Effects of wave addition. standing elastic waves

Chapter 7

Waves. wave equation

In addition to the motions we have already considered, in almost all areas of physics there is another type of motion - waves. Distinctive feature This movement, which makes it unique, is that it is not the particles of matter that propagate in the wave, but changes in their state (perturbations).

Perturbations that propagate in space over time are called waves . Waves are mechanical and electromagnetic.

elastic wavesare propagating perturbations of the elastic medium.

A perturbation of an elastic medium is any deviation of the particles of this medium from the equilibrium position. Perturbations arise as a result of deformation of the medium in any of its places.

The set of all points where the wave has reached in this moment time, forms a surface called wave front .

According to the shape of the front, the waves are divided into spherical and plane. Direction propagation of the wave front is determined perpendicular to the wave front, called beam . For a spherical wave, the rays are a radially diverging beam. For a plane wave, a ray is a beam of parallel lines.

In any mechanical wave, two types of motion simultaneously exist: oscillations of the particles of the medium and the propagation of a disturbance.

A wave in which the oscillations of the particles of the medium and the propagation of the perturbation occur in the same direction is called longitudinal (fig.7.2 a).

A wave in which the particles of the medium oscillate perpendicular to the direction of propagation of perturbations is called transverse (Fig. 7.2 b).

In a longitudinal wave, disturbances represent a compression (or rarefaction) of the medium, and in a transverse wave, they are displacements (shears) of some layers of the medium relative to others. Longitudinal waves can propagate in all media (in liquid, solid, and gaseous), while transverse waves can propagate only in solid ones.

Each wave propagates at some speed . Under wave speed υ understand the propagation speed of the disturbance. The speed of a wave is determined by the properties of the medium in which this wave propagates. AT solids the speed of longitudinal waves is greater than the speed of transverse waves.

Wavelengthλ is the distance over which a wave propagates in a time equal to the period of oscillation in its source. Since the speed of the wave is a constant value (for a given medium), the distance traveled by the wave is equal to the product of the speed and the time of its propagation. So the wavelength

It follows from equation (7.1) that particles separated from each other by an interval λ oscillate in the same phase. Then we can give the following definition of the wavelength: the wavelength is the distance between two nearest points oscillating in the same phase.

Let us derive the equation of a plane wave, which allows us to determine the displacement of any point of the wave at any time. Let the wave propagate along the beam from the source with some speed v.

The source excites simple harmonic oscillations, and the displacement of any point of the wave at any moment of time is determined by the equation

S = Asinωt (7. 2)

Then the point of the medium, which is at a distance x from the wave source, will also perform harmonic oscillations, but with a delay in time by a value, i.e. the time it takes for the vibrations to propagate from the source to that point. The displacement of the oscillating point relative to the equilibrium position at any moment of time will be described by the relation

This is the plane wave equation. This wave is characterized by the following parameters:

· S - displacement from the position of the equilibrium point of the elastic medium, to which the oscillation has reached;

· ω - cyclic frequency of oscillations generated by the source, with which the points of the medium also oscillate;

· υ - wave propagation velocity (phase velocity);

x – distance to that point of the medium where the oscillation has reached and the displacement of which is equal to S;

· t – time counted from the beginning of oscillations;

Introducing the wavelength λ into expression (7. 3), the plane wave equation can be written as follows:

(7. 4)

Rice. 7.3

where called the wave number (number of waves per unit length).

Wave interference. standing waves. The equation standing wave

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

Imagine that at the point S there is a vibrator, from which a plane wave propagates along the ray SO. 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 beam: forward and backward. These two waves are coherent, since they are generated by the same source and, superimposed on each other, will interfere with each other.

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

Let's write the equation of direct and backward traveling wave:

straight - ; reverse -

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

Thus, the standing wave equation has the form

The factor cosωt shows that all points of the medium on the SO beam perform simple harmonic oscillations with a frequency . The expression is 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,….)

from where, or (4.70)

antinodes of a standing wave .

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

Or (n=0, 1, 2,….)

from where or (4.71)

Points with such coordinates are called standing wave nodes . Comparing expressions (4.70) and (4.71), we see that the distance between neighboring antinodes and neighboring nodes is equal to λ/2.

In the figure, the solid line shows the displacement of the oscillating points of the medium at some point in time, the dotted curve shows the position of the same points through T / 2. Each point oscillates with an amplitude determined by its distance from the vibrator (x).

Unlike a traveling wave, there is no energy transfer in a standing wave. Energy simply passes from potential (with 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 a standing wave within the limits between the nodes oscillate in the same phase, and on opposite sides of the node - in antiphase.

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.

Sound. Doppler effect

Longitudinal elastic waves propagating in gas, liquid and solids are invisible. However, under certain conditions they can be heard. So, if we excite vibrations of a long steel ruler, clamped in a vise, then we will not hear the waves generated by it. But if we shorten the protruding part of the ruler and thereby increase the frequency of its oscillations, then we will find that the ruler will begin to sound.

Elastic waves that cause auditory sensations in humans are called sound waves or simply sound.

The human ear is capable of perceiving elastic mechanical waves with a frequency ν from 16 Hz to 20,000 Hz. Elastic waves with frequency ν<16Гц называют инфразвуком, а волны с частотой ν>20000 Hz - ultrasonic.

Frequencies in the range from 16 Hz to 20000 Hz are called sound. Any body (solid, liquid or gaseous) oscillating with sound frequency creates a sound wave in the environment.

In gases and liquids, sound waves propagate in the form of longitudinal compression and rarefaction waves. The compression and rarefaction of the medium, which occurs as a result of vibrations of the sound source (strings, tuning fork legs, vocal cords, etc.), after a while reach the human ear and, causing the eardrum to make forced vibrations, cause certain auditory sensations in a person.

Sound waves cannot propagate in a vacuum because there is nothing to vibrate there. This can be verified by a simple experiment. If we place an electric bell under the glass dome of an air pump, as the air is pumped out, we will find that the sound will become weaker and weaker until it stops altogether.

sound in gases. It is known that during a thunderstorm we first see a flash of lightning and only then hear thunder. This delay occurs because the speed of sound in air is much less than the speed of light. The speed of sound in air was first measured by the French scientist Marin Mersen in 1646. At a temperature of +20ºС, it is equal to 343 m/s, i.е. 1235km/h

The speed of sound depends on the temperature of the medium. It increases with increasing temperature and decreases with decreasing temperature.

The speed of sound does not depend on the density of the gas in which this sound propagates. However, it depends on the mass of its molecules. The larger the mass of gas molecules, the lower the speed of sound in it. So, at a temperature

0 ºС the speed of sound in hydrogen is 1284m/s, and in carbon dioxide- 259 m / s.

Sound in liquids. The speed of sound in liquids is generally greater than the speed of sound in gases. The speed of sound in water was first measured in 1826. The experiments were carried out on Lake Geneva in Switzerland. On one boat they set fire to gunpowder and at the same time hit the bell, lowered into the water. The sound of this bell, with the help of a special horn, also lowered into the water, was caught on another boat, which was located at a distance of 14 km from the first. The speed of sound in water was determined from the time difference between the flash of light and the arrival of the sound signal. At a temperature of 8 ºС, it turned out to be equal to 1435m/s.

In liquids, the speed of sound generally decreases with increasing temperature. Water is an exception to this rule. In it, the speed of sound increases with increasing temperature and reaches a maximum at a temperature of 74 ºС, and with a further increase in temperature, it decreases.

It must be said that the human ear does not “work” well under water. Most of the sound in this case is reflected from the eardrum and therefore does not cause auditory sensations. It was this that at one time gave reason to our ancestors to consider undersea world"world of silence". Hence the expression "mute like a fish." However, even Leonardo da Vinci suggested listening to underwater sounds by putting your ear to an oar lowered into the water. Using this method, you can make sure that the fish are actually quite talkative.

Sound in solids. The speed of sound in solids is even greater than in liquids. Only here it should be taken into account that both longitudinal and transverse waves can propagate in solids. The speed of these waves, as we know, is different. For example, in steel, transverse waves propagate at a speed of 3300 m/s, and longitudinal waves at a speed of 6100 m/s. The fact that the speed of sound in a solid is greater than in air can be verified as follows. If your friend hits one end of the rail and you put your ear on the other end, two hits will be heard. The sound will reach your ear first through the rail and then through the air.

The earth has good conductivity. Therefore, in the old days, during a siege, “hearers” were placed in the fortress walls, who, by the sound transmitted by the earth, could determine whether the enemy was digging to the walls or not. Putting your ear to the ground also made it possible to detect the approach of enemy cavalry.

In addition to audible sounds, infrasound waves propagate in the earth's crust, which the human ear no longer perceives. Such waves can occur during earthquakes.

Powerful infrasonic waves propagating both in the ground and in the air arise during volcanic eruptions and explosions of atomic bombs. The sources of infrasound can also be air vortices in the atmosphere, cargo discharges, gun shots, wind, flowing ridges sea waves, working engines of jet aircraft, etc.

Ultrasound is also not perceived by the human ear. However, some animals, such as bats and dolphins, can emit and capture it. In technology, special devices are used to produce ultrasound.

An oscillating body placed in an elastic medium is a source of vibrations that propagate from it in all directions. The process of propagation of oscillations in a medium is called wave.

When a wave propagates, the particles of the medium do not move along with the wave, but oscillate around their equilibrium positions. Together with the wave from particle to particle, only the state of oscillatory motion and its energy is transmitted. Therefore, the main property of all waves, regardless of their nature, is the transfer of energy without the transfer of matter.

Waves are transverse (oscillations occur in a plane perpendicular to the direction of propagation), and longitudinal (concentration and rarefaction of the particles of the medium occur in the direction of propagation).

When two identical waves with equal amplitudes and periods propagate towards each other, then when they are superimposed, standing waves arise. Standing waves can be obtained 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

(A very important case of interference is observed when two opposite plane waves with the same amplitude are superimposed. The resulting oscillatory process is called a standing wave. Practically standing waves arise when reflected from obstacles.)

This equation is called the wave equation. Any function that satisfies this equation describes some wave.
wave equation called an expression that gives bias fluctuating point as a function of its coordinates ( x, y, z) and time t.

This function must be periodic both with respect to time and coordinates (a wave is a propagating oscillation, hence a periodically repeating motion). In addition, points separated by a distance l oscillate in the same way.

- this is plane wave equation.
Equation (5.2.3) will have the same form if the oscillations propagate along the axis y or z
In general plane wave equation is written like this:

Expressions (5.2.3) and (5.2.4) are traveling wave equations .

Equation (5.2.3) describes a wave propagating in the direction of increase x. A wave propagating in the opposite direction has the form:

Let's introduce wave number , or in vector form:

where is the wave vector and is the normal to the wave surface.

Since , then . From here. Then plane wave equation will be written like this:

spherical wave equation:

where BUT is equal to the amplitude at a distance from the source equal to unity.

WAVE VECTOR- vector k, which determines the direction of propagation and the spatial period of a flat monochromatic. waves

where - constant amplitude and phase of the wave, - circular frequency, r is the radius vector. V. module called wave number k= , where - spatial period or wavelength. In the direction of V. c. the fastest change in the phase of the wave occurs, so it is taken as the direction of propagation. The velocity of the phase in this direction, or phase velocity, is determined through the wave number .. in.

If several waves propagate simultaneously in a medium, then the oscillations of the particles of the medium turn out to be the geometric sum of the oscillations that the particles would make during the propagation of each of the waves separately. Consequently, the waves simply overlap one another without disturbing each other. This statement is called the principle of superposition of waves. The principle of superposition states that the movement caused by the propagation of several waves at once is again a certain wave process. Such a process, for example, is the sound of an orchestra. It arises from the simultaneous excitation of sound vibrations of the air by individual musical instruments. It is remarkable that when waves are superimposed, special phenomena can arise. They are called the effects of addition or, as they say, the superposition of waves. Among these effects, the most important are interference and diffraction.

Interference is a phenomenon of time-stable redistribution of the energy of vibrations in space, as a result of which vibrations are amplified in some places, and weakened in others. This phenomenon occurs when adding waves with a phase difference that persists over time, the so-called coherent waves. The interference of a large number of waves is commonly called diffraction. There is no fundamental difference between interference and diffraction. The nature of these phenomena is the same. We confine ourselves to discussing only one very important interference effect, which is the formation of standing waves.

A necessary condition for the formation of standing waves is the presence of boundaries that reflect the waves incident on them. Standing waves are formed as a result of the addition of incident and reflected waves. Phenomena of this kind are quite common. So, each tone of the sound of any musical instrument is excited by a standing wave. This wave is formed either in a string (stringed instruments) or in a column of air (wind instruments). The reflective boundaries in these cases are the points of attachment of the string and the surfaces of the internal cavities of wind instruments.

Each standing wave has the following properties. The entire region of space in which the wave is excited can be divided into cells in such a way that oscillations are completely absent at the boundaries of the cells. The points located on these boundaries are called the nodes of the standing wave. The phases of oscillations at the internal points of each cell are the same. Oscillations in neighboring cells are made towards each other, that is, in antiphase. Within one cell, the amplitude of oscillations varies in space and reaches its maximum value in some place. The points at which this is observed are called the antinodes of the standing wave. Finally, a characteristic property of standing waves is the discreteness of their frequency spectrum. In a standing wave, oscillations can occur only with strictly defined frequencies, and the transition from one of them to another occurs in a jump.

Consider a simple example of a standing wave. Suppose that a string of limited length is stretched along the axis ; its ends are rigidly fixed, and the left end is at the origin of coordinates. Then the coordinate of the right end will be . Let's excite a wave in a string

spreading along from left to right. The wave will be reflected from the right end of the string. Let's assume that this happens without energy loss. In this case, the reflected wave will have the same amplitude and the same frequency as the incident wave. Therefore, the reflected wave should have the form:

Its phase contains a constant that determines the phase change upon reflection. Since reflection occurs at both ends of the string and without loss of energy, waves of the same frequency will simultaneously propagate in the string. Therefore, when adding, interference should occur. Let's find the resulting wave.

This is the standing wave equation. It follows from it that at each point of the string vibrations occur with a frequency. In this case, the amplitude of oscillations at a point is equal to

Since the ends of the string are fixed, there are no vibrations there. It follows from the condition that . So we end up with:

It is now clear that at points where , there are no oscillations at all. These points are the nodes of the standing wave. In the same place, where , the oscillation amplitude is maximum, it is equal to twice the value of the amplitude of the added oscillations. These points are the antinodes of the standing wave. The appearance of antinodes and knots is precisely the interference: in some places the oscillations are amplified, while in others they disappear. The distance between a neighboring node and an antinode is found from the obvious condition: . Because , then . Therefore, the distance between adjacent nodes is .

It can be seen from the standing wave equation that the factor when passing through zero, it changes sign. In accordance with this, the phase of oscillations on different sides of the node differs by . This means that the points lying on opposite sides of the node oscillate in antiphase. All points enclosed between two neighboring nodes oscillate in the same phase.

Thus, when adding the incident and reflected waves, it is indeed possible to obtain the pattern of wave motion that was characterized earlier. In this case, the cells that were discussed in the one-dimensional case are segments enclosed between neighboring nodes and having length .

Finally, let us make sure that the wave we have considered can exist only at strictly defined oscillation frequencies. Let us use the fact that there are no vibrations at the right end of the string, that is, . Hence it turns out that . This equality is possible if , where is an arbitrary positive integer.

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, we assume that the equations of these waves have the form:

This means that at the origin both waves cause oscillations 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

This equation shows that as a result of the interference of direct 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

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

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

The wave process in a medium described by formula (5.16) is called a standing wave. Graphically, a 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 displacement wave". At some point in time, when all points of the medium have maximum displacements, the direction of which, depending on the value of the x coordinate, is determined by the sign. These displacements are shown in Fig. 1.61 with solid arrows. After a quarter of the 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 dashed arrows. The points are the antinodes of the standing displacement wave; points nodes of this wave.

The characteristic features of a standing wave, in contrast to a conventional propagating or traveling wave, are as follows (meaning plane waves in the absence of attenuation):

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

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

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

When describing oscillatory processes in elastic systems, the oscillating value y can be taken not only as the displacement or velocity of the particles of the system, but also as the value of the relative deformation or the value of the stress in compression, tension, or shear, etc. At the same time, in a standing wave, in places where antinodes of particle velocities are formed, deformation nodes are located, and vice versa, velocity nodes coincide with deformation antinodes. The transformation of energy from kinetic to potential and vice versa occurs within the section of the system from the antinode to the neighboring node. We can assume 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 deformed sections of the medium occurs twice in one period.

Above, considering the interference of direct 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 vibrations propagate has limited dimensions, for example, vibrations 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, interference of waves caused by an external source and reflected from the boundaries continuously occurs.

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

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

direction. Let us find the result of interference of direct and reflected waves at a certain point of the rod having the coordinate x. For simplicity of reasoning, we assume that there is no absorption of vibrational 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 oscillating particles at a point is equal to y, at another point on the rod, the displacement caused by a direct wave will, according to the wave formula, be equal to

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 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 the 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 oscillation phase; in some cases such a phase change (called phase loss) occurs and must be taken into account.

The addition 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 oscillation amplitude at the point; it depends on the x coordinate, i.e., 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 turns to zero or one occurs at argument values that 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. Such reflections

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

Let us assume that an external source of vibrations caused waves in the rod for some time, after which the flow of vibration energy from the outside stopped. During this time, reflections occurred in the rod, where is the time during which the wave passed from one end of the rod to the other. Consequently, in the rod there will simultaneously exist waves traveling in the forward direction and waves traveling 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 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, i.e., it lags behind in phase by equaling this lag where is an integer, we get

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

Expressions (5.19) and (5.20) determine the points at which the cosine has the values or 1:

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

antinode coordinates will be obtained with even values

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

In ch. it was shown that in the absence of periodic external influences, the nature of the coding motions in the system and, above all, the main quantity - the oscillation frequency - are determined by the dimensions and physical properties of the system. Each oscillatory system has its own, inherent oscillatory motion; this fluctuation can be observed if the system is taken out of equilibrium and then external influences are eliminated.

In ch. 4 hours I considered predominantly oscillatory systems with lumped parameters, in which some bodies (point) possessed inertial mass, and other bodies (springs) possessed elastic properties. 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 discussed above, strings, 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 the distribution of nodes and antinodes, as well as the frequency of oscillations - is determined only by the dimensions and properties of the system. Standing waves can also exist in the absence of an external (periodic) action on the system; this action is necessary only to cause or maintain standing waves in the system or to change the amplitudes of oscillations. In particular, if an external action on a system with distributed parameters occurs at a frequency equal to the frequency of its natural oscillations, i.e., the frequency of a standing wave, then the resonance phenomenon takes place, which was considered in Chap. 5. for different frequencies is the same.

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; the rest) are overtones or harmonics.

Each system is characterized not only by the presence of such a spectrum of oscillations, but also by a certain distribution of energy between oscillations of different frequencies. For musical instruments, this distribution gives the sound a peculiar feature, the so-called sound timbre, 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, oscillating strings), or there are one or more points of fastening along the rod. movements are forced displacement nodes.For example,

if it is necessary to obtain standing waves in the rod at one, two, three fixing points, etc., then these points cannot be chosen arbitrarily, but must be located along the rod so that they are at the nodes of the formed 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; displacement antinodes are always formed at the free ends, and displacement nodes at the fixed ends. For oscillating air columns in pipes, displacement nodes (and velocities) are obtained at reflecting solid walls; antinodes of displacements and velocities are formed at the open ends of the tubes.

6.1 Standing waves in an elastic medium

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

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

When adding coherent waves, the phenomenon arises interference, which consists in the fact that at some points in space the waves strengthen each other, and at other points they weaken. An important case of interference is observed when two opposite 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 together, give a standing wave.

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

where - the phase of oscillations of the points of the medium during the passage of the first wave;

- the phase of oscillations of the points of 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 oscillation of the particles of the medium resulting from the addition of the considered waves will be as follows:

We transform the sum of the cosines of the angles according to the rule (4.4) and get:

Rearranging the factors, we get:

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

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

Equation (6.6) is called standing wave equation. It can be seen from it that the frequency of the standing wave is equal to the frequency of the traveling wave, and the amplitude, in contrast to the 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, the points of the medium oscillate with a frequency coinciding with the frequency of the traveling wave, and with an 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 visualize the location of the minima and maxima of the amplitude, we replace, according to (5.29), the wave number by its value:

Then expression (6.7) for the amplitude takes the form

(6.10)

From this it becomes clear that the displacement amplitude is maximum at , i.e. at points whose coordinate satisfies 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 the oscillations of the medium is maximum are called wave antinodes.

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

, (6.13)

where

From (6.13) it can be seen that the coordinates of the nodes have the values:

, (6.14)

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

Find the distance between adjacent antinodes and nodes. From (6.12) we obtain the distance between the antinodes:

(6.15)

The distance between the nodes is obtained from (6.14):

(6.16)

From the relations (6.15) and (6.16) obtained, it can be seen 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 the wavelength, we can write an expression for the length of the standing wave: it is equal to half the length of the traveling wave:

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

, (6.18)

, (6.19)

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

The nodes conditionally divide the medium into autonomous regions in which harmonic oscillations occur independently. There is no transfer of motion between the regions, and, therefore, there is no energy flow between the regions. That is, there is no transmission of perturbation along the axis. Therefore, the wave is called standing.

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 modulus and opposite in direction, and when added they give zero. Therefore, a standing wave does not transfer energy.

6.2 Examples of standing waves

6.2.1 Standing wave in a string

Consider a string of length L, fixed at both ends (Fig. 6.4).

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

Let us write down the boundary conditions for the considered string. 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 vibrations based on the written boundary conditions. We write equation (6.20) for the left end of the string, taking into account (6.21):

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

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

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

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

. (6.25)

Given that

, (6.26)

from (6.25) we get:

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

In the second case, the equality must hold:

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

We discard the value, because in this case , which would mean either zero string length ( L=0) or wave-new number k=0. Considering 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 would have to be infinite, and this would mean the absence of oscillations.

It can be seen from (6.28) that the wave number during vibrations of a string fixed at both ends can take only certain discrete values:

Taking into account (6.9), we write (6.30) as:

whence we derive the expression for the possible wavelengths in the string:

In other words, over the length of the string L must be an integer n half wave:

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

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

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

, (6.36)

Frequencies are called natural frequencies strings. frequency (when n = 1):

(6.37)

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

corresponds to the first harmonic, and the frequency :

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

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

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

Figure 6.5 shows a view of several vibration modes 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 oscillation frequency (or wavelengths). An elastic rod with one or both clamped ends behaves in the same way, as do fluctuations in the air column in pipes, which will be discussed in subsequent sections.

6.2.2 Influence of initial conditions on motion

continuous string. Fourier analysis

Vibrations of a string with clamped ends, in addition to a discrete spectrum of vibration frequencies, have one more important property: the specific form of vibrations of a string depends on the method of excitation of vibrations, i.e. from initial conditions. Let's consider in more detail.

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 is composed of all possible modes (for a string - an infinite number), then the general solution of the wave equation (5.61) is composed of an infinite number of particular solutions:

, (6.43)

where i is the oscillation mode number. Expression (6.43) is written taking into account 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;

is the wave number of the 1st mode;

Let us find the value of the initial phase for each oscillation mode. For this, at the time 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)

On fig. 6.6 shows an example of the shape of a string described by my function f 0 (x).

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

and by substituting into it t=0, we obtain an expression for the speed of the points of 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 (). With this in mind, expression (6.43), which describes the motion of the string, takes the form:

, (6.50)