Standing waves. 6.1 Standing waves in an elastic medium

6.1 Standing waves in an elastic medium

According to the principle of superposition, when several waves propagate simultaneously in an elastic medium, their superposition occurs, and the waves do not disturb each other: the oscillations of the particles of the medium are the vector sum of the oscillations that the particles would make if each wave propagated separately .

Waves that create oscillations of the medium, the phase differences between which are constant at each point in space, are called coherent.

When coherent waves are added, the phenomenon occurs interference, which consists in the fact that at some points in space the waves strengthen each other, and at other points they weaken each other. An important case of interference is observed when two counterpropagating plane waves with the same frequency and amplitude are superimposed. The resulting oscillations are called standing wave. Most often, standing waves arise when a traveling wave is reflected from an obstacle. In this case, the incident wave and the wave reflected towards it, when added, give a standing wave.

We obtain the standing wave equation. Let's take two plane harmonic waves propagating towards each other along the axis X and having the same frequency and amplitude:

Where – phase of oscillations of points of the medium during the passage of the first wave;

– phase of oscillations of points in the medium during the passage of the second wave.

Phase difference at each point on the axis X the network will not depend on time, i.e. will be constant:

Therefore, both waves will be coherent.

The vibration of the particles of the medium resulting from the addition of the waves under consideration will be as follows:

Let us transform the sum of the cosines of angles according to rule (4.4) and obtain:

Regrouping the factors, we get:

To simplify the expression, we choose the reference point so that the phase difference and the beginning of the time count so that the sum of the phases is equal to zero: .

Then the equation for the sum of waves will take the form:

Equation (6.6) is called standing wave equation. It shows that the frequency of a standing wave is equal to the frequency of a traveling wave, and the amplitude, unlike a traveling wave, depends on the distance from the origin:

. (6.7)

Taking into account (6.7), the standing wave equation takes the form:

. (6.8)

Thus, points of the medium oscillate with a frequency coinciding with the frequency of the traveling wave and amplitude a, depending on the position of the point on the axis X. Accordingly, the amplitude changes according to the cosine law and has its own maxima and minima (Fig. 6.1).

In order to visually represent the location of the amplitude minima and maxima, we replace, according to (5.29), the wave number with its value:

Then expression (6.7) for the amplitude will take the form

(6.10)

From this it becomes clear that the displacement amplitude is maximum at , i.e. at points whose coordinates satisfy the condition:

, (6.11)

Where

From here we obtain the coordinates of the points where the displacement amplitude is maximum:

; (6.12)

The points where the amplitude of vibrations of the medium is maximum are called antinodes of the wave.

The amplitude of the wave is zero at points where . The coordinates of such points, called wave nodes, satisfies the condition:

, (6.13)

Where

From (6.13) it is clear that the coordinates of the nodes have the values:

, (6.14)

In Fig. Figure 6.2 shows an approximate view of a standing wave, the location of nodes and antinodes is marked. It can be seen that neighboring nodes and displacement antinodes are spaced from each other at the same distance.

Let's find the distance between neighboring antinodes and nodes. From (6.12) we obtain the distance between the antinodes:

(6.15)

The distance between nodes is obtained from (6.14):

(6.16)

From the obtained relations (6.15) and (6.16) it is clear that the distance between neighboring nodes, as well as between neighboring antinodes, is constant and equal to ; nodes and antinodes are shifted relative to each other by (Fig. 6.3).

From the definition of wavelength, we can write an expression for the length of a standing wave: it is equal to half the length of a traveling wave:

Let us write, taking into account (6.17), expressions for the coordinates of nodes and antinodes:

, (6.18)

, (6.19)

The factor that determines the amplitude of a standing wave changes its sign when passing through the zero value, as a result of which the phase of oscillations on different sides of the node differs by . Consequently, all points lying on opposite sides of the node oscillate in antiphase. All points located between neighboring nodes oscillate in phase.

The nodes conditionally divide the environment into autonomous regions in which harmonic oscillations occur independently. There is no transfer of motion between regions, and, therefore, there is no flow of energy between regions. That is, there is no transmission of disturbance along the axis. That's why the wave is called a standing wave.

So, a standing wave is formed from two oppositely directed traveling waves of equal frequencies and amplitudes. The Umov vectors of each of these waves are equal in magnitude and opposite in direction, and when added they give zero. Consequently, a standing wave does not transfer energy.

6.2 Examples of standing waves

6.2.1 Standing wave in a string

Let's consider a string of length L, fixed at both ends (Fig. 6.4).

Let's place an axis along the string X so that the left end of the string has the coordinate x=0, and the right one – x=L. Vibrations occur in the string, described by the equation:

Let us write down the boundary conditions for the string under consideration. Since its ends are fixed, then at points with coordinates x=0 And x=L no hesitation:

(6.22)

Let us find the equation of string oscillations based on the written boundary conditions. Let us write equation (6.20) for the left end of the string taking into account (6.21):

Relation (6.23) is satisfied for any time t in two cases:

1. . This is possible if there are no vibrations in the string (). This case is not of interest, and we will not consider it.

2. . Here is the phase. This case will allow us to obtain the equation of string vibrations.

Let us substitute the resulting phase value into the boundary condition (6.22) for the right end of the string:

. (6.25)

Considering that

, (6.26)

from (6.25) we obtain:

Again, two cases arise in which relation (6.27) is satisfied. We will not consider the case when there are no vibrations in the string ().

In the second case, the equality must be satisfied:

and this is only possible when the argument of sine is a multiple of an integer:

We discard the value, because in this case, and this would mean either zero length of the string ( L=0) or wave number k=0. Taking into account the connection (6.9) between the wave number and the wavelength, it is clear that in order for the wave number to be equal to zero, the wavelength should be infinite, and this would mean the absence of oscillations.

From (6.28) it is clear that the wave number when oscillating a string fixed at both ends can take only certain discrete values:

Taking into account (6.9), we write (6.30) in the form:

from which we obtain the expression for possible wavelengths in the string:

In other words, over the length of the string L must fit into an integer n half waves:

The corresponding oscillation frequencies can be determined from (5.7):

Here is the phase velocity of the wave, depending, according to (5.102), on the linear density of the string and the tension force of the string:

Substituting (6.34) into (6.33), we obtain an expression describing the possible vibration frequencies of the string:

, (6.36)

The frequencies are called natural frequencies strings. Frequency (at n = 1):

(6.37)

called fundamental frequency(or main tone) strings. Frequencies determined at n>1 are called overtones or harmonics. The harmonic number is n-1. For example, frequency:

corresponds to the first harmonic, and frequency:

corresponds to the second harmonic, etc. Since a string can be represented as a discrete system with an infinite number of degrees of freedom, then each harmonic is fashion string vibrations. In the general case, string vibrations represent a superposition of modes.

Each harmonic has its own wavelength. For the main tone (with n= 1) wavelength:

respectively for the first and second harmonics (at n= 2 and n= 3) wavelengths will be:

Figure 6.5 shows the appearance of several modes of vibration carried out by a string.

Thus, a string with fixed ends realizes an exceptional case within the framework of classical physics - a discrete spectrum of vibration frequencies (or wavelengths). An elastic rod with one or both clamped ends and oscillations of an air column in pipes behave in the same way, which will be discussed in subsequent sections.

6.2.2 Influence of initial conditions on movement

continuous string. Fourier analysis

In addition to the discrete spectrum of oscillation frequencies, oscillations of a string with clamped ends have another important property: the specific form of oscillations of the string depends on the method of excitation of oscillations, i.e. from the initial conditions. Let's take a closer look.

Equation (6.20), which describes one mode of a standing wave in a string, is a particular solution of the differential wave equation (5.61). Since the vibration of a string consists of all possible modes (for a string there is an infinite number), then the general solution of the wave equation (5.61) consists of an infinite number of partial solutions:

, (6.43)

Where i– vibration mode number. Expression (6.43) is written taking into account the fact that the ends of the string are fixed:

and also taking into account the frequency connection i-th mode and its wave number:

(6.46)

Here – wave number i th fashion;

– wave number of the 1st mode;

Let us find the value of the initial phase for each oscillation mode. For this, at the moment t=0 let's give the string a shape described by the function f 0 (x), the expression for which we obtain from (6.43):

. (6.47)

In Fig. Figure 6.6 shows an example of the shape of a string described by the function f 0 (x).

At a moment in time t=0 the string is still at rest, i.e. the speed of all its points is zero. From (6.43) we find an expression for the speed of the string points:

and, substituting in it t=0, we obtain an expression for the speed of points on the string at the initial moment of time:

. (6.49)

Since at the initial moment of time the speed is equal to zero, then expression (6.49) will be equal to zero for all points of the string if . It follows from this that the initial phase for all modes is also zero (). Taking this into account, expression (6.43), which describes the motion of the string, takes the form:

, (6.50)

and expression (6.47), describing the initial shape of the string, looks like:

. (6.51)

A standing wave in a string is described by a function that is periodic over the interval , where it is equal to two lengths of the string (Fig. 6.7):

This can be seen from the fact that periodicity on an interval means:

Hence,

which leads us to expression (6.52).

From mathematical analysis it is known that any periodic function can be expanded with high accuracy into a Fourier series:

, (6.57)

where , , are Fourier coefficients.

When two identical waves with equal amplitudes and periods propagate towards each other, standing waves arise when they overlap. Standing waves can be produced by reflection from obstacles. Let's say the emitter sends a wave to an obstacle (incident wave). The wave reflected from it will be superimposed on the incident wave. The standing wave equation can be obtained by adding the incident wave equation

and reflected wave equations

The reflected wave moves in the direction opposite to the incident wave, so we take the distance x with a minus sign. The displacement of a point that participates simultaneously in two oscillations is equal to the algebraic sum. After simple transformations, we get

does not depend on time and determines the amplitude of any point with coordinate x. Each point performs a harmonic oscillation with a period T. The amplitude A st for each point is completely defined. But when moving from one point of the wave to another, it changes depending on the distance x. If we give x values ​​equal to etc., then when substituting into equation (8.16) we get . Consequently, the indicated points of the wave remain at rest, because the amplitudes of their oscillations are zero. These points are called standing wave nodes. The points at which oscillations occur with maximum amplitude are called antinodes. The distance between adjacent nodes (or antinodes) is called the standing wave length and is equal to

where λ is the length of the traveling wave.

In a standing wave, all points of the medium in which they propagate, located between two adjacent nodes, oscillate in the same phase. Points of the medium lying on opposite sides of the node oscillate in antiphase - their phases differ by π. those. when passing through a node, the oscillation phase changes abruptly by π. Unlike traveling waves, there is no energy transfer in a standing wave due to the fact that the forward and backward waves that form this wave transfer energy in equal quantities in both the forward and opposite directions. In the case when a wave is reflected from a medium denser than the medium where the wave propagates, a node appears at the place of reflection and the phase changes to the opposite. In this case, they say that half the wave is lost. When a wave is reflected from a less dense medium at the place of reflection, clustering appears, and there is no loss of half the wave.

Any wave is an oscillation. A liquid, an electromagnetic field, or any other medium can vibrate. In everyday life, every person daily encounters one or another manifestation of fluctuations. But what is a standing wave?

Imagine a capacious container into which water is poured - it could be a basin, bucket or bathtub. If you now pat the liquid with your palm, then wave-like ridges will run from the center of impact in all directions. By the way, that’s what they are called - traveling waves. Their characteristic feature is the transfer of energy. However, by changing the frequency of the claps, you can achieve their almost complete visible disappearance. It seems that the mass of water becomes jelly-like, and the movement occurs only down and up. A standing wave is this displacement. This phenomenon occurs because each wave moving away from the center of the impact reaches the walls of the container and is reflected back, where it intersects (interferes) with the main waves traveling in the opposite direction. A standing wave appears only if the reflected and direct waves are in phase, but different in amplitude. Otherwise, the above interference does not occur, since one of the properties of wave disturbances with different characteristics is the ability to coexist in the same volume of space without distorting each other. It can be argued that a standing wave is the sum of two counter-directed traveling ones, which leads to a drop in their speeds to zero.

Why does the water in the above example continue to oscillate in the vertical direction? Very simple! When waves with the same parameters are superimposed, at certain times the oscillations reach their maximum value, called antinodes, and at others they are completely damped (nodes). By changing the frequency of the clapping, you can either completely suppress horizontal waves or increase vertical displacements.

Standing waves are of interest not only to practitioners, but also to theorists. In particular, one of the models states that any material particle is characterized by some kind of vibration: an electron oscillates (trembles), a neutrino oscillates, etc. Further, within the framework of the hypothesis, it was assumed that the mentioned vibration is a consequence of the interference of some as yet undiscovered disturbances of the environment. In other words, the authors argue that where those amazing waves form standing waves, matter arises.

No less interesting is the phenomenon of Schumann Resonance. It lies in the fact that under certain conditions (none of the proposed hypotheses has yet been accepted as the only correct one), standing electromagnetic waves arise in the space between the earth's surface and the lower boundary of the ionosphere, the frequencies of which lie in the low and ultra-low ranges (from 7 to 32 hertz ). If the wave formed in the “surface - ionosphere” gap goes around the planet and enters resonance (phase coincidence), it can exist for a long time without attenuation, self-sustaining. The Schumann resonance is of particular interest because the frequency of the waves is almost identical to the natural alpha rhythms of the human brain. For example, research into this phenomenon in Russia is carried out not only by physicists, but also by such a large organization as the Institute of the Human Brain.

The brilliant inventor Nikola Tesla drew attention to standing ones. It is believed that he could use this phenomenon in some of his devices. Thunderstorms are considered to be one of the sources of their appearance in the atmosphere. Electrical discharges excite an electromagnetic field and generate waves.

What is a standing wave? What is a standing wave? How does it arise? What is the difference between a standing wave and a traveling wave?

1. Have you seen the slate sheet?
   The same thing happens on the surface of the water, a puddle on a windy day, for example.
2. wow, how difficult your answer was. I explain it simply as a carrot.
   What is a wave process? This is when something changes and it has a maximum and a minimum (an example of water waves when at different times at the same point the maximum of the wave (peak) changes to a minimum). When the maximum changes to a minimum, these are traveling waves. Waves can be standing. This is when the maximum does not change to the minimum, but there are different levels in different places (standing ripples on the surface of the water from the wind).
3. Oho. This is a concept that swells the brains of tens of thousands of people around the clock! A standing wave is the essence of BTG. The essence of Tesla engineering. The essence of future energy from nothing!)))
4. Standing#769;tea wave#769; oscillations in distributed oscillatory systems with a characteristic arrangement of alternating maxima (antinodes) and minima (nodes) of amplitude. In practice, such a wave arises during reflections from obstacles and inhomogeneities as a result of the superposition of the reflected wave on the incident one. In this case, the frequency, phase and attenuation coefficient of the wave at the place of reflection are extremely important.

   Examples of a standing wave include vibrations of a string, vibrations of air in an organ pipe; in nature Schumann waves.

   A purely standing wave, strictly speaking, can exist only in the absence of losses in the medium and complete reflection of the waves from the boundary. Usually, in addition to standing waves, the medium also contains traveling waves that supply energy to places of absorption or radiation.

   A Rubens tube is used to demonstrate standing waves in gas.

5. Pour water into the bath and splash your hand on the surface. Waves will spread from your hand in all directions. They are called runners. By smoothly changing the frequency of hand vibrations, you can ensure that the waves stop moving to the sides, but remain in place. The movement would only be up and down. These are standing waves.

   They are formed in this case only because the bath has walls from which reflection occurs; if there were no walls, then standing waves would not form, as, for example, on an open water surface.

   The explanation for the occurrence of standing waves is simple: when a direct wave and a wave reflected from a wall collide, they reinforce each other, and if this collision occurs all the time in the same place, then the horizontal movement of the waves disappears.

6. standing waves,
   waves arising due to the interference of waves propagating in mutually opposite directions. Almost S. century. arise when waves are reflected from obstacles and inhomogeneities as a result of the superposition of the reflected wave on the direct wave. Various sections of the North century. oscillate in the same phase, but with different amplitudes (Fig.). In the N. century. , unlike running energy, there is no flow of energy. Such waves arise, for example, in an elastic system - a rod or a column of air located inside a pipe, closed at one end, when the piston oscillates in the pipe. Traveling waves are reflected from the boundaries of the system, and as a result of the superposition of incident and reflected waves, turbulence is established in the system. In this case, along the length of the air column, so-called nodes of displacements (velocities) of the plane, perpendicular to the axis of the column, at which there are no displacements of air particles, and pressure amplitudes are maximum, and antinodes of displacements of the plane, at which displacements are maximum, and pressures are equal to zero. Displacement nodes and antinodes are located in the pipe at distances of a quarter wavelength, and a displacement node and a pressure antinode are always formed near a solid wall. A similar picture is observed if the solid wall at the end of the pipe is removed, but then the velocity antinode and the pressure node are on the plane of the hole (approximately). In any volume that has certain boundaries and a sound source, sounds are formed. , but with a more complex structure.

   Any wave process associated with the propagation of disturbances can be accompanied by the formation of a wave. They can occur not only in gaseous, liquid and solid media, but also in a vacuum during the propagation and reflection of electromagnetic disturbances, for example in long electrical lines. The antenna of a radio transmitter is often made in the form of a rectilinear vibrator or a system of vibrators, along the length of which the S.V. In sections of waveguides and closed volumes of various shapes, used as resonators in microwave technology, CVs are installed. certain types. In electromagnetic systems. electric and magnetic fields are separated in the same way as in elastic solar systems. displacement and pressure are separated.

   Pure S. v. can be established, strictly speaking, only in the absence of attenuation in the medium and complete reflection of the waves from the boundary. Usually, except for S. v. , there are also traveling waves that supply energy to the places of absorption or emission.

   In optics it is also possible to establish S. century. with visible maxima and minima of the electric field. If the light is not monochromatic, then in the North century. the antinodes of the electric field of different wavelengths will be located in different places and color separation is often observed.

Standing waves are formed by the superposition of two identical waves traveling towards each other. Everyone has probably seen standing waves in guitar strings. When a string is pulled back and released in any place, elastic transverse waves begin to scatter in different directions, which are then reflected from the ends of the string and, overlapping each other, form standing waves(if there is no attenuation during propagation and reflection). How does this happen?

When two sinusoidal waves with the same frequency and amplitude are added, but propagating in different axis directions x, we obtain a disturbance that is described by the function

F(x,t) = f 0 sin (ωt — kx +φ 1) + f 0 sin (ωt + kx + φ 2) = 2f 0 cos (kx + (φ 2 —φ 1) /2) + (φ 1 + φ 2) / 2).

This is it standing wave equation. At each point of a standing wave, oscillations occur according to the harmonic law:

F(x, t) = F 0 sin (ωt + (φ 1 + φ 2) / 2.

Amplitude of oscillations

| F 0| = 2 f 0 | cos (kx + (φ 2 —φ 1) / 2)|

depends on the coordinate x. At points where kx + Δφ / 2 = (n + 1 / 2)π (n- an integer, Δφ = φ 1 —φ 2), amplitude F 0 = 0. Such points are called standing wave nodes, there are no vibrations in them. Points for which the amplitude of oscillations | F 0 | = 2f 0 maximum is called standing wave antinodes. Distance Δx between adjacent nodes (or adjacent antinodes) is equal to half the length of the traveling waves from which the standing wave was formed:

Δx =π / k= λ / 2.

At points between two adjacent nodes, oscillations occur in the same phase, and the amplitude changes from zero to maximum (at the antinode, which is located in the middle between the nodes) and again to zero. Material from the site

When passing through a node, the oscillation phase changes to π, because the sign changes F 0. In a standing wave, the disturbance of the medium becomes zero simultaneously at all points, and at the same time at all points the disturbance reaches its maximum value. Thus, the sounding string straightens out after each half-period, and after a quarter of the period after straightening, it takes on the “most curved” shape.

If you observe oscillations at only one point, then it is impossible to say which wave is running or standing tea— caused these fluctuations. But if you monitor the oscillations at several points, the patterns of oscillations in the traveling and standing waves will be completely different. In a plane traveling wave, oscillations at different points occur with the same amplitude, but in different phases. In a standing wave, oscillations at different points occur with different amplitudes, but in the same phase. Therefore, when observing the “whole picture,” it is, of course, impossible to confuse traveling and standing waves.