Under what conditions do standing waves appear? §5 Standing waves

Consider the result of the interference of two sinusoidal plane waves of the same amplitude and frequency, propagating in opposite directions. For simplicity of reasoning, let us assume that the equations of these waves have the form:

This means that at the origin, both waves vibrate in the same phase. At point A with coordinate x, the total value of the oscillating quantity, according to the principle of superposition (see § 19), is equal to

This equation shows that as a result of the interference of forward and backward waves at each point of the medium (with a fixed coordinate, a harmonic oscillation occurs with the same frequency, but with an amplitude

depending on the value of the x coordinate. At points in the medium where there are no vibrations at all: these points are called vibration nodes.

At the points where the vibration amplitude has greatest value, equal These points are called the antinodes of the oscillations. It is easy to show that the distance between neighboring nodes or adjacent antinodes is equal to the distance between the antinodes and the nearest node is equal to When x changes to cosine in formula (5.16), it changes sign to the opposite (its argument changes to, therefore, if within one half-wave - from one node to another - the particles of the medium deviated in one direction, then within the adjacent half-wave the particles of the medium will be deflected in the opposite direction.

A wave process in a medium described by formula (5.16) is called a standing wave. Graphically standing wave can be depicted as shown in fig. 1.61. Let us assume that y has a displacement of the points of the medium from the state of equilibrium; then formula (5.16) describes a "standing wave of displacement". At a certain moment in time, when all points of the medium have maximum displacements, the direction of which, depending on the value of the coordinate x, is determined by the sign. These displacements are shown in Fig. 1.61 with solid arrows. After a quarter of a period, when the displacements of all points of the medium are equal to zero; particles of the medium pass through the line at different speeds. After another quarter of the period, when the particles of the medium will again have maximum displacements, but in the opposite direction; these offsets are shown in

rice. 1.61 with dotted arrows. The points are the antinodes of the standing displacement wave; points are the nodes of this wave.

The characteristic features of a standing wave, in contrast to an ordinary propagating, or traveling, wave are as follows (we mean plane waves in the absence of damping):

1) in a standing wave, the oscillation amplitudes are different in different places of the system; the system has nodes and antinodes of oscillations. In a "traveling" wave, these amplitudes are the same everywhere;

2) within a section of the system from one node to a neighboring one, all points of the medium oscillate in the same phase; when moving to a neighboring section, the oscillation phases are reversed. In a traveling wave, the oscillation phases, according to formula (5.2), depend on the coordinates of the points;

3) there is no one-way transfer of energy in a standing wave, as is the case in a traveling wave.

When describing oscillatory processes in elastic systems, for the oscillating value y can be taken not only the displacement or velocities of the particles of the system, but also the magnitude of the relative deformation or the magnitude of the stress for compression, tension or shear, etc. In this case, in a standing wave, in places where antinodes of particle velocities are formed, nodes of deformations are located and, conversely, nodes of velocities coincide with antinodes of deformations. The transformation of energy from kinetic to potential and vice versa occurs within a section of the system from an antinode to a neighboring node. It can be assumed that each such section does not exchange energy with neighboring sections. Note that the transformation of the kinetic energy of moving particles into the potential energy of the deformed areas of the medium occurs twice in one period.

Above, considering the interference of forward and backward waves (see expressions (5.16)), we were not interested in the origin of these waves. Let us now assume that the medium in which the oscillations propagate has limited dimensions, for example, oscillations are caused in some solid body - in a rod or string, in a column of liquid or gas, etc. A wave propagating in such a medium (body) , is reflected from the boundaries, therefore, within the volume of this body, the interference of waves caused by an external source and reflected from the boundaries continuously occurs.

Consider simplest example; for example, at a point (Fig. 1.62) of a rod or string with the help of an external sinusoidal source, an oscillatory motion with a frequency is excited; the time origin is chosen so that at this point the displacement is expressed by the formula

where the amplitude of vibrations at the point The wave caused in the rod will be reflected from the second end of the rod 0% and will go in the opposite direction

direction. Let us find the result of the interference of the direct and reflected waves at a certain point of the rod with the x coordinate. For simplicity of reasoning, let us assume that there is no absorption of vibration energy in the rod and therefore the amplitudes of the direct and reflected waves are equal.

At some point in time, when the displacement of the vibrating particles at a point is equal to y, at another point of the rod, the displacement caused by the direct wave will, according to the wave formula, be

The reflected wave also passes through the same point A. To find the displacement caused at point A by the reflected wave (at the same moment in time, it is necessary to calculate the time during which the wave will travel from to and back to the point Since the displacement caused at point by the reflected wave will be equal to

In this case, it is assumed that at the reflecting end of the rod in the process of reflection, there is no abrupt change in the phase of the oscillation; in some cases, such a phase change (called phase loss) occurs and must be accounted for.

The combination of vibrations caused at various points of the rod by direct and reflected waves gives a standing wave; really,

where is some constant phase independent of the x coordinate, and the quantity

is the amplitude of vibrations at a point, it depends on the x coordinate, that is, it is different in different places of the rod.

Let us find the coordinates of those points of the rod at which the nodes and antinodes of the standing wave are formed. The cosine becomes zero or one when the argument values ​​are multiples of

where is an integer. For an odd value of this number, the cosine vanishes and formula (5.19) gives the coordinates of the nodes of the standing wave; for even we get the coordinates of the antinodes.

Above, only two waves were added: a direct one coming from and a reflected one propagating from.However, it should be taken into account that the reflected wave at the rod boundary will be reflected again and go in the direction of the direct wave. Of such reflections

there will be many from the ends of the rod, and therefore it is necessary to find the result of the interference not of two, but of all waves simultaneously existing in the rod.

Suppose that an external source of vibrations caused waves in the rod for some time, after which the influx of vibration energy from the outside ceased. During this time, reflections occurred in the rod, where the time during which the wave passed from one end of the rod to the other. Consequently, in the rod there will be simultaneously waves traveling in the forward direction and waves going in the opposite direction.

Let us assume that as a result of the interference of one pair of waves (direct and reflected), the displacement at point A turned out to be equal to y. Let us find the condition under which all the displacements y caused by each pair of waves have the same directions at the point A of the rod and therefore add up. For this, the phases of the oscillations caused by each pair of waves at a point must differ by from the phase of the oscillations caused by the next pair of waves. But each wave again returns to point A with the same direction of propagation only after a time, that is, it lags behind in phase by c, equating this lag where is an integer, we get

that is, an integer number of half-waves must fit along the length of the rod. Note that under this condition, the phases of all waves going from in the forward direction differ from each other by where is an integer; in the same way, the phases of all waves coming from to reverse direction, differ from each other by Therefore, if one pair of waves (forward and backward) gives along the rod the distribution of displacements, determined by formula (5.17), then with the interference of pairs of such waves, the distribution of displacements will not change; only the amplitudes of the oscillations will increase. If the maximum oscillation amplitude at the interference of two waves, according to formula (5.18), is equal, then at the interference of many waves it will be greater. Let us denote it through then the distribution of the vibration amplitude along the rod, instead of expression (5.18), is determined by the formula

From expressions (5.19) and (5.20), the points are determined at which the cosine has values ​​or 1:

where an integer The coordinates of the nodes of the standing wave are obtained from this formula for odd values, then depending on the length of the rod, i.e., the value

the coordinates of the antinodes are obtained for even values

In fig. 1.63 schematically shows a standing wave in a rod, the length of which; points are antinodes, points are nodes of this standing wave.

In ch. it was shown that in the absence of periodic external influences, the nature of the codebatic movements in the system and, first of all, the main quantity - the frequency of oscillations - is determined by the size and physical properties systems. Each oscillatory system has its own inherent oscillatory motion; this oscillation can be observed if the system is brought out of equilibrium and then external influences are eliminated.

In ch. 4 h. I mainly considered oscillatory systems with lumped parameters, in which only bodies (point bodies) possessed inert mass, and elastic properties- other bodies (springs). In contrast, oscillatory systems in which mass and elasticity are inherent in each elementary volume are called systems with distributed parameters. These include the rods, strings discussed above, as well as columns of liquid or gas (in wind musical instruments), etc. For such systems, standing waves are natural vibrations; the main characteristic of these waves - the wavelength or distribution of nodes and antinodes, as well as the frequency of oscillations - is determined only by the size and properties of the system. Standing waves can also exist in the absence of external (periodic) impact on the system; this effect is necessary only in order to cause or support standing waves in the system, or to change the vibration amplitudes. In particular, if an external influence on a system with distributed parameters occurs with a frequency equal to the frequency of its natural oscillations, i.e., the frequency of a standing wave, then the phenomenon of resonance takes place, considered in Ch. 5.

It is the same for different frequencies.

Thus, in systems with distributed parameters, natural oscillations - standing waves - are characterized by a whole spectrum of frequencies that are multiples of each other. The smallest of these frequencies, corresponding to the longest wavelength, is called the fundamental frequency; others) - overtones or harmonics.

Each system is characterized not only by the presence of such a spectrum of vibrations, but also by a certain distribution of energy between vibrations of different frequencies. For musical instruments this distribution gives the sound a peculiar feature, the so-called timbre of the sound, which is different for different instruments.

The above calculations refer to a free oscillating "rod of length. However, we usually have rods fixed at one or both ends (for example, vibrating strings), or there are one or more anchorage points along the rod. movements are forced displacement nodes.

if it is necessary to obtain standing waves in the rod at one, two, three points of attachment, etc., then these points cannot be chosen arbitrarily, but must be located along the rod so that they are at the nodes of the resulting standing wave. This is shown, for example, in Fig. 1.64. In the same figure, the dotted line shows the displacements of the points of the rod during vibrations; at the free ends, displacement antinodes are always formed, at the fixed ends, displacement nodes. For oscillating air columns in pipes, the displacement (and velocity) nodes are obtained at the reflecting solid walls; antinodes of displacements and velocities are formed at the open ends of the tubes.

Standing waves are formed as a result of the interference of two counterpropagating plane waves of the same frequency ω and amplitude A.

Let us imagine that at point S (Fig. 7.4) there is a vibrator, from which a plane wave propagates along the SO beam. Having reached the obstacle at point O, the wave will be reflected and go in the opposite direction, i.e. two traveling plane waves propagate along the ray: forward and backward. These two waves are coherent, since they are generated by the same source and, superimposing on each other, will interfere with each other.

The vibrational state of the medium arising as a result of interference is called a standing wave.

Let us write the equation of the forward and backward traveling waves:

straight -
; reverse -

where S 1 and S 2 are the displacement of an arbitrary point on the SO ray. Taking into account the formula for the sine of the sum, the resulting displacement is

Thus, the standing wave equation has the form

(7.17)

The factor cosωt shows that all points of the medium on the SO beam perform simple harmonic oscillations with a frequency
... Expression
called the amplitude of the standing wave. As you can see, the amplitude is determined by the position of the point on the SO (x) ray.

Maximum value amplitudes will have points for which

or
(n = 0, 1, 2, ....)

where
, or
(7.18)

standing wave antinodes .

Minimum value, equal to zero, will have those points for which

or
(n = 0, 1, 2, ....)

where
or
(7.19)

Points with such coordinates are called standing wave knots ... Comparing expressions (7.18) and (7.19), we see that the distance between adjacent antinodes and neighboring nodes is λ / 2.

N in the figure, the solid line shows the displacement of the oscillating points of the medium at a certain moment in time, the dashed curve - the position of the same points through T / 2. Each point vibrates with an amplitude determined by its distance from the vibrator (x).

Unlike a traveling wave, a standing wave does not transfer energy. Energy simply transfers from potential (at the maximum displacement of the points of the medium from the equilibrium position) to kinetic (when the points pass through the equilibrium position) within the limits between the nodes that remain motionless.

All points of the standing wave within the limits between the nodes oscillate in the same phase, and along different sides from the node - in antiphase.

Standing waves arise, for example, in a stretched string fixed at both ends when transverse vibrations are excited in it. Moreover, in the places of anchoring, nodes of a standing wave are located.

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 node is formed at the opposite end.

Examples of problem solving

Example . Determine the speed of propagation of sound in water, if the wavelength is 2m, and the source oscillation frequency ν = 725Hz. Also determine the smallest distance between points of the medium that vibrate in the same phase.

Given : λ = 2m; ν = 725Hz.

Find : υ; X.

Solution ... The wavelength is equal to the distance over which a certain phase of the wave extends over the period T, i.e.

,

where υ is the speed of the wave; ν is the vibration frequency.

Then the required speed

Wavelength is the distance between the nearest particles of the medium, vibrating in the same phase. Consequently, the sought-for smallest distance between points of the medium, vibrating in the same phase, is equal to the wavelength, i.e.

Answer: υ = 1450 m / s; x = 2m.

Example . Determine how many times the length of the ultrasonic wave will change when it passes from copper to steel, if the speed of propagation of ultrasound in copper and steel, respectively, are υ 1 = 3.6 km / s and υ 2 = 5.5 km / s.

Given : υ 1 = 3.6 km / s = 3.6 ∙ 10 3 m / s. and υ 2 = 5.5 km / s = 5.5 ∙ 10 3 m / s.

Find :.

Solution ... When waves propagate, the frequency of oscillations does not change when they pass from one medium to another (it depends only on the properties of the wave source), i.e. ν 1 = ν 2 = ν.

Relationship between wavelength and frequency ν:

, (1)

where υ is the speed of the wave.

The sought relation, according to (1),

.

Calculating, we get
(will increase by 1.53 times).

Answer :

Example . One end of the elastic rod is connected to a source of harmonic vibrations that obey the law
and the other end is rigidly fixed. Considering that the reflection in the place of the rod anchorage comes from a denser medium, determine: 1) the equation of a standing wave; 2) coordinates of nodes; 3) coordinates of antinodes.

Given :
.

Find : 1) ξ (x, t); 2) x y; 3) x n.

Solution ... Incident Wave Equation

, (1)

where A is the amplitude of the wave; ω - cyclic frequency; υ is the speed of the wave.

According to the condition of the problem, the reflection at the place where the rod is fixed comes from a denser medium, therefore the wave changes its phase to the opposite one, and the equation of the reflected wave

Adding equations (1) and (2), we obtain the standing wave equation

(took into account
; λ = υТ).

At points in the environment where

(m = 0, 1, 2, ....) (3)

The amplitude of the oscillations vanishes (nodes are observed), at the points of the medium, where

(m = 0, 1, 2, ....) (4)

The amplitude of the oscillations reaches a maximum value equal to 2A (antinodes are observed). The sought coordinates of nodes and antinodes are found from expressions (3) and (4):

coordinates of nodes
(m = 0, 1, 2, ....);

coordinates of antinodes
(m = 0, 1, 2,….).

Answer : 1)
;
(m = 0, 1, 2, ....);
(m = 0, 1, 2,….).

Example . The distance between adjacent nodes of a standing wave created by a tuning fork in the air ℓ = 42cm. Taking the speed of sound in air υ = 332 m / s, determine the vibration frequency ν of the tuning fork.

Given : ℓ = 42cm = 0.42m; υ = 332 m / s.

Find : ν.

Solution ... In a standing wave, the distance between two neighboring nodes is ... Therefore, ℓ = , whence the traveling wavelength

The relationship between wavelength and frequency
... Substituting the value (1) into this formula, we obtain the sought-for frequency of the tuning fork

.

Answer : ν = 395 Hz.

Example . The tube ℓ = 50cm long is filled with air and is open at one end. Taking the speed υ of sound equal to 340 m / s, determine at which lowest frequency a standing sound wave will appear in the pipe. Taking the speed of sound in air υ = 332 m / s, determine the vibration frequency ν of the tuning fork.

Given : ℓ = 50cm = 0.5m; υ = 340 m / s.

Find : ν 0 .

Solution. The frequency will be the minimum provided that the standing wavelength is at its maximum.

In a pipe open from one end, on the open part there will be an antinode (reflection from a less dense medium), and on the closed part there will be a node (reflection from a denser medium). Therefore, the pipe will fit a quarter of the wavelength:

Considering that the wavelength
, we can write

,

Where is the sought lowest frequency

.

Answer : ν 0 = 170 Hz.

Example . Two electric trains are moving towards each other at speedsυ 1 = 20 m / s and υ 2 = 10 m / s. The first train blows a whistle, the pitch of which corresponds to the frequency ν 0 = 600 Hz. Determine the frequency perceived by the passenger of the second before the meeting of the trains and after their meeting. Take the speed of sound equal to υ = 332 m / s.

Given : υ 1 = 20 m / s; υ 2 = 10 m / s; ν 0 = 600 Hz; υ = 332 m / s.

Find: ν ; ν".

Solution. According to the general formula describing the Doppler effect in acoustics, the frequency of sound perceived by a moving receiver is

, (1)

where ν 0 is the frequency of the sound sent by the source; υ pr is the speed of the receiver; υ ist is the speed of the source. If the source and receiver are approaching each other, then the upper sign is taken, if they move away - the lower sign.

According to the designations given in the problem (υ pr = υ 2 and υ source = υ 1) and the above explanations, from formula (1) the sought frequencies perceived by the passenger of the second train:

Before the trains meet (electric trains approach each other):

;

After the trains meet (trains move away from each other):

Answer: ν = 658 Hz; ν "= 549 Hz.

The held end is pulled up sharply and then brought to its original position. The ridge formed on the tube moves along the tube to the wall, where it is reflected. In this case, the reflected wave has the shape of a trough, i.e., it is located below the average position of the tube, while the original antinode was above. What is the reason for this difference?

Imagine the end of a rubber tube attached to a wall. Since it is fixed, it cannot move. The upward force of the incoming impulse tends to make it move upward (see fig.). However, since it cannot move, there must be an equal and oppositely directed downward force emanating from the support and applied to the end of the rubber tube, and therefore the reflected pulse is located antinode downward. The phase difference between the reflected and original pulses is 180 °.

When the hand holding the rubber tube moves up and down and the frequency of movement gradually increases, the point is reached at which a single antinode is obtained (Fig. A). A further increase in the frequency of vibration of the hand will lead to the formation of a double antinode (Fig. 6). If you time the frequency of your hand movements, you will see that their frequency has doubled. Since it is difficult to move your hand more quickly, it is better to use a mechanical vibrator (fig. C).

The metal bar inside the electromagnetic coil vibrates at a frequency controlled by the generator. The waves formed are called standing or stationary waves... They are formed because the reflected wave is superimposed on the incident wave. This phenomenon is known as. There are two waves here: incident and reflected. They are the same, but spread in opposite directions. This traveling waves but they interfere with each other and thus create standing waves.



This has the following consequences:

a) all particles in each half of the wavelength oscillate in phase, that is, they all move in the same direction at the same time;

b) each particle has an amplitude that is different from the amplitude of the next particle;

c) the phase difference between the oscillations of the particles of one half-wave and the oscillations of the particles of the next half-wave is 180 °.

This simply means that they are either rejected as much as possible. opposite sides at the same time, or, if they find themselves in the middle position, begin to move in opposite directions. This is shown in the figure, where you can see that some particles (designated N) do not move (they have zero amplitude), since the forces acting on them are always equal and opposite.

These points are called nodules or nodes, and the distance between two subsequent nodes is half the wavelength, that is, 1/2 λ.

The maximum movement occurs at the points indicated by A, and the amplitude of these points is twice the amplitude of the incident wave. These points are called antinodes, and the distance between two subsequent antinodes is half the wavelength. The distance between the node and the next antinode is one fourth of the wavelength, i.e. 1/4 λ.

Standing wave differs from the running one. V traveling wave:

a) all particles have the same vibration amplitude;

§4 Wave interference.

Superposition principle. The concept of wave coherence

If several waves propagate in the medium simultaneously, then the oscillations of the particles of the medium are equal to the geometric sum of the oscillations that the particles would perform during the propagation of each of the waves separately. Consequently, the waves are simply superimposed, not perturbing each other - the principle of superposition (superposition) of waves.

Two waves are called coherent if their phase difference does not depend on time

-
coherence condition.

Sources of coherent waves are called coherent sources.

since for coherent sources, the difference between the initial phases, then the amplitude A rez at various points depends on the valuecalled the path difference. If

then a maximum is observed.

At

there is a minimum.

When the waves from coherent sources are superimposed, the minima and maxima of the resulting amplitude are observed, i.e. mutual amplification at some points in space and attenuation at others, depending on the relationship between the phases of these waves - the essence of the phenomenon of interference.

§5 Standing waves

A particular case of interference is standing waves - waves generated by the superposition of two traveling waves propagating towards each other waves with the same amplitudes and frequencies.

To derive the standing wave equation, let us take: 1) waves propagate in a medium without damping; 2) A 1 = A 2 = A- have equal amplitudes; 3) ω 1 = ω 2 = ω - equal frequencies; 4) φ 10 = φ 20 = 0.

The equation of a traveling wave propagating along the positive direction of the x-axis (i.e., the equation of an incident wave):

(1)

The equation of a traveling wave propagating in the negative direction of the x-axis (i.e., the equation of the reflected wave):

(2)

Adding (1) and (2), we obtain the equation of a standing wave:

A feature of a standing wave is that the amplitude depends on the coordinate X... When moving from one point to another, the amplitude changes according to the law:

Amplitude of the standing wave.

Those points of the medium at which the amplitude of the standing wave is maximum and is equal to 2 A, are called antinodes. The coordinates of the antinodes can be found from the condition that

from here

The distance between two adjacent antinodes is.

The points at which the amplitude of the standing wave is minimal and equal to 0 are called nodes. The coordinates of the nodes can be found from the condition

from here

The distance between two adjacent nodes is.

Unlike a traveling wave, all points of which oscillate with the same amplitude, but with different phases depending on the coordinate X points (), the point of a standing wave between two nodes oscillates with different amplitudes, but with the same phases (). When passing through a node, the multiplierchanges its sign, therefore the phase of oscillations on opposite sides of the node differs by π, i.e. points lying on opposite sides of the node oscillate in antiphase.

If several waves propagate simultaneously in the medium, then the oscillations of the particles of the medium turn out to be the geometric sum of the oscillations that the particles would perform during the propagation of each of the waves separately. Consequently, the waves are simply superimposed on one another, not disturbing each other. This statement is called the principle of superposition (overlapping) waves.

In the case when the oscillations caused by separate waves at each point of the medium have a constant phase difference, the waves are called coherent. (A more rigorous definition of coherence will be given in § 120.) When coherent waves are added, the phenomenon of interference arises, which consists in the fact that oscillations at some points amplify, and at other points weaken each other.

A very important case of interference occurs when two counterpropagating plane waves with the same amplitude are superimposed. The resulting oscillatory process is called a standing wave. Practically standing waves arise when waves are reflected from obstacles. A wave falling on an obstacle and a reflected wave running towards it, superimposing on each other, Give a standing wave.

Let us write the equations for two plane waves propagating along the x-axis in opposite directions:

Adding these equations together and transforming the result using the formula for the sum of cosines, we get

Equation (99.1) is a standing wave equation. To simplify it, we will choose the origin so that the difference becomes equal to zero, and the origin - so that the sum is equal to zero.In addition, we replace the wave number k with its value

Then equation (99.1) takes the form

From (99.2) it can be seen that at each point of the standing wave there are oscillations of the same frequency as that of the counterpropagating waves, and the amplitude depends on x:

the vibration amplitude reaches its maximum value. These points are called the antinodes of the standing wave. From (99.3) the values ​​of the coordinates of the antinodes are obtained:

It should be borne in mind that the antinode is not a single point, but a plane, the points of which have the values ​​of the x coordinate determined by formula (99.4).

At points whose coordinates satisfy the condition

the vibration amplitude vanishes. These points are called standing wave nodes. The points of the medium located at the nodes do not vibrate. The coordinates of the nodes matter

A node, like an antinode, is not a single point, but a plane, the points of which have x coordinate values ​​determined by formula (99.5).

From formulas (99.4) and (99.5) it follows that the distance between adjacent antinodes, as well as the distance between neighboring nodes, is equal. The bumps and nodes are shifted relative to each other by a quarter of a wavelength.

Let us turn again to equation (99.2). The multiplier changes sign when crossing the zero value. In accordance with this, the phase of oscillations on different sides of the node differs by This means that the points lying on different sides of the node oscillate in antiphase. All points enclosed between two adjacent nodes oscillate in phase (i.e., in the same phase). In fig. 99.1 shows a number of "snapshots" of deviations of points from the equilibrium position.

The first "photograph" corresponds to the moment when the deviations reach the greatest absolute value. Subsequent “photographs” are taken at quarter-period intervals. Arrows indicate particle velocities.

Differentiating equation (99.2) once with respect to t, and another time with respect to x, we find expressions for the velocity of particles and for the deformation of the medium:

Equation (99.6) describes a standing wave of velocity, and (99.7) - a standing wave of deformation.

In fig. 99.2 “snapshots” of displacement, velocity and deformation are compared for the moments of time 0 and From the graphs it is seen that the nodes and antinodes of the velocity coincide with the nodes and antinodes of the displacement; the nodes and antinodes of the deformation coincide, respectively, with the antinodes and displacement nodes. While reaching maximum values, it vanishes, and vice versa.

Accordingly, twice in a period, the energy of the standing wave is converted either completely into potential, concentrated mainly near the nodes of the wave (where the antinodes of deformation are located), then completely into kinetic, concentrated mainly near the antinodes of the wave (where the antinodes of the velocity are located). As a result, there is a transition of energy from each node to the adjacent antinodes and vice versa. The time-average energy flux in any section of the wave is zero.