(ν) - the number of complete oscillations performed at a given point per unit time
\((\rm \nu) =\dfrac(N)(t).\)
The frequency of the wave is determined by the oscillation frequency of the source;
- speed(υ) - the speed of the wave crest (this is not the speed of particles!)
- wavelength(λ) - the smallest distance between two points, oscillations in which occur in the same phase, i.e. this is the distance over which the wave propagates in a time interval equal to the period of the source oscillations
\(\lambda =\upsilon \cdot T.\)
To characterize the energy carried by waves, the concept is used wave intensity (I), defined as the energy ( W) carried by the wave per unit time ( t= 1 c) through a surface area S\u003d 1 m 2, located perpendicular to the direction of wave propagation:
\(I=\dfrac(W)(S\cdot t).\)
In other words, intensity is the power carried by waves across a surface of unit area, perpendicular to the direction of wave propagation. The SI unit of intensity is the watt per square meter (1 W/m2).
Traveling wave equation
Consider wave source oscillations occurring with cyclic frequency ω \(\left(\omega =2\pi \cdot \nu =\dfrac(2\pi )(T) \right)\) and amplitude A:
\(x(t)=A\cdot \sin \; (\omega \cdot t),\)
Where x(t) is the displacement of the source from the equilibrium position.
At a certain point in the medium, oscillations will not arrive instantly, but after a period of time determined by the wave speed and the distance from the source to the observation point. If the wave speed in a given medium is υ, then the time dependence t coordinates (offset) x oscillating point at a distance r from the source, is described by the equation
\(x(t,r) = A\cdot \sin \; \omega \cdot \left(t-\dfrac(r)(\upsilon ) \right)=A\cdot \sin \; \left(\omega \cdot t-k\cdot r \right), \;\;\; (1)\)
Where k-wave number \(\left(k=\dfrac(\omega )(\upsilon ) = \dfrac(2\pi )(\lambda ) \right), \;\;\; \varphi =\omega \cdot t-k \cdot r\) - wave phase.
Expression (1) is called traveling wave equation.
A traveling wave can be observed in the following experiment: if one end of a rubber cord lying on a smooth horizontal table is fixed and, slightly pulling the cord by hand, bring its other end into oscillatory motion in a direction perpendicular to the cord, then a wave will run along it.
Longitudinal and transverse waves
There are longitudinal and transverse waves.
- The wave is called transverse, If particles of the medium oscillate in a plane 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 (Fig. 3, a). Figure 3 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 time \(\left(t_1 = 0 \right)\) all points are in equilibrium (Fig. 3, a). If you deflect the ball 1 from the equilibrium position perpendicular to the entire chain of balls, then 2 -th ball, elastically connected with 1 -th, will begin to follow him. Due to the inertia of the movement 2 th ball will repeat the movements 1 th, but with a delay in time. Ball 3 th, elastically connected with 2 -th, will begin to move behind 2 th ball, but with an even greater delay.
After a quarter of the period \(\left(t_2 = \dfrac(T)(4) \right)\) the oscillations propagate up to 4 -th ball, 1 -th ball will have time to deviate from its equilibrium position by a maximum distance equal to the amplitude of oscillations A(Fig. 3b). After half a period \(\left(t_3 = \dfrac(T)(2) \right)\) 1 -th ball, moving down, will return to the equilibrium position, 4 -th will deviate from the equilibrium position by a distance equal to the amplitude of oscillations A(Fig. 3, c). The wave during this time reaches 7 -th ball, etc.
Through the period \(\left(t_5 = T \right)\) 1 -th ball, having made a complete oscillation, passes through the equilibrium position, and the oscillatory motion will spread to 13 th ball (Fig. 3, e). And then the movement 1 th ball begin to repeat, and more and more balls participate in the oscillatory motion (Fig. 3, e).
Examples of longitudinal waves are sound waves in air and liquid. Elastic waves in gases and liquids arise only when the medium is compressed or rarefied. Therefore, only longitudinal waves can propagate in such media.
Waves can propagate not only in a medium, but also along the interface between two media. Such waves are called surface waves. An example of this type of waves are well-known waves on the surface of the water.
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.: Adukatsy i vykhavanne, 2004. - C. 424-428.
- Zhilko, V.V. Physics: textbook. allowance for grade 11 general education. school from Russian lang. training / V.V. Zhilko, L.G. Markovich. - Minsk: Nar. Asveta, 2009. - S. 25-29.
