For experiments with a string, the device shown in Fig. 98. One of the horses of the string is fixed, and the other is thrown over the block, and one or another load can be suspended from it. Thus, we know the string tension force: it is equal to the weight of the load. The board over which the string is stretched is equipped with a scale. This allows you to quickly determine the length of the entire string or any part of it.

Rice. 98. Instrument for Investigating String Vibrations

Pulling the string in the middle and releasing it, we excite in it the oscillation shown in Fig. 99, a. At the ends of the string, knots are obtained, in the middle - antinodes.

Rice. 99. Free vibrations strings: a) with one antinode; b) with two antinodes; c) with three antinodes

Using this device, by changing the mass of the load pulling the string and the length of the string (by moving the additional clamp from the side of the fixed end), it is easy to experimentally establish what determines the natural frequency of the string. These experiments show that the frequency of string vibration is directly proportional to the square root of the string tension force and inversely proportional to the length of the string, i.e.

As for the coefficient of proportionality, it turns out that it depends only on the density of the material from which the string is made, and on the thickness of the string, namely, it is equal to . Thus, the natural frequency of the string vibrations is expressed by the formula

In string instruments, the tension force is created, of course, but by hanging loads, and by stretching the string when winding one of its ends, neither a rotating rod (peg). By turning the peg, that is, by changing the tension force, the string is also tuned to the required frequency.

Let's proceed now as follows. Let's pull one half of the string up and the other half down so that the middle point of the string does not move. Letting go both drawn points of the string at the same time (distanced from the ends of the string by a quarter of its length), we will see that an oscillation is excited in the string, which, in addition to two nodes at the ends, also has a node in the middle (Fig. 99, b) and, consequently, two antinodes . With such a free vibration, the sound of the string is twice as high (an octave higher, as they say in acoustics) than with the previous vibration with one antinode, i.e., the frequency is now equal to . The string, as it were, was divided into two shorter strings, the tension of which was the same.

You can further excite an oscillation with two nodes dividing the string into three equal parts, i.e., an oscillation with three antinodes (Fig. 99, c). To do this, you need to pull the string at three points, as shown by the arrows in Fig. 99, c. The frequency of this oscillation is . By pulling the string at several points, it is difficult to obtain vibrations with an even greater number of nodes and antinodes, but such vibrations are possible. They can be excited, for example, by drawing a bow along the string in the place where the antinode should be, and lightly holding the nearest nodal points with your fingers. Such free vibrations with four, five antinodes, etc., have frequencies, etc.

So, the string has a whole set of vibrations and, accordingly, a whole set of natural frequencies that are multiples of the lowest frequency. The frequency is called the fundamental, the oscillation with frequency is called the fundamental tone, and the oscillations with frequencies, etc., are called overtones (respectively, the first, second, etc.).

In stringed musical instruments, string vibrations are excited either by plucking or jerking the plate (guitar, mandolin), or by hitting the hammer (piano), or by bowing (violin, cello). In this case, the strings perform not one of their own vibrations, but several at once. One of the reasons why different instruments have different timbres (§ 21) is precisely the fact that the overtones accompanying the main vibration of the string are expressed differently in different instruments. (Other reasons for the difference in timbre are related to the device of the instrument body itself - its shape, dimensions, rigidity, etc.)

The presence of a whole set of natural oscillations and the corresponding set of natural frequencies is characteristic of all elastic bodies. However, in contrast to the case of a vibrating string, the frequencies of the overtones are, in general, not necessarily an integer number of times higher than the fundamental frequency.

On fig. 100 schematically shows how a plate clamped in a vise and a tuning fork vibrate during the fundamental vibration and the two nearest overtones. Of course, knots are always obtained at fixed places, and the largest amplitudes at free ends. The higher the overtone, the greater the number of additional nodes.

Rice. 100. Free oscillations at the frequency of the fundamental tone and the first two overtones: a) a plate clamped in a vise; b) tuning fork

Speaking earlier about one natural frequency of elastic vibrations of heat, we had in mind its fundamental frequency and simply kept silent about the existence of higher natural frequencies. However, when it came to vibrations of a load on a spring or torsional vibrations of a disk on a wire, i.e., elastic vibrations of systems in which almost all the mass is concentrated in one place (load, disk), and deformations and elastic forces are in another (spring, wire), then there was every reason for such a selection of the fundamental frequency. The point is that in such cases, the frequencies of overtones, starting from the first one, are many times higher than the fundamental frequency, and therefore, in experiments with the fundamental vibration, overtones practically do not appear.

Music is the mysterious arithmetic of the soul; she calculates without realizing it.

G. Leibniz

On a November morning in 1717, a baby was found on the steps of the Parisian church of Saint Jean le Rhone. He was brought up and, in honor of the saint of the church, was christened Jean le Ron. The boy early showed a brilliant mind and an avid curiosity, and soon became the pride of all France. It was Jean le Ron D "Alembert (1717-1783) - an outstanding French mathematician, philosopher, writer, member of the Paris, St. Petersburg and other academies.

The range of D'Alembert's interests was unusually wide: mechanics (D'Alembert's principle), hydrodynamics (D'Alembert's paradox), mathematics (D'Alembert's convergence sign), mathematical physics (D'Alembert's formula), philosophy, music theory. Such breadth Demanded and work together with Denis Diderot on the creation of the famous "Encyclopedia of Sciences, Arts and Crafts", and the very spirit of the era of enlightenment, when everyone was drawn to knowledge, including the "enlightened despots" Frederick II and Catherine II. "Alambera to be the tutor of her son, Tsarevich Pavel, while appointing a fabulous reward, but always received a delicate but firm refusal.

Vibrations of a string of length l. Shown are two times t 1

In 1747, d'Alembert published an article "Investigations on the curve formed by a stretched string set into vibration", where for the first time the problem of string vibration was reduced to solving differential equation in private derivatives. And although this topic is beyond the scope of school mathematics, but in knowledge, “keeping oneself within” means ruining one’s curiosity!), We will consider a simple and truly beautiful equation that describes the vibration of a string, the so-called complete equation, from which a new branch of mathematics began - mathematical physics:

(10.1)

Here t is time; x is the string coordinate in the equilibrium position; u = u(x, t) is an unknown function expressing the deviation of the point with coordinate x at time t from the equilibrium position; and 2 - coefficient of proportionality characterizing the elastic properties of the string , T is the tension force of the string, p is the density of a homogeneous string). It is assumed that the string performs small oscillations occurring in one plane. Finally, symbols denote the partial derivative of the second order, which is defined as the derivative of the derivative . Partial derivatives - , like the usual "school" derivative characterizes the rate of change of the function u(x, t) for each of the variables x or t separately, provided that the other variable does not change (functions of one variable y = y(x) have one derivative, and the function of two variables u = u(х,t) has two partial derivatives . To distinguish partial derivatives from ordinary "school" derivatives, they write not a straight letter, but a round one.

The wave equation (10.1) is nothing but a consequence of Newton's second law. The left side of (10.1) expresses the vertical acceleration of the string at the point x, and the right side - the force related to the mass of the string that causes this acceleration, which is the greater, the greater the concavity of the string.

D "Alembert found a general solution of equation (10.1)

which contains two arbitrary functions φ(х,t) and ψ(х,t). Five years later, Daniil Bernoulli (1700-1782), mathematician, mechanic, physiologist and physician, honorary member of the St. Petersburg Academy of Sciences, representative of the glorious Bernoulli family, who by now has given the world more than 100 descendants who have achieved significant results in all spheres of human activity, and first of all in the scientific, received another general solution of equation (10.1)

Comparing the solutions of D "Alembert (10.2) and D. Bernoulli (10.3), we seem to come to an absurdity: the same equation (10.1) has completely different solutions! But there is no absurdity here, this is how differential equations are arranged. They have an infinite number of solutions, which is easy to see from (10.2), where the functions φ(x - at) and ψ(x + at) are arbitrary.Under sufficiently general assumptions about the functions φ and ψ, the right-hand side of (10.2) can be represented by the series ( 10.3).

The choice of one or another particular solution of the differential equation is dictated by the conditions in which the process proceeds (these are the so-called boundary conditions), and the conditions that took place at the beginning of the process (the so-called initial conditions). Only a set of differential equations, initial and boundary conditions determines the solution of a particular physical problem. Via common solution(10.2) D "Alembert solved one of these problems: to find the vibrations of an infinite string (i.e., in the absence of boundary conditions), which at the initial time t = 0 was given some form f (x) and reported some acceleration g (x) Mathematically, the problem was formulated as follows: to find a solution to equation (10.1) that satisfies the initial conditions u(x, 0) = f(x), u(x, 0) = g(x), i.e., solve the system

(10.4)

The solution to problem (10.4) is determined by the d'Alembert formula

Formula (10.5) in the simplest case g(x) = 0, i.e. when the string is gently pulled and released without giving it additional acceleration, takes the form

and physically means that the profile f(x) imparted to the string at t=0 will propagate to the left and to the right with a velocity a. These are the so-called two traveling waves moving in opposite directions with the same speed a.

In fact, there are no infinite strings. The string has a finite length l and, as a rule, is rigidly fixed at the ends. This is how the boundary conditions arise: u(0,t) = 0 - the string is fixed on the left (x = 0); and (l,t) = 0 - the string is fixed on the right (x = l). It is clear that in this case the traveling waves will be reflected from the ends, interact with each other and form a more complex pattern of oscillations.

The problem of the vibration of a finite string was independently solved by D'Alembert and Euler, and half a century later Joseph Fourier invented a new method that made it possible to solve this and many other problems of mathematical physics. The problem of the vibration of a finite string is formulated as follows: find a solution to the wave equation (10.1), satisfying the initial conditions u(x, 0) = f(x), ut (x, 0) = g(x) and the boundary conditions u(0, t) = 0, u(l, t) = 0, i.e., .solve the system

(10.6)

Jean Baptiste Joseph Fourier (1768-1830) was not an armchair scientist. He took off on the crest of the Great French Revolution of 1789, and from the son of a provincial tailor, who was preparing to take monastic vows, he turned into a friend of Emperor Napoleon. In 1798, Fourier participated in the Egyptian campaign of Napoleon, where his life was repeatedly exposed to danger. On his return from Egypt, Fourier was engaged in administrative work, but he also found time for mathematical research. In 1807 he wrote his immortal work "The Mathematical Theory of Heat". The main mathematical result of Fourier can be described as follows: under certain restrictions, any function f (x) can be represented as a series (an infinite sum of numbers or functions), now called the Fourier series:

Fourier developed a method for solving equations of the type (10.1), called the Fourier variable separation method.

The idea behind the Fourier method is ingeniously simple. The solution of equation (10.1) is sought as a product of two functions X(x) and T(t), each of which depends only on one, "own" variable:

u(x,t) = X(x)T(t). (10.7)

Replacement (10.7) splits equation (10.1) into two differential equations in ordinary "school" derivatives:

(10.8)

where λ is an unknown auxiliary parameter. Solving equations (10.8) and satisfying the initial and boundary conditions (10.6) (of course, we omit all intermediate calculations, which here, as in the derivation of the d "Alembert formula (10.5), are enough for several pages), find the final solution of problem (10.6 ) about the oscillation of a finite string:

(10.9)

Let's find out physical meaning solution (10.9), and above all the functions u n (x, t) that make up this solution. To do this, perform an artificial transformation:

(Here we have used the formula for the sine of the sum of two arguments and the fact that Thus, u n (x, t) can be represented as

(10.10)

From formula (10.10) it can be seen that each solution u n is a harmonic oscillation (i.e., oscillation according to the sine law) with the same frequency and phase φ n . The oscillation amplitude A n (x) is different for different points of the string, i.e., it depends on the coordinate of the string point x. It can be seen from (10.10) that for x = 0 and x = l A n (0) = A n (1) = 0, i.e., the string is motionless at the ends.

So, in time, the string vibrations occur at a constant frequency ω n , the vibration amplitude for each point of the string is different. In this case, all points of the string simultaneously reach their maximum deviation in one direction or another and simultaneously pass the equilibrium positions. Such fluctuations are called standing waves.

Using the expression for the amplitude of the standing wave (10.10) and taking into account that 0≤x≤l, we find the fixed points of the standing waves:

The fixed points are called knots standing wave. It is clear that in the middle between the nodes there are points at which the deviations in the standing wave reach a maximum. These points are called antinodes standing wave.

Let's do general conclusion: the oscillation of a finite string is an infinite sum of standing waves u n (x, t), each of which has a constant oscillation frequency and an amplitude varying along the length of the string. In the kth standing wave, there are k

antinodes and (k + 1) nodes.

Let us now turn to the "musical content" of the solution (10.9) and, above all, to the oscillation frequencies. We came to the conclusion that the string oscillates not only with its entire length, but simultaneously with separate parts: halves, thirds, quarters, etc. Therefore, the string emits a sound not only of the fundamental frequency, but also frequency overtones. The tone of the fundamental frequency of the string ω 1 is called the main tone of the string, and the remaining tones corresponding to the frequencies ω 2 , ω 3 , ..., ω k , ... are called overtones(upper tones) or harmonics. The fundamental tone of the string is taken as the first overtone (the first harmonic). It is the overtones, merging in the overall sound with the main tone, that give the sound a musical coloring, called timbre.

The difference in the timbres of musical sounds is mainly due to the composition and intensity of overtones in different sources sounds. The more overtones a sound has, the more beautiful, "richer" it seems to us. By timbre, that is, by the composition of the overtones, we distinguish sounds of the same pitch and the same loudness, reproduced on the violin or piano, by voice or on the flute. Of course, the instrument itself is capable of producing various timbre colors, which primarily applies to the violin.

Violinists have a special way of producing sound that is unusual in terms of timbre - playing with harmonics. Lightly touching the string with his finger in the knots of standing waves, but so that the string does not touch the fingerboard, the violinist dampens some overtones and leaves others. The result is a soft, slightly whistling sound, reminiscent in timbre of the sound of an old woodwind instrument - a harmonica. For example, by touching the string exactly in the middle, the violinist cancels all the harmonics that have antinodes at this point, and preserves only the harmonics that have nodes at this point, i.e. even harmonics. Thus, the second overtone will become the lowest frequency. .

But it will not be exactly an octave above the fundamental tone in terms of timbre, since it will be composed only of even harmonics. Similarly, touching the string at point l/3, the violinist will leave only harmonics that are multiples of three: ω 3 , ω 6 , ..., and get a harmonic that is not like the first one, even if we do ω 2 = ω 3 . Playing with harmonics requires virtuoso precision. After all, if we don’t hit the knot exactly, then we’ll extinguish all the harmonics in general and the string simply won’t sound!

This is the enormous role played in music by the terms u n (x, t) in the solution (10.9). They are rightfully called the sound paint of a musician. But not only musicians, but also the creators of musical instruments show constant concern for these terms, on which the timbre of sound depends. Suffice it to recall the special "Italian timbre" of violins made by famous Italian masters of the 16th-18th centuries, representatives of several generations of the Amati, Guarneri, and Stradivari families.

From the solution (10.9), by setting the functions f(x) and g(x) as necessary and calculating the integrals, one can formally obtain the laws that were experimentally discovered by the English encyclopedist Thomas Young (1773 - 1829):

1. If the string is excited at any point, then an antinode arises at this point and a knot cannot form.

2. If the string is braked at some point, then a knot appears at this point and no antinode can form.

It follows from Young's first law that if a string is excited, for example, exactly in the middle, then all harmonics in it that have a node at this point, i.e., all even overtones, will be extinguished. This means that we will lose half of the overtones and the sound will become faded. It is clear that the farther from the middle we excite the string, the less the first, most important harmonics, we will lose. The timbre of the sound from this will become fuller and brighter. That's why the bow on the violin right hand on the guitar, hammers on the piano - they all excite the string approximately 1 / 7-1 / 10 of the string from the place of its fixing. This is done in order not to disturb the first overtones, and therefore not to impoverish the musical sound. As for playing the violin with harmonics, it is based on Young's second law, which is the inverse of the first law.

Before parting with Jung's laws, let's say a few words about their creator. Thomas Jung was an amazing person. "Everyone can do what others do" - that was the motto of his life. And Jung was extraordinarily successful in fulfilling this difficult rule. He was a circus actor (acrobatic and tightrope walker), an authoritative connoisseur of painting, played almost all sous. Marched in his time musical instruments, deciphered Egyptian hieroglyphs, knew a lot of languages, including Latin, Greek and Arabic. And besides all these "hobbies", Jung received brilliant results in the sciences: physics (wave theory of light), elasticity theory (Young's modulus of elasticity), optics, acoustics, astronomy, physiology, medicine. Jung wrote about 60 chapters of scientific appendices to the famous Encyclopædia Britannica.

Let's take a closer look at the main tone of the string. Remembering that , we obtain the formula for the frequency of the fundamental tone:

(10.11)

from where it is easy to see the laws of string vibration, which were experimentally discovered by the ancient Greeks and which were then rediscovered and described in his "Universal Harmony" by Maren Mersenne:

1. For strings of the same density and the same tension, the oscillation frequency is inversely proportional to the length of the string (this is nothing more than the "first law of Pythagoras - Archytas"; see p. 101).

2. For a given string length and density, its frequency is proportional to the square root of the tension.

3. For a given length and tension, the frequency of a string is inversely proportional to the square root of its density. (At constant density, the thicker the string, the lower the frequency of its oscillations, i.e., the lower the sound.)

Of course, all these laws (at least qualitatively) could be established on a monochord.

But let's get back to overtones. It is easy to see that the overtone frequencies are related as natural numbers:

Thus, the string emits a whole scale of tones, called the natural scale. Theoretically, the natural scale is infinite. In practice, the first 16 overtones matter, since the remaining overtones differ too little from each other, have too little energy and are virtually inaudible.

Natural soundtrack. Assuming ω 1 = l, the frequencies of the natural scale are expressed by the natural series of numbers (ω n = n). The natural scale contains all consonances and all intervals of pure tuning.

Indeed, from (10.12) it follows that the interval coefficient of two neighboring harmonics ω n and ω n+1 is (n = 1, 2, 3, ...). Insofar as then we easily come to the conclusion that as the number n increases, the interval between adjacent harmonics of the natural scale decreases and, in the limit, tends to a pure prime (unison).

The figure shows the first 16 harmonics of a vibrating string, forming a natural scale. The numbers on the right indicate harmonic frequencies, assuming ω 1 = 1, and the red line (hyperbola) cuts off the part of the 1/n string that oscillates at a frequency ω n = n. We see that the second overtone and the fundamental tone make up the octave interval ω 2 / ω 1 \u003d 2. The third and second overtones are the fifth interval: ω 3 / ω 2 \u003d 3/2. The fourth and third are quarts: ω 4 / ω 3 \u003d 4/3. The fifth and fourth are major thirds: ω 5 / ω 4 = 5/4. The sixth and fifth are minor thirds: ω 6 / ω 5 = 6/5. But this is nothing but a set of perfect and imperfect consonances! Thus, we have come to the solution of the "law of consonances" - "the second law of Pythagoras - Archytas" (p. 101 - 102): consonant intervals, which are mathematically expressed by the ratio

of the form (n = 1, 2, 3, 4, 5) are determined by the very nature of the vibration of the string! All consonances are contained in the first six harmonics, i.e., the first six tones of the natural scale, and as you move away from the first harmonic (fundamental tone), the degree of consonance of the interval decreases. So, the law of integer ratios for consonant intervals, which, according to legend, was experimentally discovered by Pythagoras on a monochord, is a consequence of the mathematical solution of the problem of string vibration and follows directly from the solution (10.9).

Moving on to higher harmonics, it is also easy to detect two pure tone intervals: ω 9 / ω 8 \u003d 9/8, ω 10 / ω 9 \u003d 10/9 and a pure semitone interval: ω 16 / ω 15 \u003d 16/15. In this way, all intervals of pure tuning are contained in the natural scale! That is why pure tuning is more pleasing in harmonic sounding than Pythagorean tuning.

But the tones themselves of pure tuning (8.7) are almost completely determined by the natural scale. Indeed, if we consider the octave between the 8th and 16th harmonics, taking the frequency of the 8th harmonic as one (i.e., dividing all frequencies by 8), then we will find in this octave all steps of the pure tuning, except for 4 th (4/3) and 6th (5/3). Hence, pure tuning is almost entirely contained in the natural scale.

However, this insidious "almost" is still one of the mysteries of music. Indeed, why exactly the 7th, 11th and 13th overtones (the 14th overtone is an octave repetition of the 7th) are not included in any of the musical scales? The famous "fake" 7th overtone haunts music theorists for the third century! On the one hand, it is clear that it is wrong to call this sound false, because it is given by nature itself, which is difficult to blame for being false. But on the other hand, all music theorists, starting with Rameau, were too great musicians to include the seventh harmonica in any musical system (the seventh sound clearly "cut the ear"!). However, back in the 18th century, the French musical theorist Ballier wrote with the ease inherent in a Frenchman: "The difference between antiquity and modernity lies in the fact that then they began to count dissonances from the 5th overtone, and now they begin to count them only from the 7th." Will the development of music go so that in new musical systems there is a place for both 7th, and 11th, and 13th overtones? .. In the meantime, piano hammers, following Young's first law, strike 1/8 of the string length in order to reduce the force as much as possible the ill-fated 7th overtone.

Finally, we note another important feature of the natural scale. Looking at the figure, we see that the 4th, 5th and 6th harmonics form a major sound ( do-mi-sol). And if you add the 1st and 2nd harmonics to them, you get a major triad accompanied by an octave bass! So, the major triad is composed of the nearest harmonics (4th, 5th and 6th) of the fundamental tone (bass of the major triad). Consequently, it not only consonates, but also has an acoustic unity inherent in the very nature of string vibration. This gave grounds to one of the last universal scientists - the German mathematician, physicist, physiologist and psychologist Hermann Helmholtz (1821 - 1894) to assert that "the major chord is the most natural of all chords."

How about a minor triad? Disputes about the nature of the minor do not subside to this day. They were attended by Rameau, D "Alembert, Rousseau, Goethe, Helmholtz, many of our contemporaries. Today, opinions agree that since in the minor triad (C - E-flat - G), the second sound (E-flat) lies on a semitone below the fifth harmonic of the main tone, then it forms a barely audible dissonance with it, which causes some "shadowing", "something gloomy and obscure, inexplicable for the listener" (Helmholtz).For this reason, in the music of Bach, Handel, Mozart, minor works often end with a major - the most natural, enlightened - chord.

So, in the major scale, the third degree seems to gravitate upward, while in the minor scale it gravitates downward. The upward movement is perceived by us as an ascent to the light, enlightenment, joy. On the contrary, downward movement is associated with a descent into darkness, blackout, sadness. These objective premises are supported, in addition, by a certain tradition of using major and minor. In those cases when these traditions are violated, we meet the daring song "Apple", written in minor, and the prayer " Ave Maria", which, despite its name - "Hail Mary" - and the major mode, cannot be called cheerful. Unfortunately, mixing the objective physical and mathematical laws of the structure of major and minor with their subjective aesthetic assessment gave rise to a lot of unnecessary disputes around them.

In conclusion, let us dwell on one more problem of a vibrating string. Until now, following the solution (10.9), we have tried to "disassemble like a corpse" string vibrations into the simplest harmonic components. But in fact, again, according to (10.9), the harmonics that make up the vibration of the string add up, forming a complex picture of vibrations. The nature of this pattern depends primarily on the amplitudes of the harmonics. It is not easy to solve this problem in a general way, so we will focus on a simpler problem.

Let two fluctuations of constant and identical amplitude, equal to unity for simplicity, and different frequencies ω 1

The total oscillation, using the formulas for the sum of sines and the cosine of a half angle, can be represented as

When adding two oscillations close in frequency (ω 1 \u003d 8 and ω 2 \u003d 10), beats occur - periodic amplification and attenuation of sound that occurs with a beat frequency ω \u003d ω 2 - ω 1 \u003d 2

Equality (10.13), when the frequencies ω 1 and ω 2 are close to each other with a sufficient degree of accuracy, can be interpreted as follows: the sum of two harmonic oscillations of frequencies ω 1 and ω 2 is an "almost harmonic" oscillation, the frequency of which is the arithmetic mean of these frequencies , and the amplitude changes in time with a frequency ω 2 -ω 1 and is limited from above by the function and from below by the function - . It is easy to see that the amplitude of the total oscillation pulsates with a frequency ω 2 - ω 1 from zero to the maximum value and then back to zero. For this reason, such oscillations are called beats. From (10.13) it is also seen that the maximum amplitude of the beats is twice as large as the amplitudes of the component oscillations.

So, when two oscillations close in frequency are added, beats arise, i.e., almost harmonic oscillations with a frequency equal to the average frequency of these oscillations, and an amplitude pulsating with a beat frequency that is equal to the difference in the frequencies of these oscillations. The sound emitted during beats is then periodically amplified , then freezes.

Let's move on to the musical side of the phenomenon of beats. It is known that any intermittent irritation of the nerves is perceived more strongly than a constant one. However, with an increase in the frequency of stimuli, the nerve does not have time to follow the changes, individual stimuli merge with each other and become invisible. It has been experimentally established that beats with a frequency of 4-5 Hz (oscillations per second) are most clearly audible. Beats at about 15 Hz are still discernible, and at about 30 Hz they start to merge, but create an unpleasant sensation of husky sound.

There is Helmholtz's theory, which explains the phenomena of consonance and dissonance by beats that occur between the harmonics of two sounding fundamental tones. According to Helmholtz's theory, the degree of consonance and dissonance of an interval depends on the presence of beats, their frequency and loudness (amplitude). Let's explain this theory with an example. In table 2, a note up to a small octave, the frequency of which is 131 Hz, and its five overtones are taken as the fundamental tone. The following are the first five overtones for sounds up to 1, sol, fa, mi (pure tuning), mi (Pythagorean tuning) and C-sharp (pure tuning), which form intervals of octaves, fifths, quarts, major thirds with the main tone up, respectively , the Pythagorean third and the minor second.

It is clear that for an octave the frequencies of harmonics coincide, the numbers of which are related as 2/1, for a fifth - as 3/2, etc. (see Table 2). Comparing the frequencies of the first harmonics, we see that the smaller the interval becomes, the closer the frequencies of the fundamental tones, the more distinguishable the beats will be and, consequently, the less the degree of consonance of the interval will be. Therefore, the most consonant interval is the octave, followed by the fifth and fourth. All three of these intervals do not give beats and are perfect consonances. The third gives a little more than 30 beats in the first harmonics, i.e., on the threshold of distinguishability, and therefore refers to imperfect consonances. But the small second gives 9 beats (140-131 = 9) and therefore is a clear dissonance. Note that the fourth harmonic of the Pythagorean third (663.2) and the fifth harmonic of the fundamental tone (655) give 8 beats (663-655 = 8). These beats create the unpleasant harmonic sound of the Pythagorean third. However, since they occur in the higher harmonics, i.e., much weaker than the beats in the first harmonics, it is clear that the Pythagorean third cannot be considered a dissonance along with the minor second, where the same beats occur in the first harmonics.

Thus, we have fulfilled the promise given on p. 130 and explained why the harmonic Pythagorean third sounds tense compared to the pure one. Of course, Helmholtz's theory does not solve all the musical mysteries of the vibrating string - such as the problem of the 7th overtone, for example - and there are still many points of application for the inquisitive mind.

Isn't it true, what an amazing variety of laws, properties and mysteries is fraught with a simple vibration of a simple string! The laws of Pythagoras - Archytas (especially the "law of consonances"), the laws of Mersenne and the laws of Young, the solutions of D "Alembert D. Bernoulli and Fourier, the natural scale and major triad, beats ... For the third millennium, an ordinary string has been revealing its extraordinary secrets to mankind! And maybe someone will think about these secrets next time before hitting the strings of an old guitar.

Let along the axis X two plane harmonic waves propagate towards each other with the same frequencies and amplitudes:

.

All particles of the elastic medium covered by the wave process will participate in the oscillations excited by each of the waves:

x = x 1 + x 2 = + .

Using the trigonometric formula for the sum of cosines, we obtain

where A(X) = 2Acos kx.

The resulting expression shows that the particles of an elastic medium, covered by two wave processes, perform harmonic oscillations with a frequency w.

The amplitude of oscillations of particles of the medium depends on the coordinate X.

At points whose coordinates meet the condition kx = ± n p, where n= 0, 1, 2, 3...cos kx= ±1 and the amplitude of oscillations of the particles of the medium is maximum. Such points are called antinodes. The antinode coordinates are determined by the relation .

At the points that meet the condition, the amplitude is zero, i.e., the particles of the medium at these points do not oscillate at all. Such points are called knots. Node coordinates are determined by the relation .

Since the amplitude of oscillations of the particles of the medium is determined by their coordinate and does not depend on time, the position of the nodes and antinodes does not change. Knots and antinodes remain in one place. Therefore, the wave resulting from the superposition of counterpropagating waves of the same frequency is called standing.

Consider a stretched string, the ends of which are rigidly fixed. Let the string length be l.

Let us assume that oscillations are excited in this string.

A string can be thought of as a collection of infinitely small interconnected elements. Vibrations of one such element must involve other elements of the string in the oscillatory process. Therefore, if vibrations are excited in a string, then an elastic wave will arise in it.

The end of the string is rigidly fixed and cannot oscillate. Therefore, it cannot excite vibrations in the medium to which it is attached. Therefore, the wave that has reached the end of the string will be completely reflected.

This means that two counterpropagating waves will propagate along the string and .

As shown above, when such waves are superimposed, a standing wave arises. This means that a standing wave can arise on a string with fixed ends.

Since we are talking about a string with fixed ends, there must always be knots at the ends of the string.

From the expressions for calculating the coordinates of nodes and antinodes, it can be seen that neighboring nodes (as well as antinodes) are separated from each other by l / 2.

Therefore, the length of the string must be such that half the wavelength fits on it an integer number of times:

where n = 1, 2, 3...

This, in turn, means that on a string of long l standing waves of only certain frequencies can occur

These frequencies are called natural frequencies strings, or frequencies of normal vibrations. Oscillations with such frequencies are called harmonics (oscillation with a frequency corresponding to n= 1 is called the first harmonic, n= 2 - the second harmonic, etc.).

group speed

In science and technology, waves are widely used to transmit information. However, a harmonic wave is capable of conveying information only that there is a source of the wave somewhere.

In order to be able to transmit with the help of waves required amount information, they need to be changed (for example, to emit waves in the form of impulses, or to change the amplitude of the wave, its frequency, the initial phase). Such a wave is called modulated.

With the help of modulated elastic waves, the depth of the seas and oceans (echo sounder) is determined, and modulated electromagnetic waves allow radio and television broadcasting.

But if modulated waves differ from harmonic ones in their ability to carry information, then perhaps they have other differences as well.

Let's explore one of the aspects of this problem - we will find the speed with which the modulated wave transfers energy.

To do this, consider two identically directed plane transverse traveling waves, the oscillations of which occur in the same plane, the amplitudes of which are equal, and the frequencies are almost the same.

.

This wave can be represented as

,

,

that is, it is a wave with a slowly changing amplitude, or modulated, the same as in the figure.

The picture shown here corresponds to some point in time. The next moment it will move to the right.

Let's find the speed with which the modulated wave will propagate. For simplicity, consider the point at which the amplitude is maximum - the speed of movement of this point is equal to the speed of the modulated wave.

The behavior of the point with the maximum amplitude is described by the expression . But this expression can be interpreted as the equation of a traveling wave with a cyclic frequency d w = w 1 –w 2 and wave number dk = k 1 – k 2 .

For any traveling wave , and w= kv. Then the speed of the point with the maximum amplitude will be equal to

,

where v 1 and v 2 – phase velocity of waves with cyclic frequencies w 1 and w 2, respectively.

If there is no dispersion, then v 1 = v 2 = v and , i.e., the "crest" of such a wave moves with the phase velocity.

If the medium is dispersive, then the velocity . This means that the "comb" is moving at a speed different from v 1 and v 2 .

If we remember that the energy of oscillations is proportional to the square of the amplitude, then it is easy to see that most of the energy carried by such a wave is concentrated where the wave amplitude is large. This means that the resulting speed u is the energy transfer rate.

This speed u and call group:

It is important to note that the wave front propagates with group velocity.

Electromagnetic waves

By the middle of the XIX century. discovered a number of important laws in the field of electricity and magnetism. A significant part of the discoveries in this area belongs to Michael Faraday.

This outstanding scientist, who is rightfully considered the founder of modern electrodynamics, oddly enough, did not know mathematics.

Therefore, the phenomena he discovered did not have a mathematical description.

In 1854 James Clerk Maxwell, who had just graduated, was hired by the University of Cambridge. He chose the mathematical description of Faraday's discoveries as the main goal of his activity.

He succeeded (see Sections 5.6, 5.7). One of the results of Maxwell's activity is the prediction of the existence of electromagnetic waves.

About twenty years later, electromagnetic waves were obtained experimentally by the German physicist Heinrich Hertz.

Consider the mechanism of occurrence and some features of electromagnetic waves.

Let us assume that the electric field in vacuum is created by a charge that performs harmonic oscillations.

The electric field created by such a charge must also change over time according to the harmonic law.

The density of the bias current created by the changing electric field, is equal to . Since the derivative of the harmonic function is a harmonic function, the displacement current will also change according to the harmonic law.

Displacement current creates a magnetic field

.

The integral of a harmonic function is also a harmonic function. Consequently, the magnetic field created by the bias current will change according to the harmonic law.

It is important to note that the change in electric and magnetic fields is described by the same harmonic function.

The bias current is in the same direction as the vector ¶ E .

The magnetic field induction vector is always perpendicular to the current that created it.

This means that the magnetic field created by the changing electric field will be perpendicular to it.

In accordance with the Maxwell equation about the circulation of the vector E , a changing magnetic field generates an electric field. Moreover, the generated electric field will be perpendicular to the changing magnetic field.

This, in turn, means that even if the charge that created the changing electric field disappears, the changing electric and magnetic field will continue to propagate in space in the form of an electromagnetic wave.

A more rigorous analysis allows us to show that the changing electric and magnetic fields are described by the wave equations:

where With is the speed of light in vacuum (if an electromagnetic wave propagates in a medium, then the speed of light in this medium is used).

The solution of these equations has the following form:

,

where the amplitudes E and H related by the ratio .

It can also be shown that if the vector E parallel to axis X, and the vector B is parallel to the axis at, then the electromagnetic wave propagates along the axis z(see pic-nok). In other words, the vectors E , H and the velocity vector of the electromagnetic wave c form the right triple.

It is important to note that fluctuations E and H in-phase.

Laboratory work

Investigation of the dependence of the string oscillation frequency on the length tension force and the linear density of the string material. Equipment: Installation including a string tensioning device with a dynamometer, a measuring ruler with movable thresholds, an electric light bulb with a holder, a photocell, a low-frequency amplifier, an oscilloscope and a universal counter; rubber mallet; string set. String vibrations as an example of a standing wave In practice, standing waves arise when waves are reflected from obstacles: a wave incident on an obstacle and a wave traveling to it ...

Lab #25

STRING VIBRATIONS

Objective:

Study of the oscillatory motion of a string. Investigation of the dependence of the string oscillation frequency on the tension force, length and linear density of the string material.

Equipment:

Installation, including a device for string tension with a dynamometer, a measuring ruler with movable thresholds, an electric light bulb with a holder, a photocell, a low-frequency amplifier, an oscilloscope and a universal counter; rubber mallet; string set.

Duration of work - 4 hours.

Theoretical part.

1. Elastic waves

elastic wave is the process of propagation of a perturbation in an elastic medium, accompanied by the transfer of energy. A special role in the theory of waves is played byharmonic waves, in which the change in the state of the medium occurs according to the sine or cosine law.

wave surfaceis the locus of points oscillating in the same phase. V plane wave wave surfaces are a set of planes parallel to each other.

Consider a harmonic plane wave propagating along the axis x . Let us introduce the notation: - deviation from the equilibrium position of the point of the medium with the coordinate x at the time t . Figure 1 shows the graph of the function for some fixed moment t .

Fig.1 - Type of function for a fixed moment t .

Wavelength λ called the distance over which the wave propagates in a time equal to the period of oscillation:

where v is the wave propagation speed, and T is the period of oscillation. As can be seen in Fig. 1, the wavelength can also be defined as the distance between the nearest points of the medium, oscillating with a phase difference of 2π . Given the relationship between period and frequency, we get:

(1)

Let the source of vibrations located at the point x =0 fluctuates according to the law, where a is the displacement amplitude;ω is the cyclic frequency. Then oscillations at the point with coordinate x will be delayed by the time required for the wave to travel from the source to the given point:

Taking into account relation (1), we obtain:

The value is called wave number . With this designation in mind:

(2)

This expression is calledplane wave equation. If the wave propagates in the direction of negative axis values x , then its equation will take the form:

(3)

The equation of any wave is the solution of a differential equation calledwave equation. For a plane harmonic wave propagating along the axis x , the wave equation has the form:

(4)

It is easy to verify the validity of this statement by simply substituting the plane wave equation (2) into the wave equation (4).

2. Standing waves

standing wave called an oscillatory process that occurs as a result of the superposition of two counterpropagating plane waves with the same frequency and amplitude.

Using this definition, we derive the standing wave equation. Equations for two plane waves propagating along an axis x in opposite directions:

When these waves are superimposed, an oscillatory process occurs:

Transforming this expression according to the formula for the sum of cosines, we get:

(5)

That's what it is standing wave equation. The factor describes the harmonic oscillations. However, as can be seen from formula (5), the amplitude of these oscillations depends on the coordinate x in law. Figure 2 (a) shows the standing wave function for several fixed successive times t . Figure 2 (b) also shows the form of a similar function for a conventional traveling wave. Comparing these figures, we can conclude that a standing wave is a special type of oscillatory motion and, despite the name, is not a wave in the strict sense of the word, since a standing wave does not transfer energy in space.

Rice. 2 - Type of function of standing (a) and traveling (b) waves for several fixed successive moments of time t .

The points at which the amplitude of the standing wave oscillations vanishes are called nodes . At the nodes, the points of the medium do not oscillate (see Fig. 2, a). Node coordinates must satisfy the condition:

(6)

The points at which the oscillation amplitude is maximum (see Fig. 2, a) are called antinodes . Accordingly, the antinode coordinates satisfy the condition:

(7)

3. Vibrations of a string as an example of a standing wave

In practice, standing waves arise when waves are reflected from obstacles: a wave incident on an obstacle and a reflected wave running towards it, superimposed on each other, give a standing wave.

Another example of standing waves is the vibrations of a stretched string fixed at both ends. The ends of the string cannot oscillate, which means that the standing wave must have nodes at these points. Consequently, only such oscillations can be excited, the wavelength of which allows this condition to be realized. In other words, half the wavelength must fit the length of the string an integer number of times, as shown in Fig. 3. We number these vibrations, starting with the largest wavelength, and write down the relationship between the length of the string and the wavelength of the vibration with the number n (see Fig. 3). In general, this ratio has the form:

Or (8)

Wavelengths (8) correspond to frequencies:

where V is phase velocitywaves - the speed at which vibrations propagate along the string. These frequencies are callednatural frequencies. Harmonic oscillations with natural frequencies is own (normal) oscillations or harmonics. Frequency v corresponding to n =1 is called fundamental frequency:

(9)

Rice. 3 - Natural vibrations of the string

The phase velocity of the wave is constant in time and is determined by the densityρ the material of the string and the force of its tension F (See Appendix):

(10)

Let us substitute into the expression for the fundamental frequency (9):

(11)

Experimental verification of this ratio is the main content of this laboratory work.

Installation description

Appearance and installation diagram are shown in Fig. 4.

The string (1) is stretched between the peg (2), which regulates the tension force, and the spring dynamometer (3) that measures it. The string rests on two movable triangular nut (4). Its length is regulated by moving these thresholds along the measuring ruler (5). The string is located between the bulb (6) and the photocell with a slit aperture (7).

The vibrations of the string are excited by a light blow of a rubber mallet. As a result, the illumination of the photocell and the signal generated by it change at the same frequency as the string vibrates. The signal from the photocell through the amplifier (8) goes to the oscilloscope (9) and the universal counter (10), which measures the frequency of the signal.

Rice. 4 - Appearance (a) and diagram (b) of the installation for measuring the frequency of string vibrations

experimental part

Exercise 1. Measuring the dependence of the string vibration frequency on the tension force.

1. Before stringing, the dynamometer must be zeroed. If the string is already tense, turn the peg to loosen the tension until it is completely gone. After loosening the fixing screw on the side surface of the cylindrical body of the dynamometer, ensure that the edge of the body coincides with the zero division of the dynamometer scale. Fasten the fixing screw.
2. For experimental verification of relation (17) between the frequency of vibrations of the string and the tension force, a constantan wire with a cross-sectional diameter d \u003d 0.4 mm (wire No. 1). Install it between the hook of the dynamometer and the hook with the thread attached to the peg. In this case, the string should lie on triangular sills. Slowly rotate the peg to set the string tension F \u003d 10 N.
3. By moving the nut along the ruler, set the length of the string l=50 see Here and below, under the string length we mean the distance between the upper corners of the nut. It can be measured both by the position of the thresholds on the ruler (pos. 5 Fig. 4), and directly using a metal ruler.

1. Turn on the oscilloscope. Adjustment " VOLTS/DIV » of the corresponding channel, set to « 1 » (see Fig. 5). Adjustment " TIME/DIV » set to « 2 ms".
2. Turn on the amplifier (the switch is located on the rear panel of the device). Set the setting " Amplification" to position "10 2 » (see Fig. 6). Button " AC/DC ” must be depressed, which corresponds to the variable input signal. Set the amplitude control to the middle position.
3. Turn on the universal meter (the switch is located on the back of the device). With the button " mode » switch it to « analog » (see Fig. 7). With the button " function » set frequency measurement mode « freq". Press the "Set" button » set mode « Digits ". Click the button start ". The light above this button will light up, indicating that the meter is ready to measure the frequency.

Rice. 5 - Appearance of the oscilloscope screen.

Rice. 6 - Appearance of the front panel of the amplifier.

Rice. 7 - Appearance of the front panel of the universal meter.

1. Just before measuring the frequency, you need to make sure that:

• The length of the string corresponds to the desired value (the nut may move slightly when the string tension changes);
• The tension corresponds to the desired value (the tension may change when the thresholds are moved);
• The shadow from the string coincides with the slit of the photodetector slit. To do this, you need to either look at the photodetector from below, or use a mirror.

These checks must be repeated before each subsequent measurement.

1. Vibrate the string with a light blow of a rubber mallet. The counter does not start measuring the frequency immediately, but after the damping of high harmonics. This process can be controlled using an oscilloscope: at the moment of measurement, oscillations close to sinusoidal should be observed on its screen (see Fig. 5).
2. Repeat the measurements, gradually increasing the tension until F \u003d 40 N, in increments of 5 N.

Do not try to set the tension force of a constantan string with a diameter of 0.4 mm greater than 40 N! This may lead to its rupture.

Enter the measurement results in the table:

Table 1.

F , N	v, Hz	ν 2, Hz 2	Hz 2

1. Fill in the remaining cells of Table 1. Take the frequency measurement error equal, and the tension force measurement error - . Plot the squared frequency versus tension force. According to (11), this dependence is linear:

(12)

Determine the slope of the straight line from the graph. Using this value, calculate the linear density of the wireρ l and its error.

1. Linear and volumetric densities are related by the relation:

, (13)

where S and d are the area and diameter of the cross section of the wire, respectively. Using this formula, determine the bulk density of constantanρ and the error of this value. The error in measuring the length of the string is taken equal, and the error in measuring the diameter of the string is .

Exercise 2. Measuring the dependence of the frequency of vibrations of a string on its length.

1. For experimental verification of relation (11) between the frequency of vibrations of the string and its length, as in the first exercise, a constantan wire with a cross-sectional diameter d \u003d 0.4 mm (wire No. 1).
2. Set string length l=30 cm and string tension F =30 N.
3. Measure the frequency of the string.
4. Gradually increase the length of the string to l=60 cm in steps of 5 cm, measure the dependence of the frequency of vibration of the string on its length. Record the results in the table:

Table 2.

l , cm	v, Hz	MS	MS

1. After filling in the remaining cells of Table 2, build a graph of the dependence of the period of oscillation of the string on its length. According to (17), this dependence should be linear:

(14)

Determine the slope of the straight line from the graph. Using this value, calculate the linear density of the wire, the bulk density of constantan and the errors of these values.

Exercise 3. Measuring the dependence of the string vibration frequency on the linear density of the material.

1. The data on the wires used for the experimental verification of relation (11) between the frequency of the string vibrations and the linear density of the material are given in Table 3.

Table 3

	Material	Cross section diameter d, mm	Bulk densityρ, g/cm 3	Line Densityρ l , g/m
	Tantalum			0.471±0.002
	Constantan			0.620±0.002
	Nickel			0.647±0.002
	Copper			1.110±0.002
	Copper			1.794±0.003

These wires do not withstand high tensions. Therefore, do not attempt to set the tension force above 20 N!

1. By setting alternately for each string the length l=50 cm and string tension F=20 H, measure the frequencies of their oscillations. Record the results in a table.
2. Plot the experimental points on the theoretical graph of the frequency of the string vibrations on the linear density, constructed during the execution of the calculation task. Make a conclusion.

Preparation for work.

1. Physical concepts, quantities, laws, the knowledge of which is necessary for the successful completion of work:

• harmonic oscillations, amplitude, phase, frequency;
• waves in an elastic medium;
• wave speed, frequency, wavelength;
• plane wave equation;
• wave equation;
• standing waves, knots and antinodes;
• string vibrations, harmonics;

2. Derive formula (11)

3. Estimated task.Using formula (11), for a string of length l=50 see, strung with force F \u003d 20 N, plot on graph paper the dependence of the oscillation frequency on the linear density in the rangeρ l from 0 to 2 g/m in steps of 0.2 g/m.

4. Formulate the main goal of the work and the order of its implementation.

Appendix 1. Derivation of the formula for the speed of a wave in a string.

On Fig. 1-1 schematically shows a stretched string. Let's select a small fragment of it and write down Newton's second law for it:

where and are the forces acting on the left and right ends of the string, respectively.

Rice. 1-1 - To the derivation of the wave equation of string vibrations.

In projections on the vertical axisξ :

At small displacements, and the tangent of the slope of the curve, in turn, is equal to the derivative of the function: . In this way, .

The mass per unit length of the string is called the linear density. Then the mass of the fragment can be expressed in terms of its length: . For small vibrations, the length of a string fragment can be taken equal to its projection onto the axis X : . As a result, we get:

Ignoring the change in force F along the cord, we get:

This is nothing more than a wave equation describing the propagation of a wave with a speed

Literature

1. Irodov I.E. Wave processes. Basic laws.– M.: BINOM. Basic Knowledge Laboratory, 2004, §§1.1 – 1.6.

2. Saveliev I.V. Course of general physics. Waves. Optics - M .: Astrel AST, 2003; §§1.1, 1.4, 1.5, 1.7, 1.8.

Page 5

EMBED Equation.3

EMBED Equation.3

α2

Embed Equation.3 Embed Equation.3

EMBED Equation.3

α 1

λ= VT

260.5Hz

Counter

Amplifier

Oscilloscope

b) Traveling wave

a) standing wave

antinodes

Knots

As well as other works that may interest you
38790.		DYNAMICS OF VALUE ORIENTATIONS OF YOUTH IN RELATION TO FAMILY AND MARRIAGE IN CONDITIONS OF MODERNIZATION OF RUSSIAN SOCIETY	758KB
	Theoretical and methodological foundations of the study and value orientations of young people in relation to family and marriage. Some theoretical approaches to the study of young people's value orientations in relation to family and marriage. Factors of formation and trends in the development of value orientations of modern Russian youth in relation to family and marriage.
38791.		Effect of reduced glutathione and catalase inhibitor on peroxide resistance and erythrocyte lysis rate under the action of ferric chloride	650KB
	It has been established that when catalase activity is inhibited by sodium azide, including the action of ferric chloride, the rate of erythrocyte hemolysis increases. Iron (III) chloride at a concentration of 0.5% caused complete lysis of human erythrocytes in 5 minutes of incubation with a maximum of lysis from 1.5 to 3.5 minutes of incubation, regardless of the pretreatment of erythrocytes
38792.		Methods for assessing the creditworthiness of a commercial bank borrower	1.08MB
	Credit is the backbone of the modern economy, an integral element of economic development. It is used by both large enterprises and associations, as well as small industrial, agricultural and trade structures; both states, governments and individual citizens. It becomes an inevitable attribute of the commodity economy.
38793.		Forest natural-reserved territories as the middle of the evolutionary conservation of forest dendrophytorism	403KB
	The current camp of forest genetic resources and strategies for their conservation. Strategies for the conservation of the genetic sluggishness of the fox dendroflora. Come to the conservation of the genetic fragility of the forest gene pool. Preservation of views in the villages of Roslin in situ.
38794.		INCREASING THE EFFICIENCY OF AN ENTERPRISE IN MARKET CONDITIONS (ON THE EXAMPLE OF DABAN LLC)	856.5KB
	Analysis of the volume and range of products. Analysis of the structure of products and the impact of structural shifts on the change in the cost of products. The concept of efficiency The goal of any industrial enterprise is the release of certain products, the performance of work, the provision of services of a specified volume and quality within a certain time frame. But when establishing the scale of production, one should proceed not only from the national economic and individual needs for this product, but also from the need to take into account the achievement of the maximum ...
38795.		ECONOMIC EFFICIENCY OF THE ORGANIZATION'S ACTIVITY AND WAYS TO INCREASE IT ON THE EXAMPLE OF RUE "KLIMOVICH DISTILLERY"	742KB
	These include, in particular: an increase in the sale of finished product residues in warehouses; the sale of unused equipment; a decrease in the cost of production as a result of the acquisition of new equipment. For successful functioning, each economic entity should strive to improve the efficiency of its activities based on the rational use of the resource potential, increase the profitability of production, improve the quality of products sold. The basis of this concept is the limited resources, the desire to save ...
38796.		The study of accounting and analysis of funds on the example of a commercial organization VOOI "Sintez"	555.5KB
	Theoretical and organizational bases of accounting and analysis of funds. Types of funds of the organization. Classification of cash flows. Regulatory framework for accounting and analysis of funds.
38797.		Criminal - legal qualification of fraud	325KB
	The relevance of the research topic lies in the fact that there are a number of debatable problems in it, in particular regarding the objective and subjective nature of signs of fraud. In the context of an insufficiently deep study of the signs and specifics of fraud, the presence in the theory of criminal law of many contentious issues Difficulties and errors in qualification often arise on this problem ...
38798.		Calculation of the automated electric drive of the cross feed of the surface grinding machine 3E711	3.36MB
	To increase the grinding accuracy in this course project, it is necessary to pay special attention to the vertical feed drive, so we will consider several options for its implementation: Based on the use of a brushless motor: The brushless motor can be connected using the MC 33033 and MC 33039 microcircuits on the basis of a stepper motor: The main functional units of an open-loop stepper drive are shown in fig. The principle of its operation is that when the frequency changes ...

Figure 3 shows typical dependences of the square of the string vibration frequencies on the tension force for various harmonics n. Observation of natural vibrations of the string is difficult, since they decay relatively quickly. Therefore, in the work, oscillations excited by a constantly acting periodic driving force are considered.

Experimental setup

The unit (Fig. 4) consists of a metal frame consisting of two guide tubes (1) fixed at certain distances with the help of bars (2) . On one of the bars (2) rack installed (3) designed to secure one end of the string (4). On another bar (2) device installed A, which serves to change the tension of the string and consists of a spring dynamometer (5) and node of its movement (6) . The other end of the string is attached to the dynamometer spring (4) . The tension force is changed by the handle (7) , and is measured by a spring dynamometer (5) . On guide pipes (2) bars with elements installed on them are strengthened at certain distances. Racks (8) set the working length of the string (4) . The length of the string between its two fixed ends, equal to the distance between the uprights (8) measured with a ruler (9) located on one of the pipes. Vibrations of the string are excited by an electromagnetic vibrator (10) powered by alternating current from a generator (11) which has a built-in frequency counter. Electromagnetic vibrator (10) causes the string to perform forced vibrations with the frequency of the generator (11) . The oscillation amplitude is recorded by an electromagnetic sensor (12) connected to a voltmeter (13) . The magnitude of the signal produced by an electromagnetic pickup depends on its distance from the string. This change is made with a screw (14) . A similar device is used to adjust the distance between the vibrator and the string. The distance between the string and the vibrator is changed with a screw (15) , while the amplitude of the forced vibrations of the string changes.

Conducting an experiment
Exercise 1. Establishment of the dependence of natural vibration frequencies on the string tension force.
Tension force P determines the velocity of perturbation propagation along the string (2) and, consequently, the frequency of natural oscillations (19). In this exercise, the nature of the dependence is experimentally determined v n from string tension force P .

measurements

Racks (8) set the string length to the maximum multiple of 10 cm. Tension the string with a force of 2 kG (1 kG = 9.8 N). Set the vibrator to a position 10 cm from the fixed end of the string. Place the pickup about 10 cm from the middle of the string.

By changing the frequency of the generator with the "rough" knob (starting from zero according to its school), fix the maximum deviation of the voltmeter needle, which registers the amplitude of the string vibrations. In this case, the frequency of vibrations of the string, set on a scale built into the generator, is a "rough" value of the experimentally established resonant frequency. To determine the exact value of the value v exp use the "smooth" scale of the generator. By turning the handle of the generator to the right or left "smoothly" achieve the maximum deviation of the arrow (if the arrow goes beyond the scale, increase the measuring range of the voltmeter with the "range" knob). Record the reading of the built-in frequency counter. This is the value of the resonant frequency.

Determine which of the harmonics corresponds to this oscillation. To do this, without changing the frequency of the generator, moving the sensor along the string, determine the number of nodal points (when the sensor is under the nodal point, its signal is zero). Harmonic number n fluctuations is determined by the formula n = N + 1, where N- number of nodes (not counting anchor points).

Increasing the oscillation frequency in the manner described above to set the resonant frequencies for the next four harmonics. Experimentally established values v exp enter in the table. one.

Set the normal vibration values ​​of the first harmonics for different string tensions P. To do this, in the frequency range of normal vibrations for the corresponding harmonics, set the frequencies at which the maximum vibrations (according to the voltmeter) of the string are observed for the forces of its tension equal to 2, 3, 4, 5, 6 and 7 kGs. Experimentally established values v exp enter in the table. one.

Results processing

Using expression (19), determine the theoretical values ​​of the frequencies v theor normal vibrations for the first five harmonics at string tensions equal to 2, 3, 4, 5, 6 and 7 kG. Enter the calculation results in Table 1.

Plot Theoretical Frequency Squared Relationships v 2 theor from tensile force P for the first five harmonics of oscillations. They should be similar to those shown in Fig.3.

Mark on the theoretical dependences the squares of the experimentally established values ​​of the frequencies of the first five harmonics of normal vibrations for different values P. Compare experimental and theoretical values v 2 n for normal vibrations.

Table 1

P , kG	1st harmonic		2nd harmonic		3rd harmonic		4th harmonic		5th harmonic
	v exp	v theor	v exp	v theor	v exp	v theor	v exp	v theor	v exp	v theor
2
3
4
5
6
7

Exercise 2. Determination of the dependence of the vibration harmonic number on the string tension.

From fig. 3 shows that the value v2(and hence the frequencies of normal vibrations) for different harmonics can take the same values ​​for certain values ​​of the string tension P. Therefore, by changing the string tension, one can observe different harmonics of normal vibrations at the same frequency. In this exercise, by changing the tension force of the string, various harmonics of normal vibrations at the same frequency are observed.
measurements

Stretch the string with a force of 2 kgf and find the 5th harmonic using the method described in exercise 1.

Without changing the frequency of the generator and increasing the tension of the string, the values ​​are determined P, at which the maximum values ​​of oscillation amplitudes are observed. According to the method described in Exercise 1, the number of nodal points and, accordingly, the numbers of harmonics for these normal vibrations are set.

Enter the found values ​​of the tension forces and the harmonic numbers corresponding to them in Table 2.

table 2

Results processing
Plot Dependency Plot n from P .

Exercise 3 Determination of the dependence of natural vibration frequencies on the string length.

measurements

Set the string tension to 3kG.

Using the technique described in Exercise 1, determine the frequencies of the 1st and 2nd harmonics of natural oscillations. Record the results in Table 3

By changing the length of the string (reducing its length by about 20% each time), determine the frequencies of the 1st and 2nd harmonics of its natural vibrations. Enter the results of the experiment in Table 3

Table 3

L , cm	1/L , cm -1	1st harmonic		2nd harmonic
		v exp	v theor	v exp	v theor

Results processing

Using expression (19), determine the theoretical values ​​of the frequencies of the 1st and 2nd harmonics of the natural vibrations of the string at those string lengths for which the experimental results were obtained. Record the results in Table 3.

Plot Theoretical Dependencies v 1.2 on the reciprocal of the string length 1/L .

Mark on the theoretical dependences the experimentally established values ​​of the frequencies of the 1st and 2nd harmonics of normal vibrations for different values 1/L. Compare experimental and theoretical values v 1.2 for normal vibrations.

The main results of the work
In the course of the work, the dependences of the natural vibration frequencies of the string on the tension force and length should be experimentally obtained. The results should be compared with the theoretically calculated dependences for the known linear density of the string.
Control questions

What are free, forced, natural and normal vibrations of a system?

How many degrees of freedom does a stretched string have, how many normal vibrations can be excited in it?

Derive the wave equation.

Derive the relationship between the frequency of the normal vibration, the length of the string, and the speed of wave propagation in the string.

What happens in the string when the frequency of the external signal is chosen arbitrarily (not necessarily equal to one of the natural frequencies)?

Literature

Strelkov S.P. Mechanics, M. Nauka, 1975, ch.15, § 143.

Sivukhin D.V. General course of physics. v.1. Mechanics. M. Nauka, 1989, § 84.