In what media do standing waves occur? §5 Standing waves

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

Practically standing waves arise when waves are reflected from obstacles. The wave incident on the barrier and the reflected wave running towards it, superimposed on each other, give a standing wave.

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

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

The equations for these oscillations have the form:

.

Adding both equations and transforming the result, according to the formula for the sum of sines, we get:

- standing wave equation.

Comparing this equation with the equation of harmonic oscillations, we see that the amplitude of the resulting oscillations is equal to:

.

Since , and , then .

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At the points of the medium, where , there are no oscillations, i.e. . These points are called standing wave nodes.

At points where , the oscillation amplitude has the greatest value, equal to . These points are called antinodes of a standing wave. The antinode coordinates are found from the condition , because , then .

From here:

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

.

Where:

.

From the formulas for the coordinates of nodes and antinodes, it follows that the distance between neighboring antinodes, as well as the distance between neighboring nodes, is equal to . The antinodes and nodes are shifted relative to each other by a quarter of the wavelength.

Let us compare the nature of oscillations in a standing and traveling wave. In a traveling wave, each point oscillates, the amplitude of which does not differ from the amplitude of other points. But fluctuations of various points occur with different phases.

In a standing wave, all particles of the medium located between two neighboring nodes oscillate in the same phase, but with different amplitudes. When passing through the node, the phase of the oscillations changes abruptly to , because the sign changes.

Graphically, a standing wave can be depicted as follows:

At the time when , all points of the medium have maximum displacements, the direction of which is determined by the sign . These displacements are shown in the figure by solid arrows.

After a quarter of the period, when , the displacements of all points are equal to zero. Particles pass through the line at different speeds.

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

When describing oscillatory processes in elastic systems, not only the displacement, but also the particle velocity, as well as the magnitude of the relative deformation of the medium, can be taken as an oscillating value.

To find the law of change in the speed of a standing wave, we differentiate by the displacement equation of a standing wave, and to find the law of change in deformation, we differentiate by the equation of a standing wave.

.

Analyzing these equations, we see that the nodes and antinodes of the velocity coincide with the nodes and antinodes of the displacement; the nodes and antinodes of the deformation coincide, respectively, with the antinodes and nodes of the velocity and displacement.

string vibrations

In a string stretched at both ends, when transverse vibrations are excited, standing waves are established, and knots must be located at the places where the string is fixed. Therefore, only such oscillations are excited in the string, half of the length of which fits on the length of the string an integer number of times.

From this follows the condition:

where is the string length.

Or otherwise. These wavelengths correspond to frequencies , where is the phase velocity of the wave. Its value is determined by the tension force of the string and its mass.

At is the fundamental frequency.

At - natural vibration frequencies of the string or overtones.

Doppler effect

Let us consider the simplest cases, when the source of waves and the observer move relative to the medium along one straight line:

1. The sound source moves relative to the medium with a speed , the sound receiver is at rest.

In this case, for the period of oscillations sound wave will move away from the source at a distance , and the source itself will move at a distance equal to .

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

If the sound source is brought closer to the receiver, i.e. move in the direction of wave propagation, then .

The frequency of the sound perceived by the receiver is:

Substitute instead of their values ​​for both cases:

Taking into account the fact that , where is the oscillation frequency of the source, the equality takes the form:

Divide both the numerator and denominator of this fraction by , then:

2. The sound source is stationary, and the receiver is moving relative to the medium at a speed.

In this case, the wavelength in the medium does not change and is still equal to . At the same time, two successive amplitudes that differ in time by one period of oscillations , having reached the moving receiver, will differ in time at the moments of the meeting of the wave with the receiver by a time interval , the value of which is greater or less, depending on whether the receiver is moving away or approaching the source sound. During the time the sound propagates over a distance , and the receiver will move over a distance . The sum of these quantities gives us the wavelength:

The period of oscillations perceived by the receiver is related to the frequency of these oscillations by the ratio:

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

.

Because , where is the oscillation frequency of the source, and , then:

3. The sound source and receiver are moving relative to the medium. Combining the results obtained in the two previous cases, we get:

sound waves

If elastic waves propagating in air have a frequency in the range from 20 to 20,000 Hz, then, having reached human ear they evoke the sensation of sound. Therefore, waves lying in this frequency range are called sound waves. Elastic waves with a frequency of less than 20 Hz are called infrasound . Waves with a frequency of more than 20,000 Hz are called ultrasound. Ultrasounds and infrasounds cannot be heard by the human ear.

Sound sensations are characterized by pitch, timbre and loudness. The pitch of the sound is determined by the frequency of vibrations. However, the sound source emits not one, but a whole spectrum of frequencies. The set of vibrational frequencies present in a given sound is called its acoustic spectrum. The vibration energy is distributed among all frequencies of the acoustic spectrum. The pitch of the sound is determined by one - the fundamental frequency, if this frequency accounts for significantly large quantity energy than at a fraction of other frequencies.

If the spectrum consists of a set of frequencies that are in the frequency range from to , 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 ruled(example - musical sounds).

The acoustic spectrum of sound, depending on its nature and on the distribution of energy between frequencies, determines the originality of the sound sensation, called the timbre of sound. Different musical instruments have different acoustic spectrum, i.e. differ in tone.

The intensity of sound is characterized by various quantities: oscillations of the particles of the medium, their velocities, pressure forces, stresses in them, etc.

It characterizes the amplitude of oscillations of each of these quantities. However, since these quantities are interrelated, it is advisable to introduce a single energy characteristic. Such a characteristic for waves of any type was proposed in 1877. ON THE. Umov.

Let us mentally cut out a platform from the front of the traveling wave. In time, this area will move a distance , where is the speed of the wave.

Denote by the energy of the unit volume of the oscillating medium. Then the energy of the entire volume will be equal to .

This energy was transferred over time by a wave propagating through the area.

Dividing this expression by and , we obtain the energy transferred by the wave through unit area per unit time. This value is denoted by a letter and is called Umov vector

For sound field Umov vector is called the power of sound.

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

With a very large sound strength of the order, the sound is perceived except for the ear by the organs of touch, and causes pain in the ears.

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

A person has a rather complex apparatus for the perception of sounds. Sound vibrations are collected auricle and through the auditory canal act on the eardrum. Its vibrations are transmitted to a small cavity called the cochlea. Inside the snail is a large number of fibers having different lengths and tensions and, consequently, different natural vibration frequencies. When sound is applied, each of the fibers resonates to a tone whose frequency coincides with the natural frequency of the fiber. The set of resonant frequencies in the hearing aid determines the area of ​​sound vibrations perceived by us.

The volume, subjectively assessed by our ear, increases much more slowly than the intensity of sound waves. While the intensity increases exponentially - the volume increases exponentially. arithmetic progression. On this basis, the loudness level is defined as the logarithm of the ratio of the intensity of a given sound to the intensity taken as the original

The unit of volume level is called white. Smaller units are also used - decibels(10 times less than white).

where is the sound absorption coefficient.

The value of the sound absorption coefficient increases in proportion to the square of the frequency of the sound, so low sounds propagate farther than high ones.

In architectural acoustics for large rooms, a significant role is played by reverberation or the loudness of the premises. Sounds, experiencing multiple reflections from enclosing surfaces, are perceived by the listener for some quite large gap time. This increases the strength of the sound that reaches us, however, if the reverberation is too long, the individual sounds are superimposed on each other and the speech is no longer perceived articulately. Therefore, the walls of the halls are covered with special sound-absorbing materials to reduce reverberation.

Any vibrating body can serve as a source of sound vibrations: a bell reed, a tuning fork, a violin string, a column of air in wind instruments, etc. these same bodies can also serve as receivers of sound when they are set in motion by vibrations of the environment.

Ultrasound

To get directional, i.e. close to flat, the wave dimensions of the emitter must be many times greater than the wavelength. Sound waves in air have a length of up to 15 m, in liquid and solid bodies the wavelength is even longer. Therefore, it is practically impossible to build an emitter that would create a directed wave of this length.

Ultrasonic vibrations have a frequency of over 20,000 Hz, so their wavelength is very small. As the wavelength decreases, the role of diffraction in the process of wave propagation also decreases. Therefore, ultrasonic waves can be obtained in the form of directed beams, similar to light beams.

Two phenomena are used to excite ultrasonic waves: reverse piezoelectric effect and magnetostriction.

The inverse piezoelectric effect is that the plate of some crystals (Rochelle salt, quartz, barium titanate, etc.) is slightly deformed under the influence of an electric field. By placing it between metal plates, to which an alternating voltage is applied, it is possible to cause forced vibrations of the plate. These vibrations are transmitted to the environment and generate an ultrasonic wave in it.

Magnetostriction lies in the fact that ferromagnetic substances (iron, nickel, their alloys, etc.) are deformed under the influence of a magnetic field. Therefore, by placing a ferromagnetic rod in an alternating magnetic field, it is possible to excite mechanical vibrations.

High values ​​of acoustic velocities and accelerations, as well as well-developed methods for studying and receiving ultrasonic vibrations, made it possible to use them to solve many technical problems. Let's list some of them.

In 1928, the Soviet scientist S.Ya. Sokolov suggested using ultrasound for the purposes of flaw detection, i.e. for detection of hidden internal defects such as shells, cracks, ripples, slag inclusions, etc. in metal products. If the size of the defect exceeds the wavelength of ultrasound, then the ultrasonic pulse is reflected from the defect and returned back. By sending ultrasonic pulses into the product and recording the reflected echo signals, it is possible not only to detect the presence of defects in products, but also to judge the size and location of these defects. This method is currently widely used in industry.

Directed ultrasonic beams found wide application for location purposes, i.e. to detect objects in the water and determine the distance to them. For the first time, the idea of ​​ultrasonic location was expressed by an outstanding French physicist P. Langevin and developed by him during the First World War to detect submarines. Currently, the principles of sonar are used to detect icebergs, schools of fish, etc. these methods can also determine the depth of the sea under the bottom of the ship (echo sounder).

Large-amplitude ultrasonic waves are currently widely used in engineering for mechanical processing of solid materials, cleaning of small objects (parts of clockwork, pipelines, etc.) placed in a liquid, degassing, etc.

Creating strong pressure pulsations in the medium during their passage, ultrasonic waves cause a number of specific phenomena: grinding (dispersion) of particles suspended in a liquid, formation of emulsions, acceleration of diffusion processes, activation of chemical reactions, impact on biological objects, etc.

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

1. Have you seen the slate sheet?
   The same thing on the surface of the water, a puddle on a windy day, for example.
2. oh how hard you answered. I explain simply like a gingerbread man.
   What is a wave process. This is when something changes and it has a maximum and a minimum (an example of water waves is when at different points in time at the same point the wave maximum (peak) changes to a minimum). When the maximum is replaced by a minimum, these are traveling waves. Waves are standing. This is when the maximum does not change to the minimum, but there are different levels in different places (standing ripples on the surface of the water from the wind).
3. Oho. This is such a concept that swells the brain of tens of thousands of people around the clock! The standing wave is the essence of BTG. The essence of Tesla. The essence of the future energy from nothing!)))
4. standinǵtea wavé oscillations in distributed oscillatory systems with a characteristic arrangement of alternating maxima (antinodes) and minima (nodes) of amplitude. In practice, such a wave arises during reflections from obstacles and inhomogeneities as a result of the superposition of the reflected wave on the incident one. In this case, the frequency, phase and attenuation coefficient of the wave at the place of reflection are extremely important.

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

   A purely standing wave, strictly speaking, can exist only if there are no losses in the medium and the waves are completely reflected from the boundary. Usually, in addition to standing waves, there are also traveling waves in the medium, which supply energy to the places of its absorption or emission.

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

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

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

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

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

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

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

   In optics, it is also possible to establish S. century. with visible maxima and minima of the electric field. If the light is not monochromatic, then in S. century. antinodes of the electric field different lengths waves will be located in different places and it is not uncommon to see separation of colors.

standing wave- the phenomenon of interference of waves propagating in opposite directions, in which the transfer of energy is weakened or absent.

standing wave(electromagnetic) - periodic change amplitude the strength of the electric and magnetic fields along the direction of propagation, caused by the interference of the incident and reflected waves.

For example, a standing wave occurs when a wave is reflected from obstacles and inhomogeneities as a result of the interaction (interference) of the incident and reflected waves. The result of interference is affected by the frequency of oscillations, the modulus and phase of the reflection coefficient, the directions of propagation of the incident and reflected waves relative to each other, the change or preservation of the polarization of the waves during reflection, the attenuation coefficient of the waves in the propagation medium. Strictly speaking, a standing wave can exist only if there are no losses in the propagation medium (or in the active medium) and the incident wave is completely reflected. In a real medium, however, the mode of mixed waves is observed, since there is always a transfer of energy to the places of absorption and emission. If, when a wave falls, it is completely absorption, then the reflected wave is absent, there is no wave interference, the amplitude of the wave process in space is constant. Such a wave process is called a traveling wave.

Examples of a standing wave are string vibrations, air vibrations in an organ pipe; in nature - Schumann's waves. A Rubens tube is used to demonstrate standing waves in a gas.

Standing waves are solutions of wave equations. They can be thought of as a superposition of waves propagating in opposite directions.

When there is a standing wave in the medium, there are points where the oscillation amplitude is equal to zero. These points are called knots standing wave. The points at which the oscillations have the maximum amplitude are called antinodes.

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  For example, the different modes of vibration of a string clamped at the ends determine its fundamental tone and overtones.

  Mathematical description of standing waves

  In the one-dimensional case, two waves of the same frequency, wavelength, and amplitude propagating in opposite directions (for example, towards each other) will interact, resulting in a standing wave. For example, a harmonic wave propagating to the right, reaching the end of a string, produces a standing wave. The wave that is reflected from the end must have the same amplitude and frequency as the incident wave.

  Consider the incident and reflected waves in the form:

  y 1 = y 0 sin ⁡ (k x − ω t) (\displaystyle y_(1)\;=\;y_(0)\,\sin(kx-\omega t)) y 2 = y 0 sin ⁡ (k x + ω t) (\displaystyle y_(2)\;=\;y_(0)\,\sin(kx+\omega t))

  Therefore, the resulting equation for a standing wave y will be in the form of a sum y 1 and y2:

  y = y 0 sin ⁡ (k x − ω t) + y 0 sin ⁡ (k x + ω t) . (\displaystyle y\;=\;y_(0)\,\sin(kx-\omega t)\;+\;y_(0)\,\sin(kx+\omega t).)

  Using trigonometric relations, this equation can be rewritten as:

  y = 2 y 0 cos ⁡ (ω t) sin ⁡ (k x) . (\displaystyle y\;=\;2\,y_(0)\,\cos(\omega t)\;\sin(kx).)

  If we look at fashion x = 0 , λ / 2 , 3 λ / 2 , . . . (\displaystyle x=0,\lambda /2,3\lambda /2,...) and anti-fashions x = λ / 4 , 3 λ / 4 , 5 λ / 4 , . . . (\displaystyle x=\lambda /4,3\lambda /4,5\lambda /4,...), then the distance between neighboring modes / antimodes will be equal to half the wavelength

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



and reflected wave equations


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

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

where λ is the length of the traveling wave.

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

The content of the article

SOUND AND ACOUSTICS. Sound is vibrations, i.e. periodic mechanical perturbation in elastic media - gaseous, liquid and solid. Such a perturbation, which is some physical change in the medium (for example, a change in density or pressure, displacement of particles), propagates in it in the form of a sound wave. The field of physics that deals with the origin, propagation, reception and processing of sound waves is called acoustics. A sound may be inaudible if its frequency lies beyond the sensitivity of the human ear, or if it propagates in a medium such as a solid that cannot have direct contact with the ear, or if its energy is rapidly dissipated in the medium. Thus, the usual process of sound perception for us is only one side of acoustics.

SOUND WAVES

Consider a long pipe filled with air. From the left end, a piston tightly attached to the walls is inserted into it (Fig. 1). If the piston is sharply moved to the right and stopped, then the air in its immediate vicinity will be compressed for a moment (Fig. 1, a). Then the compressed air will expand, pushing the air adjacent to it on the right, and the area of ​​compression, which initially appeared near the piston, will move through the pipe at a constant speed (Fig. 1, b). This compression wave is the sound wave in the gas.

A sound wave in a gas is characterized by excess pressure, excess density, displacement of particles and their speed. For sound waves, these deviations from the equilibrium values ​​are always small. Thus, the excess pressure associated with the wave is much less than the static pressure of the gas. Otherwise, we are dealing with another phenomenon - a shock wave. In a sound wave corresponding to ordinary speech, the excess pressure is only about one millionth of atmospheric pressure.

It is important that the substance is not carried away by the sound wave. A wave is only a temporary perturbation passing through the air, after which the air returns to an equilibrium state.

Wave motion, of course, is not unique to sound: light and radio signals travel in the form of waves, and everyone is familiar with waves on the surface of water. All types of waves are mathematically described by the so-called wave equation.

harmonic waves.

The wave in the pipe in Fig. 1 is called a sound pulse. A very important type of wave is generated when the piston vibrates back and forth like a weight suspended from a spring. Such oscillations are called simple harmonic or sinusoidal, and the wave excited in this case is called harmonic.

With simple harmonic vibrations, the movement is periodically repeated. The time interval between two identical states of motion is called the oscillation period, and the number of complete periods per second is called the oscillation frequency. Let's denote the period by T, and the frequency through f; then one can write that f= 1/T. If, for example, the frequency is 50 periods per second (50 Hz), then the period is 1/50 of a second.

Mathematically simple harmonic oscillations are described by a simple function. Piston displacement with simple harmonic oscillations for any moment of time t can be written in the form

Here d- displacement of the piston from the equilibrium position, and D is a constant multiplier, which is equal to the maximum value of the quantity d and is called the displacement amplitude.

Assume that the piston oscillates according to the harmonic oscillation formula. Then, when it moves to the right, compression occurs, as before, and when moving to the left, the pressure and density will decrease relative to their equilibrium values. There is not compression, but rarefaction of the gas. In this case, the right will propagate, as shown in Fig. 2, a wave of alternating compressions and rarefactions. At each moment in time, the pressure distribution curve along the length of the pipe will have the form of a sinusoid, and this sinusoid will move to the right at the speed of sound v. The distance along the pipe between the same wave phases (for example, between adjacent maxima) is called the wavelength. It is commonly referred to Greek letter l(lambda). Wavelength l is the distance traveled by the wave in time T. That's why l = TV, or v = lf.

Longitudinal and transverse waves.

If the particles oscillate parallel to the direction of wave propagation, then the wave is called longitudinal. If they oscillate perpendicular to the direction of propagation, then the wave is called transverse. Sound waves in gases and liquids are longitudinal. In solids, there are waves of both types. A transverse wave in a solid is possible due to its rigidity (resistance to shape change).

The most significant difference between these two types of waves is that a shear wave has the property polarization(oscillations occur in a certain plane), but the longitudinal one does not. In some phenomena, such as the reflection and transmission of sound through crystals, much depends on the direction of particle displacement, just as in the case of light waves.

The speed of sound waves.

The speed of sound is a characteristic of the medium in which the wave propagates. It is determined by two factors: elasticity and density of the material. The elastic properties of solids depend on the type of deformation. So, elastic properties metal rod are not the same in torsion, compression and bending. And the corresponding wave oscillations propagate at different speeds.

An elastic medium is one in which the deformation, be it torsion, compression, or bending, is proportional to the force causing the deformation. Such materials are subject to Hooke's law:

Voltage = Cґ Relative deformation,

where FROM is the modulus of elasticity, depending on the material and type of deformation.

Sound speed v for of this type elastic deformation is given by the expression

where r is the density of the material (mass per unit volume).

The speed of sound in a solid rod.

A long rod can be stretched or compressed by force applied to the end. Let the length of the rod be L applied tensile force F, and the increase in length is D L. Value D L/L we will call the relative deformation, and the force per unit area of ​​the cross section of the rod will be called the stress. So the voltage is F/A, where BUT - sectional area of ​​the rod. As applied to such a rod, Hooke's law has the form

where Y is Young's modulus, i.e. modulus of elasticity of the rod for tension or compression, which characterizes the material of the rod. Young's modulus is low for easily tensile materials such as rubber and high for rigid materials such as steel.

If we now excite a compression wave in it by hitting the end of the rod with a hammer, then it will propagate with a speed , where r, as before, is the density of the material from which the rod is made. Wave velocities for some typical materials are given in table. one.

Table 1. VELOCITY OF SOUND FOR DIFFERENT TYPES OF WAVES IN SOLID MATERIALS
Material	Longitudinal waves in extended solid samples (m/s)	Shear and torsion waves (m/s)	Compression waves in rods (m/s)
Aluminum
Brass
Lead
Iron
Silver
Stainless steel
Flintglass
Crown glass
plexiglass
Polyethylene
Polystyrene

The considered wave in the rod is a compression wave. But it cannot be considered strictly longitudinal, since the movement of the lateral surface of the rod is associated with compression (Fig. 3, a).

Two other types of waves are also possible in the rod - a bending wave (Fig. 3, b) and a torsion wave (Fig. 3, in). Bending deformations correspond to a wave that is neither purely longitudinal nor purely transverse. Torsion deformations, i.e. rotation around the axis of the rod, give a purely transverse wave.

The speed of a bending wave in a rod depends on the wavelength. Such a wave is called "dispersive".

The torsion waves in the rod are purely transverse and non-dispersive. Their speed is given by the formula

where m is the shear modulus characterizing the elastic properties of the material with respect to shear. Some typical shear wave velocities are given in Table 1. one.

Velocity in extended solid media.

In solid media of large volume, where the influence of boundaries can be neglected, two types of elastic waves are possible: longitudinal and transverse.

The deformation in a longitudinal wave is a plane deformation, i.e. one-dimensional compression (or rarefaction) in the direction of wave propagation. The deformation corresponding to a transverse wave is a shear displacement perpendicular to the direction of wave propagation.

The velocity of longitudinal waves in solid materials is given by the expression

where C-L- modulus of elasticity for simple plane deformation. It is related to the bulk modulus AT(which is defined below) and the shear modulus m of the material as C L = B + 4/3m . In table. 1 shows the values ​​of the velocities of longitudinal waves for various solid materials.

The speed of shear waves in extended solid media is the same as the speed of torsion waves in a rod of the same material. Therefore, it is given by the expression . Its values ​​for conventional solid materials are given in Table. one.

speed in gases.

In gases, only one type of deformation is possible: compression - rarefaction. Corresponding modulus of elasticity AT is called the bulk modulus. It is determined by the relation

-D P = B(D V/V).

Here D P– pressure change, D V/V is the relative change in volume. The minus sign indicates that as pressure increases, volume decreases.

Value AT depends on whether or not the temperature of the gas changes during compression. In the case of a sound wave, it can be shown that the pressure changes very quickly and the heat released during compression does not have time to leave the system. Thus, the change in pressure in the sound wave occurs without heat exchange with the surrounding particles. Such a change is called adiabatic. It has been established that the speed of sound in a gas depends only on temperature. At a given temperature, the speed of sound is approximately the same for all gases. At a temperature of 21.1 ° C, the speed of sound in dry air is 344.4 m / s and increases with increasing temperature.

Velocity in liquids.

Sound waves in liquids are waves of compression - rarefaction, as in gases. Speed ​​is given by the same formula. However, a liquid is much less compressible than a gas, and therefore the quantity AT, more and density r. The speed of sound in liquids is closer to the speed in solids than in gases. It is much smaller than in gases and depends on temperature. For example, the speed in fresh water is 1460 m/s at 15.6°C. sea ​​water normal salinity at the same temperature is 1504 m/s. The speed of sound increases with increasing water temperature and salt concentration.

standing waves.

When a harmonic wave is excited in a confined space so that it bounces off boundaries, so-called standing waves occur. A standing wave is the result of the superposition of two waves traveling one in a straight line and the other in a straight line. reverse direction. There is a pattern of oscillations that does not move in space, with alternating antinodes and nodes. At the antinodes, the deviations of the oscillating particles from their equilibrium positions are maximum, and at the nodes they are equal to zero.

Standing waves in a string.

In a stretched string, transverse waves arise, and the string is displaced relative to its original, rectilinear position. When photographing waves in a string, the nodes and antinodes of the fundamental tone and overtones are clearly visible.

The picture of standing waves greatly facilitates the analysis of oscillatory motions of a string of a given length. Let there be a string of length L attached at the ends. Any kind of vibration of such a string can be represented as a combination of standing waves. Since the ends of the string are fixed, only such standing waves are possible that have nodes at the boundary points. The lowest frequency of vibration of a string corresponds to the maximum possible wavelength. Since the distance between nodes is l/2, the frequency is minimal when the string length is equal to half the wavelength, i.e. at l= 2L. This is the so-called fundamental mode of string vibration. Its corresponding frequency, called the fundamental frequency or fundamental tone, is given by f = v/2L, where v is the speed of wave propagation along the string.

There is a whole sequence of higher frequency oscillations that correspond to standing waves with more nodes. The next higher frequency, which is called the second harmonic or first overtone, is given by

f = v/L.

The sequence of harmonics is expressed by the formula f = nv/2L, where n= 1, 2, 3, etc. This is the so-called. eigenfrequencies of the string vibrations. They increase in proportion to the natural numbers: higher harmonics in 2, 3, 4...etc. times the fundamental frequency. Such a series of sounds is called the natural or harmonic scale.

All this is of great importance in musical acoustics, which will be discussed in more detail below. For now, we note that the sound produced by a string contains all natural frequencies. The relative contribution of each of them depends on the point at which the vibrations of the string are excited. If, for example, a string is plucked in the middle, then the fundamental frequency will be most excited, since this point corresponds to the antinode. The second harmonic will be absent, since its node is located in the center. The same can be said about other harmonics ( see below musical acoustics).

The speed of the waves in the string is

where T - string tension, and rL - mass per unit length of the string. Therefore, the natural frequency spectrum of the string is given by

Thus, an increase in string tension leads to an increase in vibration frequencies. To lower the frequency of oscillations at a given T you can, taking a heavier string (large r L) or increasing its length.

Standing waves in organ pipes.

The theory stated in relation to a string can also be applied to air vibrations in an organ-type pipe. An organ pipe can be simplistically viewed as a straight pipe in which standing waves are excited. The pipe can have both closed and open ends. An antinode of a standing wave occurs at the open end, and a knot occurs at the closed end. Therefore, a pipe with two open ends has a fundamental frequency at which half the wavelength fits along the length of the pipe. A pipe, on the other hand, in which one end is open and the other is closed, has a fundamental frequency at which a quarter of the wavelength fits along the length of the pipe. Thus, the fundamental frequency for a pipe open at both ends is f =v/2L, and for a pipe open at one end, f = v/4L(where L is the length of the pipe). In the first case, the result is the same as for the string: the overtones are double, triple, and so on. value of the fundamental frequency. However, for a pipe open at one end, the overtones will be greater than the fundamental frequency by 3, 5, 7, etc. once.

On fig. Figures 4 and 5 schematically show the standing waves of the fundamental frequency and the first overtone for the pipes of the two considered types. For reasons of convenience, the offsets are shown here as transverse, but in fact they are longitudinal.

resonant oscillations.

Standing waves are closely related to the phenomenon of resonance. The natural frequencies discussed above are also the resonant frequencies of a string or organ pipe. Suppose that a loudspeaker is placed near the open end of the organ pipe, emitting a signal of one specific frequency, which can be changed at will. Then, if the frequency of the loudspeaker signal coincides with the main frequency of the pipe or with one of its overtones, the pipe will sound very loud. This is because the loudspeaker excites vibrations of the air column with a significant amplitude. The trumpet is said to resonate under these conditions.

Fourier analysis and frequency spectrum of sound.

In practice, sound waves of a single frequency are rare. But complex sound waves can be decomposed into harmonics. This method is called Fourier analysis after the French mathematician J. Fourier (1768–1830), who was the first to apply it (in the theory of heat).

A graph of the relative energy of sound vibrations versus frequency is called the frequency spectrum of sound. There are two main types of such spectra: discrete and continuous. The discrete spectrum consists of separate lines for frequencies separated by empty spaces. All frequencies are present in the continuous spectrum within its band.

Periodic sound vibrations.

Sound vibrations are periodic if the oscillatory process, no matter how complex it may be, is repeated after a certain time interval. Its spectrum is always discrete and consists of harmonics of a certain frequency. Hence the term "harmonic analysis". An example is rectangular oscillations (Fig. 6, a) with a change in amplitude from +A before - BUT and period T= 1/f. Another simple example is the triangular sawtooth oscillation shown in Fig. 6, b. An example of periodic fluctuations over complex shape with the corresponding harmonic components is shown in fig. 7.

Musical sounds are periodic vibrations and therefore contain harmonics (overtones). We have already seen that in a string, along with oscillations of the fundamental frequency, other harmonics are excited to one degree or another. The relative contribution of each overtone depends on the way the string is excited. The set of overtones is largely determined by timbre musical sound. These issues are discussed in more detail below in the section on musical acoustics.

The spectrum of a sound pulse.

The usual variety of sound is the sound of short duration: clapping hands, knocking on the door, the sound of an object falling on the floor, cuckoo cuckooing. Such sounds are neither periodic nor musical. But they can also be decomposed into a frequency spectrum. In this case, the spectrum will be continuous: to describe the sound, all frequencies are needed within a certain band, which can be quite wide. Knowing such a frequency spectrum is necessary to reproduce such sounds without distortion, since the corresponding electronic system should equally well "pass" all these frequencies.

The main features of a sound pulse can be clarified by considering a pulse of a simple form. Let us assume that the sound is an oscillation of duration D t, at which the change in pressure is as shown in Fig. eight, a. An approximate frequency spectrum for this case is shown in Fig. eight, b. The center frequency corresponds to the vibrations that we would have if the same signal were extended indefinitely.

The length of the frequency spectrum is called the bandwidth D f(Fig. 8, b). Bandwidth is the approximate range of frequencies needed to reproduce the original pulse without excessive distortion. There is a very simple fundamental relation between D f and D t, namely

D f D t" one.

This relationship is valid for all sound pulses. Its meaning is that the shorter the pulse, the more frequencies it contains. Let us assume that a sonar is used to detect a submarine, emitting ultrasound in the form of a pulse with a duration of 0.0005 s and a signal frequency of 30 kHz. The bandwidth is 1/0.0005 = 2 kHz, and the frequencies actually contained in the spectrum of the locator pulse lie in the range from 29 to 31 kHz.

Noise.

Noise refers to any sound produced by multiple, uncoordinated sources. An example is the sound of tree leaves being swayed by the wind. Jet engine noise is due to the turbulence of the high-velocity exhaust stream. Noise as an annoying sound is considered in Art. ACOUSTIC POLLUTION OF THE ENVIRONMENT.

Sound intensity.

Sound volume may vary. It is easy to see that this is due to the energy carried by the sound wave. For quantitative comparisons of loudness, it is necessary to introduce the concept of sound intensity. The intensity of a sound wave is defined as the average energy flux through a unit area of ​​the wave front per unit time. In other words, if we take a single area (for example, 1 cm 2), which would completely absorb sound, and place it perpendicular to the direction of wave propagation, then the sound intensity is equal to the acoustic energy absorbed in one second. The intensity is usually expressed in W/cm2 (or W/m2).

We give the value of this value for some familiar sounds. Amplitude overpressure that occurs during a normal conversation is approximately one millionth of atmospheric pressure, which corresponds to an acoustic intensity of the sound of the order of 10 -9 W / cm 2. The total power of the sound emitted during a normal conversation is on the order of only 0.00001 watts. The ability of the human ear to perceive such small energies testifies to its amazing sensitivity.

The range of sound intensities perceived by our ear is very wide. The intensity of the loudest sound that the ear can bear is about 1014 times the minimum that it can hear. The full power of sound sources covers an equally wide range. Thus, the power emitted during a very quiet whisper can be of the order of 10–9 W, while the power emitted by a jet engine reaches 10–5 W. Again, the intensities differ by a factor of 10 14.

Decibel.

Since sounds vary so much in intensity, it is more convenient to think of it as a logarithmic value and measure it in decibels. The logarithmic value of the intensity is the logarithm of the ratio of the considered value of the quantity to its value, taken as the original. Intensity level J with respect to some conditionally chosen intensity J 0 is

Sound intensity level = 10 lg ( J/J 0) dB.

Thus, one sound that is 20 dB more intense than another is 100 times more intense.

In the practice of acoustic measurements, it is customary to express the sound intensity in terms of the corresponding overpressure amplitude P e. When pressure is measured in decibels relative to some conventionally selected pressure R 0 , get the so-called sound pressure level. Since the sound intensity is proportional to the magnitude P e 2 , and lg( P e 2) = 2lg P e, the sound pressure level is determined as follows:

Sound pressure level = 20 lg ( P e/P 0) dB.

Nominal pressure R 0 = 2×10–5 Pa corresponds to the standard hearing threshold for sound with a frequency of 1 kHz. In table. 2 shows sound pressure levels for some common sound sources. These are integral values ​​obtained by averaging over the entire audible frequency range.

Table 2. TYPICAL SOUND PRESSURE LEVELS
Sound source	Sound pressure level, dB (rel. 2H 10–5 Pa)
stamping shop
Engine room on board
Spinning and weaving shop
In a subway car
In a car while driving in traffic
Typewriting Bureau
Accounting
Office
living quarters
Residential area at night
broadcasting studio

Volume.

The sound pressure level is not associated with a simple relationship with the psychological perception of loudness. The first of these factors is objective, and the second is subjective. Experiments show that the perception of loudness depends not only on the intensity of the sound, but also on its frequency and experimental conditions.

The volumes of sounds that are not tied to the conditions of comparison cannot be compared. Still, the comparison of pure tones is of interest. To do this, determine the sound pressure level at which a given tone is perceived as equally loud as a standard tone with a frequency of 1000 Hz. On fig. 9 shows equal loudness curves obtained in the experiments of Fletcher and Manson. For each curve, the corresponding sound pressure level of a standard tone of 1000 Hz is indicated. For example, at a tone frequency of 200 Hz, a sound level of 60 dB is needed to be perceived as equal to a tone of 1000 Hz with a sound pressure level of 50 dB.

These curves are used to define the hum, a unit of loudness that is also measured in decibels. The background is the sound volume level for which the sound pressure level of an equally loud standard pure tone (1000 Hz) is 1 dB. So, a sound with a frequency of 200 Hz at a level of 60 dB has a volume level of 50 phons.

The lower curve in fig. 9 is the hearing threshold curve of a good ear. The range of audible frequencies extends from about 20 to 20,000 Hz.

Propagation of sound waves.

Like the waves from a pebble thrown into still water, sound waves propagate in all directions. It is convenient to characterize such a propagation process as a wave front. A wave front is a surface in space, at all points of which oscillations occur in the same phase. Wave fronts from a pebble that has fallen into the water are circles.

Flat waves.

The wave front of the simplest form is flat. A plane wave propagates in only one direction and is an idealization that is only approximately realized in practice. A sound wave in a pipe can be considered approximately flat, just like a spherical wave at a great distance from the source.

spherical waves.

To simple types waves can also be attributed to a wave with a spherical front, emanating from a point and propagating in all directions. Such a wave can be excited using a small pulsating sphere. A source that excites a spherical wave is called a point source. The intensity of such a wave decreases as it propagates, as the energy is distributed over a sphere of ever larger radius.

If a point source producing a spherical wave radiates a power of 4 pQ, then, since the surface area of ​​a sphere with a radius r equals 4 p r 2 , the sound intensity in a spherical wave is equal to

J = Q/r 2 ,

where r is the distance from the source. Thus, the intensity of a spherical wave decreases inversely with the square of the distance from the source.

The intensity of any sound wave during its propagation decreases due to the absorption of sound. This phenomenon will be discussed below.

Huygens principle.

The Huygens principle is valid for wave front propagation. To clarify it, let us consider the shape of the wave front known to us at some point in time. It can be found even after a while D t, if each point of the initial wave front is considered as a source of an elementary spherical wave propagating over this interval to a distance v D t. The envelope of all these elementary spherical wave fronts will be the new wave front. Huygens' principle makes it possible to determine the shape of the wavefront throughout the propagation process. It also implies that waves, both plane and spherical, retain their geometry during propagation, provided that the medium is homogeneous.

sound diffraction.

Diffraction is the wave bending around an obstacle. Diffraction is analyzed using the Huygens principle. The degree of this bending depends on the relationship between the wavelength and the size of the obstacle or hole. Since the wavelength of a sound wave is many times longer than that of light, the diffraction of sound waves surprises us less than the diffraction of light. So, you can talk to someone standing around the corner of the building, although he is not visible. The sound wave easily bends around the corner, while the light, due to the smallness of its wavelength, creates sharp shadows.

Consider the diffraction of a plane sound wave incident on a solid flat screen with a hole. To determine the shape of the wavefront on the other side of the screen, you need to know the relationship between the wavelength l and hole diameter D. If these values ​​are approximately the same or l a lot more D, then complete diffraction is obtained: the wavefront of the outgoing wave will be spherical, and the wave will reach all points behind the screen. If l somewhat less D, then the outgoing wave will propagate predominantly in the forward direction. And finally, if l much less D, then all its energy will propagate in a straight line. These cases are shown in Fig. ten.

Diffraction is also observed when there is an obstacle in the path of sound. If the dimensions of the obstacle are much larger than the wavelength, then the sound is reflected, and an acoustic shadow zone is formed behind the obstacle. When the size of the obstacle is comparable to or less than the wavelength, the sound diffracts to some extent in all directions. This is taken into account in architectural acoustics. So, for example, sometimes the walls of a building are covered with protrusions with dimensions of the order of the wavelength of sound. (At a frequency of 100 Hz, the wavelength in air is about 3.5 m.) In this case, the sound, falling on the walls, is scattered in all directions. In architectural acoustics, this phenomenon is called sound diffusion.

Reflection and transmission of sound.

When a sound wave traveling in one medium is incident on an interface with another medium, three processes can occur simultaneously. The wave can be reflected from the interface, it can pass into another medium without changing direction, or it can change direction at the interface, i.e. refract. On fig. 11 shown simplest case when a plane wave is incident at right angles to a flat surface separating two different substances. If the intensity reflection coefficient, which determines the proportion of reflected energy, is equal to R, then the transmission coefficient will be equal to T = 1 – R.

For a sound wave, the ratio of excess pressure to vibrational volumetric velocity is called acoustic impedance. The reflection and transmission coefficients depend on the ratio of the wave impedances of the two media, the wave impedances, in turn, are proportional to the acoustic impedances. The wave resistance of gases is much less than that of liquids and solids. So if a wave in the air hits a thick solid object or the surface of deep water, the sound is almost completely reflected. For example, for the boundary of air and water, the ratio of wave resistances is 0.0003. Accordingly, the energy of sound passing from air into water is equal to only 0.12% of the incident energy. The reflection and transmission coefficients are reversible: the reflection coefficient is the transmission coefficient in the opposite direction. Thus, sound practically does not penetrate either from the air into the water basin, or from under the water to the outside, which is well known to everyone who swam under water.

In the case of reflection considered above, it was assumed that the thickness of the second medium in the direction of wave propagation is large. But the transmission coefficient will be significantly greater if the second medium is a wall separating two identical media, such as a solid partition between rooms. The fact is that the wall thickness is usually less than the wavelength of sound or comparable to it. If the wall thickness is a multiple of half the wavelength of sound in the wall, then the transmission coefficient of the wave at perpendicular incidence is very large. The baffle would be absolutely transparent to the sound of this frequency if it were not for absorption, which we neglect here. If the wall thickness is much less than the wavelength of sound in it, then the reflection is always small, and the transmission is large, unless special measures are taken to increase the absorption of sound.

refraction of sound.

When a plane sound wave is incident at an angle on an interface, the angle of its reflection is equal to the angle of incidence. The transmitted wave deviates from the direction of the incident wave if the angle of incidence is different from 90°. This change in the direction of the wave is called refraction. The geometry of refraction at a flat boundary is shown in Fig. . 12. The angles between the direction of the waves and the normal to the surface are indicated q 1 for the incident wave and q 2 - for the refracted past. The relation between these two angles includes only the ratio of the speeds of sound for the two media. As in the case of light waves, these angles are related to each other by the Snell (Snell) law:


Thus, if the speed of sound in the second medium is less than in the first, then the angle of refraction will be less than the angle of incidence; if the speed in the second medium is greater, then the angle of refraction will be greater than the angle of incidence.

Refraction due to temperature gradient.

If the speed of sound in an inhomogeneous medium changes continuously from point to point, then the refraction also changes. Since the speed of sound in both air and water depends on temperature, in the presence of a temperature gradient, sound waves can change their direction of movement. In the atmosphere and ocean, due to horizontal stratification, vertical temperature gradients are commonly observed. Therefore, due to changes in the speed of sound along the vertical, due to temperature gradients, the sound wave can be deflected either up or down.

Let us consider the case when the air is warmer in some place near the Earth's surface than in the higher layers. Then, as the altitude increases, the air temperature here decreases, and with it, the speed of sound also decreases. Sound emitted by a source near the surface of the Earth will go up due to refraction. This is shown in fig. 13, which shows sound "beams".

The deflection of the sound rays shown in fig. 13, in general form described by Snell's law. If through q, as before, denote the angle between the vertical and the direction of radiation, then the generalized Snell's law has the form of equality sin q/v= const referring to any point of the beam. Thus, if the beam passes into the region where the speed v decreases, then the angle q should also decrease. Therefore, sound beams are always deflected in the direction of decreasing sound speed.

From fig. 13 it can be seen that there is a region located at some distance from the source, where sound rays do not penetrate at all. This is the so-called zone of silence.

It is quite possible that somewhere at a height greater than that shown in Fig. 13, due to the temperature gradient, the speed of sound increases with height. In this case, the initially deviated upward sound wave will deviate here to the Earth's surface at a great distance. This happens when a layer of temperature inversion forms in the atmosphere, as a result of which it becomes possible to receive ultra-long-range sound signals. At the same time, the reception quality at remote points is even better than near. There have been many examples of ultra-long-range reception in history. For example, during the First World War, when atmospheric conditions favored appropriate sound refraction, cannonades on the French front could be heard in England.

Refraction of sound under water.

Sound refraction due to vertical temperature changes is also observed in the ocean. If the temperature, and therefore the speed of sound, decreases with depth, the sound rays are deflected downward, resulting in a zone of silence similar to that shown in Fig. 13 for atmosphere. For the ocean, the corresponding picture will turn out if this picture is simply turned over.

The presence of silence zones makes it difficult to detect submarines with sonar, and refraction, which deflects sound waves downward, significantly limits their propagation range near the surface. However, upward deflection is also observed. She can create more favorable conditions for hydrolocation.

Interference of sound waves.

The superposition of two or more waves is called wave interference.

Standing waves as a result of interference.

The above standing waves are a special case of interference. Standing waves are formed as a result of the superposition of two waves of the same amplitude, phase and frequency, propagating in opposite directions.

The amplitude at the antinodes of a standing wave is equal to twice the amplitude of each of the waves. Since the intensity of the wave is proportional to the square of its amplitude, this means that the intensity at the antinodes is 4 times greater than the intensity of each of the waves, or 2 times greater than the total intensity of the two waves. There is no violation of the law of conservation of energy here, since the intensity at the nodes is zero.

beats.

Interference of harmonic waves of different frequencies is also possible. When two frequencies differ little, so-called beats occur. Beats are changes in the amplitude of sound that occur at a frequency equal to the difference between the original frequencies. On fig. 14 shows the beat waveform.

It should be borne in mind that the beat frequency is the frequency of the amplitude modulation of the sound. Also, beats should not be confused with the difference frequency resulting from the distortion of a harmonic signal.

Beats are often used when tuning two tones in unison. The frequency is adjusted until the beats are no longer audible. Even if the beat frequency is very low, the human ear is able to pick up the periodic rise and fall in the volume of the sound. Therefore, beats are a very sensitive tuning method in the audio range. If the setting is not accurate, then the frequency difference can be determined by ear by counting the number of beats in one second. In music, beats of higher harmonic components are also perceived by ear, which is used when tuning the piano.

Absorption of sound waves.

The intensity of sound waves in the process of their propagation always decreases due to the fact that a certain part of the acoustic energy is scattered. Due to the processes of heat transfer, intermolecular interaction and internal friction, sound waves are absorbed in any medium. The intensity of absorption depends on the frequency of the sound wave and on other factors such as pressure and temperature of the medium.

The absorption of a wave in a medium is quantitatively characterized by the absorption coefficient a. It shows how quickly the excess pressure decreases depending on the distance traveled by the propagating wave. Decreasing amplitude of overpressure –D P e when passing distance D X proportional to the amplitude of the initial overpressure P e and distance D X. In this way,

-D P e = a P e D x.

For example, when we say that the absorption loss is 1 dB/m, this means that at a distance of 50 m the sound pressure level is reduced by 50 dB.

Absorption due to internal friction and heat conduction.

During the movement of particles associated with the propagation of a sound wave, friction between different particles environment. In liquids and gases, this friction is called viscosity. Viscosity, which determines the irreversible conversion of acoustic wave energy into heat, is main reason sound absorption in gases and liquids.

In addition, absorption in gases and liquids is due to heat loss during compression in the wave. We have already said that during the passage of the wave, the gas in the compression phase heats up. In this fast-flowing process, heat usually does not have time to be transferred to other regions of the gas or to the walls of the vessel. But in reality, this process is not ideal, and part of the released thermal energy leaves the system. Associated with this is sound absorption due to heat conduction. Such absorption occurs in compression waves in gases, liquids and solids.

Sound absorption, due to both viscosity and thermal conductivity, generally increases with the square of the frequency. Thus, high frequency sounds are absorbed much more strongly than low frequency sounds. For example, when normal pressure and temperature, the absorption coefficient (due to both mechanisms) at a frequency of 5 kHz in air is about 3 dB/km. Since the absorption is proportional to the square of the frequency, the absorption coefficient at 50 kHz is 300 dB/km.

Absorption in solids.

The mechanism of sound absorption due to thermal conductivity and viscosity, which takes place in gases and liquids, is also preserved in solids. However, here new absorption mechanisms are added to it. They are associated with defects in the structure of solids. The point is that polycrystalline hard materials consist of small crystallites; when sound passes through them, deformations occur, leading to the absorption of sound energy. Sound is also scattered at the boundaries of crystallites. In addition, even single crystals contain dislocation-type defects that contribute to sound absorption. Dislocations are violations of the coordination of atomic planes. When the sound wave causes the atoms to vibrate, the dislocations move and then return to their original position, dissipating energy due to internal friction.

Absorption due to dislocations explains, in particular, why the lead bell does not ring. Lead is a soft metal with a lot of dislocations, and therefore sound vibrations in it decay extremely quickly. But it will ring well if it is cooled with liquid air. At low temperatures, dislocations are "frozen" in a fixed position, and therefore do not move and do not convert sound energy into heat.

MUSICAL ACOUSTICS

Musical sounds.

Musical acoustics studies the features of musical sounds, their characteristics related to how we perceive them, and the mechanisms of the sound of musical instruments.

Musical sound or tone is a periodic sound, i.e. fluctuations that repeat over and over again after a certain period. It was said above that periodic sound can be represented as the sum of oscillations with frequencies that are multiples of the fundamental frequency f: 2f, 3f, 4f etc. It was also noted that vibrating strings and columns of air emit musical sounds.

Musical sounds are distinguished by three characteristics: loudness, pitch and timbre. All these indicators are subjective, but they can be associated with the measured values. Loudness is related mainly to the intensity of the sound; the pitch of the sound, which characterizes its position in the musical system, is determined by the frequency of the tone; the timbre, by which one instrument or voice differs from another, is characterized by the distribution of energy over the harmonics and the change in this distribution over time.

Sound pitch.

The pitch of a musical sound is closely related to the frequency, but not identical to it, since the assessment of the pitch is subjective.

So, for example, it was found that the estimate of the pitch of a single-frequency sound somewhat depends on the level of its loudness. With a significant increase in volume, say 40 dB, the apparent frequency can decrease by 10%. In practice, this dependence on loudness does not matter, since musical sounds are much more complex than single-frequency sound.

On the question of the relationship between pitch and frequency, something else is more significant: if musical sounds are made up of harmonics, then with what frequency is the perceived pitch associated? It turns out that this may not be the frequency that corresponds to the maximum energy, and not the lowest frequency in the spectrum. So, for example, a musical sound consisting of a set of frequencies of 200, 300, 400 and 500 Hz is perceived as a sound with a height of 100 Hz. That is, the pitch is associated with the fundamental frequency of the harmonic series, even if it is not in the spectrum of the sound. True, most often the fundamental frequency is present to some extent in the spectrum.

Speaking about the relationship between the pitch and its frequency, one should not forget about the features of the human hearing organ. This is a special acoustic receiver that introduces its own distortions (not to mention the fact that there are psychological and subjective aspects of hearing). The ear is able to select some frequencies, in addition, the sound wave undergoes non-linear distortions in it. Frequency selectivity is due to the difference between the loudness of the sound and its intensity (Fig. 9). It is more difficult to explain non-linear distortions, which are expressed in the appearance of frequencies that are absent in the original signal. The non-linearity of the ear reaction is due to the asymmetry of the movement of its various elements.

One of the characteristic features of a nonlinear receiving system is that when it is excited by sound with a frequency f 1 harmonic overtones are excited in it 2 f 1 , 3f 1 ,..., and in some cases also subharmonics of type 1/2 f one . In addition, when a nonlinear system is excited by two frequencies f 1 and f 2, the sum and difference frequencies are excited in it f 1 + f 2 and f 1 - f 2. The greater the amplitude of the initial oscillations, the greater the contribution of "extra" frequencies.

Thus, due to the non-linearity of the acoustic characteristics of the ear, frequencies that are absent in the sound may appear. Such frequencies are called subjective tones. Let's assume that the sound consists of pure tones with frequencies of 200 and 250 Hz. Due to the non-linearity of the response, additional frequencies will appear 250 - 200 = 50, 250 + 200 = 450, 2' 200 = 400, 2' 250 = 500 Hz, etc. It will seem to the listener that there is a whole set of combination frequencies in the sound, but their appearance is actually due to the non-linear response of the ear. When a musical sound consists of a fundamental frequency and its harmonics, it is obvious that the fundamental frequency is effectively amplified by the difference frequencies.

True, studies have shown that subjective frequencies arise only at a sufficiently large amplitude of the original signal. Therefore, it is possible that in the past the role of subjective frequencies in music was greatly exaggerated.

Musical standards and measuring the pitch of musical sound.

In the history of music, sounds of different frequencies were taken as the main tone, which determines the entire musical structure. Now the generally accepted frequency for the note "la" of the first octave is 440 Hz. But in the past it has changed from 400 to 462 Hz.

The traditional way to determine the pitch of a sound is to compare it to the tone of a standard tuning fork. The deviation of the frequency of a given sound from the standard is judged by the presence of beats. Tuning forks are still used today, although now there are more convenient devices for determining the pitch, such as a stable frequency reference oscillator (with a quartz resonator), which can be smoothly tuned within the entire sound range. True, the exact calibration of such a device is quite difficult.

The stroboscopic method of measuring pitch is widely used, in which the sound of a musical instrument sets the frequency of flashes of a strobe lamp. The lamp illuminates a pattern on a disk rotating at a known frequency, and the fundamental frequency of the tone is determined from the apparent frequency of movement of the pattern on the disk under stroboscopic illumination.

The ear is very sensitive to pitch change, but its sensitivity depends on the frequency. It is maximum near the lower threshold of audibility. Even an untrained ear can detect only 0.3% difference in frequencies between 500 and 5000 Hz. Sensitivity can be increased by training. Musicians have a very developed sense of pitch, but this does not always help in determining the frequency of the pure tone produced by the reference oscillator. This suggests that when determining the frequency of a sound by ear, its timbre plays an important role.

Timbre.

Timbre refers to those features of musical sounds that give musical instruments and voices their unique specificity, even if we compare sounds of the same pitch and loudness. This is, so to speak, the sound quality.

The timbre depends on the frequency spectrum of the sound and its change over time. It is determined by several factors: the distribution of energy over overtones, the frequencies that arise at the moment the sound appears or stops (the so-called transitional tones) and their decay, as well as the slow amplitude and frequency modulation of the sound (“vibrato”).

overtone intensity.

Consider a stretched string, which is excited by a pinch in its middle part (Fig. 15, a). Since all even harmonics have nodes in the middle, they will be absent, and the oscillations will consist of odd harmonics of the fundamental frequency equal to f 1 = v/2l, where v- the speed of the wave in the string, and l is its length. Thus, only frequencies will be present f 1 , 3f 1 , 5f 1 etc. The relative amplitudes of these harmonics are shown in Figs. fifteen, b.

This example allows us to draw the following important general conclusion. The set of harmonics of a resonant system is determined by its configuration, and the distribution of energy over harmonics depends on the method of excitation. When the string is excited in its middle, the fundamental frequency dominates and the even harmonics are completely suppressed. If the string is fixed in its middle part and plucked in some other place, then the fundamental frequency and odd harmonics will be suppressed.

All of this applies to other well-known musical instruments, although the details can be very different. Instruments usually have an air cavity, soundboard, or horn to emit sound. All this determines the structure of overtones and the appearance of formants.

Formants.

As mentioned above, the sound quality of musical instruments depends on the distribution of energy among the harmonics. When changing the pitch of many instruments, and especially the human voice, the distribution of harmonics changes so that the main overtones are always located in approximately the same frequency range, which is called the formant range. One of the reasons for the existence of formants is the use of resonant elements to amplify sound, such as soundboards and air resonators. The width of natural resonances is usually large, due to which the radiation efficiency at the corresponding frequencies is higher. For brass instruments, the formants are determined by the bell from which the sound is emitted. The overtones that fall within the formant range are always strongly emphasized, as they are emitted with maximum energy. Formants largely determine the characteristic qualitative features of the sounds of a musical instrument or voice.

Changing tones over time.

The tone of the sound of any instrument rarely remains constant over time, and timbre is essentially related to this. Even when the instrument sustains a long note, there is a slight periodic modulation of frequency and amplitude, enriching the sound - "vibrato". This is especially true for stringed instruments such as the violin and for the human voice.

For many instruments, such as the piano, the duration of the sound is such that a constant tone does not have time to form - the excited sound quickly increases, and then its rapid decay follows. Since the decay of overtones is usually due to frequency-dependent effects (such as acoustic radiation), it is clear that the overtone distribution changes over the course of a tone.

The nature of the change in tone over time (the rate of rise and fall of the sound) for some instruments is schematically shown in Fig. 18. As you can see, stringed instruments (plucked and keyboards) have almost no constant tone. In such cases, it is possible to speak about the spectrum of overtones only conditionally, since the sound changes rapidly in time. The rise and fall characteristics are also an important part of the timbre of these instruments.

transitional tones.

The harmonic composition of a tone usually changes rapidly in a short time after sound excitation. In those instruments in which the sound is excited by striking the strings or plucking, the energy attributable to higher harmonics (as well as to numerous non-harmonic components) is maximum immediately after the sound begins, and after a fraction of a second these frequencies fade. Such sounds, called transitional, give a specific coloring to the sound of the instrument. In the piano, they are caused by the action of the hammer striking the string. Sometimes musical instruments with the same overtone structure can only be distinguished by transitional tones.

THE SOUND OF MUSICAL INSTRUMENTS

Musical sounds can be excited and changed different ways, in connection with which musical instruments are distinguished by a variety of forms. Tools for the most part were created and improved by the musicians themselves and by skilled craftsmen who did not resort to scientific theory. Therefore, acoustic science cannot explain, for example, why a violin has such a shape. However, it is quite possible to describe the sound properties of a violin in terms of general principles games on it and its designs.

The frequency range of an instrument is usually understood as the frequency range of its fundamental tones. The human voice covers about two octaves, and a musical instrument - at least three (a large organ - ten). In most cases, the overtones extend to the very edge of the audible sound range.

Musical instruments have three main parts: an oscillating element, a mechanism for its excitation, and an auxiliary resonator (horn or soundboard) for acoustic communication between the oscillating element and the surrounding air.

Musical sound is periodic in time, and periodic sounds are composed of a series of harmonics. Since the natural frequencies of the vibrations of strings and air columns of fixed length are harmonically related, in many instruments the main vibrating elements are strings and air columns. With a few exceptions (the flute is one of them), one-frequency sound cannot be taken on instruments. When the main vibrator is excited, a sound containing overtones arises. Some vibrators resonant frequencies are not harmonic components. Instruments of this kind (for example, drums and cymbals) are used in orchestral music for special expressiveness and emphasis on rhythm, but not for melodic development.

Stringed instruments.

By itself, a vibrating string is a poor emitter of sound, and therefore a stringed instrument must have an additional resonator to excite sound of noticeable intensity. It can be a closed volume of air, a deck, or a combination of both. The nature of the sound of the instrument is also determined by the way the strings are excited.

We saw earlier that the fundamental frequency of oscillation of a fixed string of length L is given by

where T is the tensile force of the string, and r L is the mass per unit length of the string. Therefore, we can change the frequency in three ways: by changing the length, tension, or mass. Many instruments use a small number of strings of the same length, the fundamental frequencies of which are determined by the proper choice of tension and mass. Other frequencies are obtained by shortening the length of the string with your fingers.

Other instruments, such as the piano, have one of many pre-tuned strings for each note. Tuning a piano where the frequency range is large is not an easy task, especially in the low frequency region. The tension force of all piano strings is almost the same (about 2 kN), and the variety of frequencies is achieved by changing the length and thickness of the strings.

A stringed instrument can be excited by a pluck (for example, on a harp or banjo), a blow (on a piano), or with a bow (in the case of musical instruments of the violin family). In all cases, as shown above, the number of harmonics and their amplitude depend on the way the string is excited.

piano.

A typical example of an instrument where the excitation of a string is produced by a blow is the pianoforte. The large soundboard of the instrument provides a wide range of formants, so its timbre is very uniform for any excited note. The maxima of the main formants occur at frequencies of the order of 400–500 Hz, and at lower frequencies the tones are especially rich in harmonics, and the amplitude of the fundamental frequency is less than that of some overtones. In the piano, the hammer strike on all but the shortest strings falls on a point located 1/7 of the length of the string from one of its ends. This is usually explained by the fact that in this case the seventh harmonic, which is dissonant with respect to the fundamental frequency, is significantly suppressed. But due to the finite width of the malleus, other harmonics located near the seventh are also suppressed.

Violin family.

In the violin family of instruments, long sounds are produced by a bow, which applies a variable driving force to the string, which keeps the string vibrating. Under the action of a moving bow, the string is pulled to the side due to friction until it breaks due to an increase in the tension force. Returning to its original position, it is again carried away by the bow. This process is repeated so that a periodic external force acts on the string.

In order of increasing size and decreasing frequency range, the main bowed string instruments are arranged as follows: violin, viola, cello, double bass. The frequency spectra of these instruments are especially rich in overtones, which undoubtedly gives a special warmth and expressiveness to their sound. In the violin family, the vibrating string is acoustically connected to the air cavity and the body of the instrument, which mainly determine the structure of the formants, which occupy a very wide frequency range. Large representatives of the violin family have a set of formants shifted towards low frequencies. Therefore, the same note taken on two instruments of the violin family acquires a different timbre coloration due to the difference in the structure of the overtones.

The violin has a pronounced resonance near 500 Hz, due to the shape of its body. When a note whose frequency is close to this value is played, an unwanted vibrating sound called "wolf tone" can be produced. The air cavity inside the violin body also has its own resonant frequencies, the main of which is located near 400 Hz. Due to its special shape, the violin has numerous closely spaced resonances. All of them, except for the wolf tone, do not really stand out in the general spectrum of the extracted sound.

Wind instruments.

Woodwind instruments.

Natural vibrations of air in a cylindrical pipe of finite length were discussed earlier. Natural frequencies form a series of harmonics, the fundamental frequency of which is inversely proportional to the length of the pipe. Musical sounds in wind instruments arise due to the resonant excitation of the air column.

Air vibrations are excited either by vibrations in the air jet falling on the sharp edge of the resonator wall, or by vibrations of the flexible surface of the tongue in the air flow. In both cases, periodic pressure changes occur in a localized area of ​​the tool barrel.

The first of these methods of excitation is based on the occurrence of "edge tones". When a stream of air comes out of the slot, broken by a wedge-shaped obstacle with a sharp edge, vortices periodically appear - first on one side, then on the other side of the wedge. The frequency of their formation is greater, the greater the speed of the air flow. If such a device is acoustically coupled to a resonating air column, then the edge tone frequency is “captured” by the resonant frequency of the air column, i.e. the frequency of vortex formation is determined by the air column. Under such conditions, the main frequency of the air column is excited only when the air flow velocity exceeds a certain minimum value. In a certain range of speeds exceeding this value, the frequency of the edge tone is equal to this fundamental frequency. At an even higher air flow velocity (near the one at which the edge frequency in the absence of communication with the resonator would be equal to the second harmonic of the resonator), the edge frequency doubles abruptly and the pitch emitted by the entire system turns out to be an octave higher. This is called overflow.

Edge tones excite air columns in instruments such as the organ, flute, and piccolo. When playing the flute, the performer excites the edge tones by blowing from the side into a side hole near one of the ends. Notes of one octave, starting from "D" and above, are obtained by changing the effective length of the barrel, opening the side holes, with a normal edge tone. Higher octaves are overblown.

Another way to excite the sound of a wind instrument is based on the periodic interruption of the air flow by an oscillating tongue, which is called a reed, since it is made of reeds. This method is used in various woodwind and brass instruments. There are options with a single reed (as, for example, in the clarinet, saxophone and accordion-type instruments) and with a symmetrical double reed (as, for example, in the oboe and bassoon). In both cases, the oscillatory process is the same: air is blown through a narrow gap, in which the pressure decreases in accordance with Bernoulli's law. At the same time, the cane is drawn into the gap and covers it. In the absence of flow, the elastic cane straightens and the process is repeated.

In wind instruments, the selection of the notes of the scale, as on the flute, is carried out by opening the side holes and overblow.

Unlike a pipe that is open at both ends, which has a full set of overtones, a pipe that is open at only one end has only odd harmonics ( cm. above). This is the configuration of the clarinet, and therefore even harmonics are weakly expressed in it. Overblowing in the clarinet occurs at a frequency 3 times higher than the main one.

In the oboe, the second harmonic is quite intense. It differs from the clarinet in that its bore has a conical shape, while in the clarinet the cross section of the bore is constant over most of its length. The frequencies in a conical barrel are more difficult to calculate than in a cylindrical pipe, but there is still a full range of overtones. In this case, the oscillation frequencies of a conical tube with a closed narrow end are the same as those of a cylindrical tube open at both ends.

Brass wind instruments.

Brass, including horn, trumpet, cornet-a-piston, trombone, horn and tuba, are excited by the lips, the action of which, in combination with a specially shaped mouthpiece, is similar to that of a double reed. The air pressure during sound excitation is much higher here than in woodwinds. Brass wind instruments, as a rule, are a metal barrel with cylindrical and conical sections, ending with a bell. The sections are selected so that the full range of harmonics is provided. The total length of the barrel ranges from 1.8 m for the pipe to 5.5 m for the tuba. The tuba is snail-shaped for ease of handling, not for acoustic reasons.

With a fixed length of the barrel, the performer has at his disposal only notes determined by the natural frequencies of the barrel (moreover, the fundamental frequency is usually “not taken”), and higher harmonics are excited by increasing air pressure in the mouthpiece. Thus, only a few notes (second, third, fourth, fifth and sixth harmonics) can be played on a fixed-length bugle. On other brass instruments, the frequencies lying between the harmonics are taken with a change in barrel length. The trombone is unique in this sense, the length of the barrel of which is regulated by the smooth movement of the retractable U-shaped wings. Enumeration of notes of the entire scale is provided by seven different positions of the wings with a change in the excited overtone of the trunk. In other brass instruments, this is achieved by effectively increasing the overall length of the barrel with three lateral channels of different lengths and in different combinations. This gives seven different barrel lengths. As with the trombone, the notes of the entire scale are played by excitation of different series of overtones corresponding to these seven stem lengths.

The tones of all brass instruments are rich in harmonics. This is mainly due to the presence of a bell, which increases the efficiency of sound emission at high frequencies. The trumpet and horn are designed to play a much wider range of harmonics than the bugle. The part of the solo trumpet in the works of I. Bach contains many passages in the fourth octave of the series, reaching the 21st harmonic of this instrument.

Percussion instruments.

Percussion instruments make sound by striking the body of the instrument and thereby exciting its free vibrations. From the piano, in which vibrations are also excited by a blow, such instruments differ in two respects: a vibrating body does not give harmonic overtones, and it itself can radiate sound without an additional resonator. To percussion instruments include drums, cymbals, xylophone and triangle.

The oscillations of solids are much more complex than those of an air resonator of the same shape, since there are more types of oscillations in solids. So, waves of compression, bending and torsion can propagate along a metal rod. Therefore, a cylindrical rod has many more vibration modes and, therefore, resonant frequencies than a cylindrical air column. In addition, these resonant frequencies do not form a harmonic series. The xylophone uses the bending vibrations of solid bars. The overtone ratios of the vibrating xylophone bar to the fundamental frequency are: 2.76, 5.4, 8.9 and 13.3.

A tuning fork is an oscillating curved rod, and its main type of oscillation occurs when both arms simultaneously approach each other or move away from each other. The tuning fork has no harmonic series of overtones, and only its fundamental frequency is used. The frequency of its first overtone is more than 6 times the fundamental frequency.

Another example of an oscillating solid body that produces musical sounds is a bell. The sizes of bells can be different - from a small bell to multi-ton church bells. The larger the bell, the lower the sounds it makes. The shape and other features of the bells have undergone many changes in the course of their centuries-old evolution. Very few enterprises are engaged in their manufacture, which requires great skill.

The initial overtone series of the bell is not harmonic, and the overtone ratios are not the same for different bells. So, for example, for one large bell, the measured ratios of overtone frequencies to the fundamental frequency were 1.65, 2.10, 3.00, 3.54, 4.97 and 5.33. But the distribution of energy over the overtones changes rapidly immediately after the bell is struck, and the shape of the bell seems to be chosen in such a way that the dominant frequencies are related to each other approximately harmonically. The pitch of the bell is determined not by the fundamental frequency, but by the note that is dominant immediately after the strike. It corresponds approximately to the fifth overtone of the bell. After some time, the lower overtones begin to predominate in the sound of the bell.

In the drum, the vibrating element is a leather membrane, usually round, which can be considered as a two-dimensional analogue of a stretched string. In music, the drum is not as important as the string, because its natural set of natural frequencies is not harmonic. The exception is the timpani, the membrane of which is stretched over an air resonator. The drum overtone sequence can be made harmonic by changing the thickness of the head in the radial direction. An example of such a drum is tabla used in classical Indian music.