Elastic waves. Standing waves

If there are several waves in the medium simultaneously, the fluctuations of the medium are the geometric amount of oscillations, which would perform particles when the propagation of each of the waves is separate. Consequently, the waves simply superimpore one to another, not peppering each other. This statement is called the principle of superposition (imposition) of waves.

In the case when the oscillations caused by individual waves in each of the points of the medium, have a constant phase difference, the waves are called coherent. (More stringent determination of coherence will be given in § 120.) When the coherent waves are addition, an interference appears, which consists in the fact that fluctuations at one points are enhanced, and at other points weaken each other.

A very important case of interference is observed when two oncoming plane waves with the same amplitude are observed. The resulting oscillatory process is called a standing wave. Practically standing waves occur when the waves reflect the obstacles. The wave falling to the barrier and the reflected wave reflected by her towards each other, give a standing wave.

We will write equations of two flat waves propagating along the x axis in opposite directions:

Folding together these equations and converting the result by the formula for the amount of cosine, we get

Equation (99.1) is the equation of a standing wave. To simplify it, choose the beginning of the reference so that the difference becomes equal to zero, and the beginning of the reference - so that it turns out to be zero the amount in addition, we will replace the wave number k its value

Then equation (99.1) will take a view

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

the amplitude of oscillations reaches the maximum value. These points are called the beaches of a standing wave. From (99.3), the coordinate values \u200b\u200bare obtained:

It should be borne in mind that a piggy is not one single point, and the plane whose points have the values \u200b\u200bof the coordinate X formula (99.4).

At points whose coordinates satisfy the condition

the amplitude of oscillations appeals to zero. These points are called the standing wave nodes. The points of the medium in nodes are not performed. Nodes coordinates matter

The node, like a piggyback, is not one point, and the plane, the points of which have the values \u200b\u200bof the coordinate X, defined by formula (99.5).

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

Turn again to equation (99.2). The multiplier when moving through the zero value changes the sign. In accordance with this, the oscillation phase on different sides of the node differs on this means that the points lying along different sides of the node fluctuate in antiphase. All points concluded between two adjacent nodes fluctuate the simphang (i.e. in the same phase). In fig. 99.1 Dan a number of "instant photos" of deviations of points from the position of equilibrium.

The first "photo" corresponds to the moment when deviations achieve the greatest absolute value. Subsequent "photos" were made at intervals in a quarter of a period. Arrows show particle speeds.

Differentiating equation (99.2) once per T, and another time in x, we find expressions for the velocity of particles and for deformation of the medium:

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

In fig. 99.2 The "instant photos" of offset, speed and deformation for moments of time 0 and from the graphs it can be seen that the nodes and the velocity beams coincide with the nodes and the bias; The nodes and the vagidity of the deformation coincide according to the beams and bias nodes. While reaching the maximum values, turns to zero, and vice versa.

Accordingly, twice for the period there is a transformation of the energy of a standing wave that completely in the potential, focused mainly near the wave nodes (where the deformation beads are located), then completely in the kinetic, focused on the waves, which are located). As a result, the energy transition occurs from each node to neighboring the beams and back. The average stream of energy in any section of the wave is zero.

6.1 Standing waves in an elastic environment

According to the principle of superposition, with a propagation in an elastic medium at the same time, several waves of the waters of their imposition, and the waves do not outraget each other: the fluctuations of the particles of the medium are the vector sum of the oscillations that the particles would perform the particles in each of the waves each .

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

In addition to coherent waves, a phenomenon occurs interferenceThe fact that in some points of the wave space enhance each other, and at other points - weaken. An important case of interference is associated with the imposition of two oncoming plane waves with the same frequency and amplitude. Arising from this oscillations call standing wave. More often all-th standing waves occur when reflected by the running will from the barrier. In this case, the falling wave and the wave reflected towards her, when adding, gives a standing wave.

We obtain the equation of a standing wave. Take two flat harmonic waves that apply to each other along the axis X. and having the same frequency and amplitude:

where - phase of oscillations of the points of the medium in the pro-walking of the first wave;

- phase of oscillations of the points of the medium during the pro-walking of the second wave.

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

Consequently, both waves will be coherent.

The fluctuation of particles of the medium appeared as a result of the addition of the waves under consideration will be as follows:

We transform the amount of cosine angles according to rule (4.4) and get:

Rearring multipliers, we get:

To simplify the expression, choose the beginning of the reference so that the phase difference and the beginning of the countdown of time so that the phase amount is zero: .

Then the equation for the wave amount will take the form:

Equation (6.6) is called equation of standing ox. It can be seen from it that the frequency of the standing wave is equal to the frequency of the running wave, and the amplitude, in contrast to the running wave, depends on the distance from the beginning of the reference:

. (6.7)

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

. (6.8)

Thus, the points of the medium fluctuate the frequency that coincides with the frequency of the running wave, and the amplitude a.depending on the position of the point on the axis X.. Accordingly, the amplitude varies according to the law of cosine and has its own maxima and minima (Fig. 6.1).

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

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

(6.10)

From here it becomes clear that the amplitude of the displacement of Mac-Symalne at . At the points, the coordinate of which satisfies the condition:

, (6.11)

where

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

; (6.12)

Points where the amplitude of the fluctuations of the medium is maximum called poams of waves.

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

, (6.13)

where

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

, (6.14)

In fig. 6.2 shows an exemplary view of a standing wave, the location of the nodes and beatships. It can be seen that the co-gray nodes and bouquencies of the displacement will take apart from each other for the same distance.

We will find the distance between adjacent beams and Uz-la. From (6.12) we get the distance between the puffs:

(6.15)

Distance between nodes we get from (6.14):

(6.16)

Of the obtained relations (6.15) and (6.16), it can be seen that the distance between adjacent nodes, as well as between neighborhoods, is constantly equal; Nodes and beams shifted relative to each other on (Fig. 6.3).

From the determination of the wavelength, it is possible to write the expression for the length of the standing wave: it is equal to half the long wavelength:

We write, taking into account (6.17), expressions for the coordinates of ultrasound and beatities:

, (6.18)

, (6.19)

The multiplier determining the amplitude of the amplitude of the wave changes its sign when moving through a zero value, as a result of which the oscillation phase on different hundred rubles differs on the node. Consequently, all points lying along different sides of the node fluctuate in pro-tivofase. All points between adjacent UZ-LAMI fluctuate the simplicity.

Nodes conventionally divide the environment on autonomous regionsin which harmonic oscillations are committed independent. There is no transmission of the movement between the areas, and therefore the flow of energy between the regions is not. That is, there is no release of perturbation along the axis. Therefore, the wave is called standing.

So, the standing wave is formed from two opposite-directional traveling waves of equal frequencies and AMP-litud. The units of each of these waves are equal to the Mo-Duoul and are opposite to the direction, and with a symbol, they give zero. Consequently, standing wave of energy does not tolerate.

6.2 Examples of standing waves

6.2.1 Standing wave in the string

Consider a string length L.enshrined with both con-cows (Fig. 6.4).

Place along the axis string X. So that the left end of the string has a coordinate x \u003d 0., and right - x \u003d L.. In the string there are oscillations described by equation:

We write the boundary conditions for the stream under consideration. Since its ends are fixed, then at points with coor-dinatam x \u003d 0. and x \u003d L. oscillations No:

(6.22)

We find the string oscillation equation based on the records of the boundary conditions. We write equation (6.20) for the left end of the string with regard to (6.21):

The ratio (6.23) is performed for any time t. In two cases:

1. . This is possible in the event that there are no kolas in the string (). This case of interest does not represent, and we will not consider it.

2.. Here is the phase. This case will allow us to get the string oscillation equation.

We substitute the obtained phase value in the boundary condition (6.22) for the right end of the string:

. (6.25)

Considering that

, (6.26)

from (6.25) we get:

Two cases appear again in which the relation (6.27) is satisfied. The case when the oscillations in the string are-sufficient (), we will not consider.

In the second case, equality should be performed:

and this is possible only when the argument of the sinus is Kathen's number:

We discard the value, because At the same time, this would mean or zero string length ( L \u003d 0.) or a new number k \u003d 0.. Considering the link (6.9) between the wave number and the wavelength, it can be seen that in order for the will-new number to be zero, the wavelength should be infinite, and this would mean the absence of oscillations.

From (6.28) it can be seen that the wave number when the strings enshrined at both ends can take only certain discrete values:

Considering (6.9), we write (6.30) in the form:

where we gain expression for possible wavelengths in the string:

In other words, on the length of the string L. should fit the whole number n. Semi-fell:

The corresponding frequencies of oscillations can be determined from (5.7):

Here - the phase wave velocity, depending, consonant (5.102), from the linear density of the string and strength of the string:

Substituting (6.34) in (6.33), we obtain the expression, describing the possible frequencies of the oscillations of the string:

, (6.36)

Frequencies are called own frequencies stream. Frequency (as n. = 1):

(6.37)

call main frequency (or main tone) Strings. Frequencies defined by n\u003e 1. called obrafton or harmonies. The harmonic number is equal n-1. For example, frequency:

corresponds to the first harmonic, and the frequency:

communicates the second harmonic, etc. Since the string can be represented in the form of a discrete system with the faceless number of degrees of freedom, each harmonic is modoy Wasters string. In the general case, the strings of the strings are a mod superposition.

Each harmonic corresponds to its wavelength. For main tone (with n \u003d1) Wavelength:

respectively for the first and second harmonics (when n \u003d2 I. n \u003d3) wavelengths will be:

Fig. 6.5 shows the type of several modes of oscillations carried out by the string.

Thus, the string with fixed ends is in lisks within the framework of classical physics, an exceptional case is a discrete spectrum of frequency of oscillations (or wavelengths). In the same way, the elastic erases with one or both clamped ends and the oscillations of the air post in the pipes, which will be considered in the subsequent sections.

6.2.2 Effect of initial movement conditions

continuous string. Fourier analysis

The fluctuations of the string with groped ends, in addition to the disc-spectrum, the oscillation frequencies have another important property: a specific shape of the string oscillation depends on the method of excitation of oscillations, i.e. from obvious conditions. Consider more details.

Equation (6.20), which describes one-fashionable Waway in the string, is a particular solution of a differential wave equation (5.61). Since the fluctuate of the streams is consisted of all possible mod (for a string - a devontal amount), then common decision The wave equation (5.61) consists of an infinite number of private solutions:

, (6.43)

where i. - Fashion number of oscillations. The expression (6.43) is not allowed, but taking into account the fact that the ends of the strings are fixed:

as well as taking into account the connection of frequency i.Fashion and its wave number:

(6.46)

Here - Wave number i.fashion;

- wave number of the 1st fashion;

We find the magnitude of the initial phase for each oscillation fashion. For this at the time of time t \u003d 0. give the string form described by the function f. 0 (x), the expression for which we get from (6.43):

. (6.47)

In fig. 6.6 shows an example of a string form, describing my function f. 0 (x).

At the time of time t \u003d 0. The string is still resting, i.e. The speed of all its points is zero. From (6.43) We will find an expression for the speed of the strings:

and, substituting into it t \u003d 0., I get an expression for the speedpoint of the string at the initial moment of time:

. (6.49)

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

, (6.50)

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

. (6.51)

Standing wave in the string is described by the function, perio-wild on the interval, where is equal to two string lengths (Fig. 6.7):

This is seen from the fact that frequency at the interval means:

Hence,

what leads us to expression (6.52).

It is known from mathematical analysis that any ne-rhodic function can be decomposed with a high accuracy in the Fourier series:

, (6.57)

where, - Fourier coefficients.

Consider the result of the interference of two sinusoidal flat 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 beginning of the coordinates both waves cause oscillations in the same phase. At point A with the coordinate x the total value of the oscillating value, according to the principle of superposition (see § 19), equal

This equation shows that as a result of the interference of the direct and reverse 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 coordinate x. At the points of the medium in which the oscillations are missing at all: these points are called oscillation nodes.

At points where the amplitude of oscillations is of the greatest value equal to these points are called oscillations. It is easy to show that the distance between adjacent nodes or adjacent beams is equal to the distance between the beaciousness and the nearest node is equal to the change of x on the cosine in formula (5.16) changes the sign to the opposite (its argument varies on therefore if within one half-wave - from one node to another - The particles of the medium rejected in one direction, then within the next half-wave particle of the medium will be rejected in the opposite direction.

The wave process in the medium described by formula (5.16) is called a standing wave. A graphically standing wave can be depicted as it is shown in Fig. 1.61. Suppose that there is a displacement of the points of the medium from the state of equilibrium; Then formula (5.16) describes the "standing wave of offset". At some point 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 of these offsets are shown in Fig. 1.61 solid arrows. After a quarter of the period, when the offset of all points of the medium is zero; Medium particles pass through a line with different speeds. After another quarter of the period, when the medium particles again will have maximum displacements, but the opposite direction; These offsets are shown on

fig. 1.61 dotted arrows. Point of the essence of the beacon standing wave of offset; Points of the nodes of this wave.

The characteristic features of the standing wave, in contrast to the usual propagating, or running, the waves are the following (meaning flat waves in the absence of attenuation):

1) in the standing wave the amplitude of oscillations are different in various places of the system; The system has knots and beacons of oscillations. In the "Running" wave, these amplitudes are the same everywhere;

2) within the site of the system from one node to neighboring all points of the medium fluctuate in the same phase; When moving to the adjacent area, the oscillation phases change to the inverse. In the running wave of the oscillation phase, according to formula (5.2), depend on the coordinates of the points;

3) There is no one-sided transfer of energy in a standing wave, as it takes place in the running wave.

When describing the vibrational processes in elastic systems, it is possible to take not only the displacement or velocity of the system particles, but also the value of the relative deformation or the magnitude of the stress on compression, stretching or shift, etc. In this case, in the standing wave, where Puffy velocities of particles are formed, the deformations nodes are located and, on the contrary, the speed nodes coincide with the beams of deformations. The conversion of energy from the kinetic form to potential and back occurs within the system of the system from the beafness to the adjacent node. It can be assumed that each such area does not exchange energy with neighboring sites. Note that the conversion of the kinetic energy of moving particles into the potential energy of the deformed areas of the medium in one period occurs twice.

Above, considering the interference of direct and reverse waves (see expressions (5.16)), we were not interested in the origin of these waves. Suppose that the medium in which the fluctuations spread has limited sizes, such as oscillations are caused in some solid body - in a rod or string, in a pole of a liquid or gas, etc. The wave propagating in such an environment (tele) It is reflected from the borders, therefore, within the amount of this body, the interference of waves caused by an external source and reflected from the borders continuously occurs.

Consider the simplest example; Suppose, an oscillatory movement with a frequency is excited by an external sinusoidal source at the point (Fig. 1.62). The beginning of the countdown of time will choose so that at this point the displacement was expressed by the formula

where the amplitude of oscillations at the point caused in the rod wave will reflect from the second end of the rod 0% and will go in the opposite

direction. Find the result of the interference of the straight and reflected waves at some point of the rod having a coordinate x. For simplicity of reasoning, assume that there is no absorption of oscillations in the rod and therefore the amplitudes of the straight and reflected waves are equal.

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

Through the same point and the reflected wave also passes. To find the offset caused at the point A reflected wave (at the same time it is necessary to calculate the time during which the wave will pass from to and back to the point as the displacement caused by the point of reflected wave will be equal to

It is assumed that in the reflecting end of the rod in the process of reflection, there is no jump-like changing phase of the oscillation; In some cases, such a change in the phase (called the phase loss) takes place and must be taken into account.

The complexity of oscillations caused at various points of the rod straight and reflected by the waves gives a standing wave; really,

where some permanent phase, independent of the coordinate x, and the value

it is an amplitude of oscillations at the point it depends on the coordinate x, i.e. is different in various places of the rod.

We will find the coordinates of those rod points in which the nodes and the beacons of the standing wave are formed. The circulation of cosine into zero or unit occurs at the values \u200b\u200bof the argument, multiple

where an integer. With the odd value of this number, the cosine appeals to zero and formula (5.19) gives the coordinates of the standing wave nodes; With even we get the coordinates of the beatships.

Above, only two waves were added: a direct, from and reflected, propagating from however, should be taken into account that the reflected wave on the border of the rod will re-reflect and go in the direction of a straight wave. Such reflections

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

Suppose that the external source of oscillations caused the wave rod for some time after which the flow of oscillation 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 another. Consequently, the rod will simultaneously exist waves going in direct and waves going in the opposite directions.

Suppose that as a result of the interference of one pair of waves (direct and reflected), the displacement in exactly and turned out to be equal to. Find a condition in which all shifts y, caused by each pair of waves, have the same directions at a point and the rod and therefore develop. For this phase of oscillations caused by each pair of waves at the point should differ on the oscillation phases caused by the next pair of waves. But each wave again returns to the point A with the same direction of distribution only after time, i.e. lags behind the phase on with equating this lag where the integer, we get

i.e. along the length of the rod must fit the integer number of half feet. Note that this condition of the phase of all waves coming from in the forward direction differ from each other on where an integer; In the same way, the phases of all waves coming from in the opposite direction differ from each other on it therefore, if one pair of waves (direct and reverse) gives along the rod distribution of displacements, determined by formula (5.17), then during interference of pairs of such waves, the distribution of displacements is not will change; Only oscillation amplitudes will increase. If the maximum amplitude of oscillations during interference of two waves, according to formula (5.18), is equal to the interference of many waves it will be more. Denote it through then the distribution of the amplitude of oscillations along the rod instead of expression (5.18) is determined by the formula

From expressions (5.19) and (5.20), the points in which the cosine is determined or 1:

where the integer number of coordinates of the standing wave nodes is obtained from this formula with odd values, then depending on the length of the rod, i.e. the values

poofing coordinates will result from even values

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

In ch. It was shown that in the absence of periodic external influences, the nature of the codebage movements in the system and above all the main value - the frequency of oscillations - are determined by the sizes and physical properties Systems. Each oscillatory system has its own, its inherent vibrational movement; This oscillation can be observed if you derive a system from an equilibrium state and then eliminate external influences.

In ch. 4 hours I considered predominantly oscillatory systems with concentrated parameters, in which the inert mass had some bodies (point), and elastic properties - other bodies (springs). In contrast, the oscillatory systems in which the mass and elasticity are inherent in each elementary volume, are called systems with distributed parameters. These include the above rods, strings, as well as fluid or gas pillars (in wind musical instruments), etc. For such systems, there are standing waves; The main characteristic of these waves is the wavelength or distribution of nodes and beatships, as well as the oscillation frequency - is determined only by the size and properties of the system. Standing waves may exist in the absence of an external (periodic) impact on the system; This impact is necessary only to cause or maintain standing waves in the system or change the amplitudes of oscillations. In particular, if the external impact on the system with distributed parameters occurs with the frequency, equal frequency Her own oscillations, i.e. the frequency of the standing wave, then there is a phenomenon of resonance, considered in ch. 5. For different frequencies the same.

Thus, in systems with distributed parameters, its own oscillations are standing waves - characterized by a whole spectrum of frequencies, multiple among themselves. The smallest of these frequencies corresponding to the greatest wavelength is called the main frequency; The rest) - overtones or harmonics.

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

The above calculations refer to the free oscillating "rod length. However, we usually have rods attached at one or both ends (for example, the oscillating strings), or along the rod there are one or more consolidation points. The fastener location, where the particles of the system cannot perform oscillatory Movements are internally displacement nodes. For example,

if it is necessary to obtain standing waves in the rod at one, two, three points of consolidation, etc., these points cannot be selected arbitrarily, but should be located along the rod so that they are in 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 displacement of the rod points during oscillations; At the free ends, the beafness of the displacement is always formed, on the fixed - bias nodes. For the oscillating air columns in the pipes, the displacement nodes (and speed) are obtained in reflective solid walls; At the open ends of the tubes, the beams of displacements and speeds are formed.

A very important case of interference is observed when laying flat waves with the same amplitude. The resulting oscillatory process is called standing wave.

Practically standing waves occur when the waves reflect the obstacles. The wave falling to the barrier and the reflected wave reflected by her towards each other, give a standing wave.

Consider the result of the interference of two sinusoidal flat waves of the same amplitude propagating in opposite directions.

For simplicity of reasoning, we assume that both waves cause oscillations at the beginning of the coordinates in the same phase.

The equations of these oscillations are:

Folding both equations and converting the result, by the formula for the sum of the sinuses we will get:

- the equation of standing wave.

Compare this equation with the equation of harmonic oscillations, we see that the amplitude of the resulting oscillations is:

Since, but, then.

At the ambient points, where, there are no oscillations, i.e. . These points are called standing waves nodes.

At points, where, the amplitude of oscillations is of the greatest value equal to. These points are called puzzles standing wave. Beam coordinates are from the condition, because then.

From here:

Similarly, the coordinates of the nodes are from the condition:

From:

From the formulas of the coordinates of nodes and beatities, it follows that the distance between adjacent beams, as well as the distances between adjacent nodes, is equal. Puffy and nodes shifted relative to each other by a quarter of the wavelength.

Compare the character of oscillations in a standing and running wave. In the running wave, each point performs oscillations whose amplitude is not different from the amplitude of other points. But the oscillations of different points occur with various phases.

In a standing wave, all particles of the medium in between two adjacent nodes fluctuate in the same phase, but with different amplitudes. When switching through the node, the oscillation phase vibrations varies on, because Changes the sign.

A graphically standing wave can be depicted as follows:

At the time when, all the points of the medium have maximum displacements, on-board of which is determined by the sign. These offsets are shown in the drawing with solid arrows.

After a quarter of the period, when, offset of all points is zero. Particles pass through a line with different speeds.

After another quarter of the period, when, particles will again have maximum offsets, but the opposite direction (dotted arrows).

When describing the oscillatory processes in elastic systems, not only the displacement, but also the speed of particles, as well as the value of the relative deformation of the medium, can be taken.

To find the law of changing the speed of the standing wave, in relation to the equation of a stagnation of the standing wave and to find the law of changing the deformation infeentially by the equation of a standing wave.

Analyzing these equations, we see that the nodes and the beams of speed coincide with the nodes and the bias of the displacement; The nodes and beacons of deformation coincide according to the beams and nodes of speed and offset.

Wasters string

In the strained string attached with both ends, standing waves are installed during the excitation of transverse oscillations, and nodes must be placed in the plugs. Therefore, only such oscillations are excited in the string, half of the length of which is placed on the length of the string an integer number.

Hence the condition:

where is the length of the string.

Or otherwise. These wavelengths correspond to the frequency, where the phase velvene speed. It is determined by the strength of the string tension and its mass.

When - the main frequency.

When - own frequencies of string oscillations or ochtons.

Doppler effect

Consider the simplest cases when the source of the waves and the observer move relative to the medium along one straight line:

1. The sound source moves relative to the medium at speeds, the sound receiver rests.

In this case, for the period of oscillations, the sound wave will depart from the sour-nick to the distance, and the source itself will shift to the distance equal.

If the source is removed from the receiver, i.e. Move in the opposite direction of the wave propagation, then the wavelength.

If the sound source is closer to the receiver, i.e. Move in the direction of distribution of the wave, then.

The sound frequency is perceived by the receiver is:

Substitute instead of their meaning for both cases:

Taking into account the fact that, where - the frequency of the oscillations of the source, the equality will take the form:

We divide the numerator and denominator of this fraction on, then:

2. The sound source is fixed, and the receiver moves relative to the medium at speeds.

In this case, the wavelength in the medium does not change and is still equal. At the same time, two consecutive amplitudes, differing in time for one period of oscillations, reaching the moving receiver, will differ in time at the moments of the wave of the wave with the receiver for a period of time, the magnitude of which is greater or less depending on whether the receiver is deleted or approaches the source. Sound. During the time the sound applies to the distance, and the receiver will shift over the distance. The sum of these values \u200b\u200band gives us the wavelength:

The period of oscillations perceived by the receiver is associated with the frequency of these oscillations by the ratio:

Substituting instead of its expression from equality (1), we get:

Because , where - the frequency of source oscillations, and, then:

3. The source and receiver sounds are moving relative to the medium. Connecting the results obtained in the two previous cases, we obtain:

Sound waves

If a elastic wavesspreading in air have a frequency ranging from 20 to 20,000 Hz, then reaching the human ear, they cause a feeling of sound. Therefore, the waves lying in this range of frequencies are called sound. Elastic waves with a frequency of less than 20 Hz are called infrasound . Waves with frequency over 20,000 Hz are called ultrasound. Ultrasound and infrasound human ear does not hear.

Sound sensations are characterized by sound height, timbre and loudness. Sound height is determined by the frequency of oscillations. However, the sound source emits not one, but a whole spectrum of frequencies. A set of frequencies of oscillations present in this sound is called it. acoustic spectrum. The energy of oscillations is distributed between all frequencies of the acoustic spectrum. The height of the sound is determined by one - the main frequency if this frequency accounts for a much larger amount of energy than the share of other frequencies.

If the spectrum consists of a plurality of frequencies in the frequency range from before, then such a spectrum is called solid (example - noise).

If the spectrum consists of a set of oscillations of discrete frequencies, then such a spectrum is called linely (Example - musical sounds).

The acoustic spectrum of sound depending on its nature and the distribution of energy between frequencies determines the originality of the sound sensation, called the voice temperature. Various musical instruments have a different acoustic spectrum, i.e. Different with a tempor of sound.

The intensity of the sound is characterized by one-personal values: oscillations of the medium particles, their speeds, pressure forces, voltages in them, etc.

It characterizes the amplitude of oscillations of each of these quantities. However, since these values \u200b\u200bare interrelated, it is advisable to introduce a single energy characteristic. This characteristic for waves of any type was proposed in 1877. ON THE. Umovy

I snatch mentally from the front of the running wave the platform. During this platform, it moves to the distance, where - the speed of the wave.

Denote by the energy of the amount of the amount of the oscillating medium. Then the energy of all volume will be equal.

This energy was transferred during the wave propagating through the platform.

Sharing this expression on and, we obtain the energy carried by the wave through the area of \u200b\u200bthe square per unit of time. This value is indicated by the letter and is called vector Melova.

For sound field vector Umova. Wears the name of the sound.

The power of the sound is the physical characteristic of the intensity of the sound. We evaluate it subjectively as volume Sound. The human ear perceives the sounds whose strength exceeds some minimum value different for different frequencies. This value is called threshold hearingness Sound. For the average frequencies of the order of Hz threshold of hearing order.

With very large power The sound of order sound is perceived except the ear of the tangible organs, and in the ears it causes a painful feeling.

The value of the intensity in which this happens is called threshold of pain. The threshold of the pain, as well as the threshold of hearingness depends on the frequency.

A person has a pretty complex apparatus for the perception of sounds. Sound oscillations are collected by ear shell and through the auditory canal affect the eardrum. Its oscillations are transmitted to a small cavity, called snail. Inside the snail is located a large number of Fibers having different lengths and tension and, therefore, various oscillation frequencies. When the sound action, each of the fibers resonates to that tone, the frequency of which coincides with its own fiber frequency. A set of resonant frequencies in a rumor apparatus and determines the area of \u200b\u200bthe sound oscillations perceived by us.

Subjectively estimated by our ear volume increases much slower than the intensity of sound waves. While the intensity increases in geometric progression - the volume increases in arithmetic progression. On this basis, the volume level is defined as a logarithm of the intensity ratio of this sound to the intensity adopted for the original

The volume of the volume is called george. Use and smaller units - decybel(10 times less than Belarus).

where is the sound absorption coefficient.

The magnitude of the absorption coefficient of sound increases in proportion to the square of the sound frequency, so low sounds apply further to high.

In architectural acoustics for large premises significant role Playing reverberation or humidity of the premises. Sounds, experiencing repeated reflections from enclosing surfaces, are perceived by the listener for a certain rather large period of time. This increases the power of a sound reaching us, however, with too long reverb, individual sounds are superimposed on each other and the speech ceases to be perceived by the self-consistent. Therefore, the walls of the halls are covered with special sound-absorbing materials to reduce reverb.

A source of sound oscillations can serve any oscillating body: a bell tongue, akton, a string of violin, an air column in wind instruments, etc. The same bodies can serve as sound receivers when they come into motion under the action of environmental oscillations.

Ultrasound

To get directed, i.e. Close to flat, wave the dimensions of the emitter must be many times the wavelength. Sound waves in the air have a length of up to 15 m, in liquid and solid bodies The wavelength is even more. Therefore, to build a radiator, which would create a directional wave of similar length, is practically no possible.

Ultrasonic oscillations have a frequency of over 20,000 Hz, so the wavelength is very small. With a decrease in wavelength, the role of diffraction during the spread of waves is also reduced. Therefore, ultrasound waves can be obtained in the form of directional beams like light beams.

Two phenomena use two phenomena to excite ultrasonic waves: reverse piezoelectric effectand magnetostriction.

The reverse piezoelectric effect is that the plate of some crystals (ferroed salt, quartz, titanate barium, etc.) under the action electric field Lightly deformed. By placing it between the metal plates, which is supplied to an alternating voltage, you can cause forced oscillations of the plate. These oscillations are transmitted environment And they give rise to an ultrasonic wave.

Magnetostriction is that ferromagnetic substances (iron, nickel, their alloys, etc.) under the action magnetic field deform. Therefore, by placing a ferromagnetic rod into an alternating magnetic field, mechanical oscillations can be excited.

High values \u200b\u200bof acoustic speeds and accelerations, as well as well-developed methods of studying and receiving ultrasound fluctuations, allowed to use them to solve many technical tasks. List some of them.

In 1928, Soviet scientist S.Ya. Sokolov offered to use ultrasound for defectoscopy purposes, i.e. To detect hidden internal defects such as shell, cracks, rylot, slag inclusions, etc. in metal products. If the dimensions of the defect exceed the length of the ultrasound wave, then the ultrasonic pulse is reflected from the defect and returns back. Sending ultrasonic impulses in the product, and registering reflected echo signals, you can not only detect the presence of defects in products, but also judge the size and location of these defects. Currently, this method is widely used in industry.

The directional ultrasound beams were widely used for location purposes, i.e. To detect items in water and determine the distance to them. For the first time the idea of \u200b\u200bultrasonic location was punishable by an outstanding French physicist P. Lanzhen and developed by him during the First World War to detect submarines. Currently, the principles of hydrolections are used to detect icebergs, fish shocks, etc. These methods may also define the depth of the sea under the bottom of the ship (echo sounder).

Ultrasonic waves of large amplitudes are widely used in the technique for mechanical processing of solid materials, cleaning small items (parts of hourly mechanisms, pipelines, etc.) placed in liquid, aduggation, etc.

When creating strong pressure pulsations in the medium, ultrasonic waves determine a number of specific phenomena: grinding (dispersion) of particles suspended in fluid, formation of emulsions, acceleration of diffusion processes, activation of chemical reactions, impact on biological objects, etc.

If there are several waves in the medium simultaneously several waves, the fluctuations of the medium are the geometric amount of oscillations that would perform particles in the propagation of each of the waves separately. Consequently, the waves simply superimpore one to another, not peppering each other. This statement is called the principle of the superposition of the waves. The principle of the superposition claims that the movement caused by the spread of several waves at once is again some wave process. Such a process, for example, is the sound of the orchestra. It arises from the simultaneous excitement of air fluctuations by separate musical instruments. It is wonderful that when the waves are applied, special phenomena may occur. They are called the effects of addition or, as they say, the superposition of the waves. Among these effects, interference and diffraction are the most important.

The interference is the phenomenon of the respontestation of the energy of oscillations in space, as a result of which the oscillations are enhanced in some places, and others are weakened. This phenomenon occurs with the addition of waves with the difference in the time difference phases, so-called coherent waves. The interference of a large number of waves is called diffraction. There is no fundamental difference between interference and diffraction. The nature of these phenomena is the same. We restrict ourselves to the discussion of only one very important interference effect, which is to form standing waves.

Prerequisite The formation of standing waves is the presence of boundaries reflecting the waves falling on them. Standing waves are formed as a result of the addition of falling and reflected waves. Phenomena of this kind are quite common. So, every tone of the sound of any musical instrument is excited by a standing wave. This wave is formed either in the string (string tools), or in the air column (brass tools). Reflective boundaries in these cases are points of fastening the string and the surface of the inner cavities of the wind instruments.

Each standing wave has the following properties. The entire area of \u200b\u200bspace in which the wave is excited can be broken into the cells in such a way that the oscillation cells are completely absent at the boundaries. The points located on these boundaries are called the standing wave nodes. Phases of oscillations in the inner points of each cell are the same. Oscillations in neighboring cells are made towards each other, that is, in antiphase. Within one cell, the oscillation amplitude varies in space and in some place reaches the maximum value. The points in which it is observed is called the beaches of a standing wave. Finally, the characteristic property of standing waves is the discreteness of the spectrum of their frequencies. In a standing wave, oscillations can only be performed with strictly certain frequencies, and the transition from one of them to the other occurs with a jump.

Consider a simple example of a standing wave. Suppose that the strings of limited length is stretched along the axis; Its ends are rigidly fixed, and the left end is at the beginning of the coordinates. Then the coordinate of the right end will be. Excit the wave in the string

,

spreading along left to right. From the right end of the wave string will affect. Suppose it will happen without energy loss. In this case, the reflected wave will have the same amplitude and the same frequency as the incident. Therefore, the reflected wave should be:

Its phase contains a constant, determining the change in the phase when reflected. Since the reflection occurs at both ends of the string and without loss of energy, the waves of the same frequencies will simultaneously spread in the string. Therefore, when adding and should be interference. Find a resulting wave.

This is the equation of a standing wave. It follows from it that at each point of the string there are oscillations with a frequency. At the same time, the amplitude of oscillations at the point is equal

.

Since the ends of the strings are fixed, there are no oscillations. From the condition it follows that. Therefore, we finally get:

.

Now it is clear that at points in which there are no oscillations at all. These points are the nodes of a standing wave. There, where, the amplitude of oscillations is maximal, it is equal to the double value of the amplitude of the folded oscillations. These points are the beaches of a standing wave. In the appearance of beatities and nodes, interference is: in some places, the oscillations are enhanced, and others disappear. The distance between adjacent nodes and the beafness is from the obvious condition :. Since, then. Consequently, the distance between adjacent nodes.

From the equation of the standing wave it is clear that the multiplier When switching through the zero value changes the sign. In accordance with this, the oscillation phases on different sides of the node differs on. This means that points lying along different sides of the node fluctuate in antiphase. All points concluded between two adjacent nodes fluctuate in the same phase.

Thus, when adding incident and reflected waves, it is really possible to obtain a picture of a wave motion that was characterized earlier. At the same time, the cells that were discussed in the one-dimensional case are segments concluded between adjacent nodes and having a length.

Let us finally be convinced that the wave considered by us can exist only with strictly certain frequencies of oscillations. We use the fact that the oscillations on the right end of the string are absent, that is. From here it turns out that. This equality is possible if, where - a whole arbitrary positive number.