LECTURE 3 ACOUSTICS. SOUND

1. Sound, types of sound.

2. Physical characteristics of sound.

3. Characteristics of the auditory sensation. Sound measurements.

4. Passage of sound through the interface.

5. Sound research methods.

6. Factors Determining Noise Prevention. Noise protection.

7. Basic concepts and formulas. Tables.

8. Tasks.

Acoustics. In a broad sense - a branch of physics that studies elastic waves from the lowest frequencies to the highest. In a narrow sense - the teaching of sound.

Sound in a broad sense - elastic vibrations and waves propagating in gaseous, liquid and solid substances; in a narrow sense, a phenomenon subjectively perceived by the hearing organs of humans and animals.

Normally, the human ear hears sound in the frequency range from 16 Hz to 20 kHz. However, with age, the upper end of this range decreases:

Sound with a frequency below 16-20 Hz is called infrasound, above 20 kHz - ultrasound, and the highest-frequency elastic waves in the range from 10 9 to 10 12 Hz - hypersound.

Sounds found in nature are divided into several types.

Tone - it is a sound that is a periodic process. The main characteristic of a tone is frequency. Simple tone created by a body vibrating according to a harmonic law (for example, a tuning fork). Difficult tone created by periodic vibrations that are not harmonic (for example, the sound of a musical instrument, the sound created by the human speech apparatus).

Noise is a sound that has a complex non-repeating time dependence and is a combination of randomly changing complex tones (rustling of leaves).

Sonic boom- this is a short-term sound effect (clap, explosion, bang, thunder).

A complex tone, as a periodic process, can be represented as a sum of simple tones (decomposed into component tones). Such a decomposition is called spectrum.

The acoustic spectrum of a tone is the collection of all its frequencies, with an indication of their relative intensities or amplitudes.

The lowest frequency in the spectrum (ν) corresponds to the fundamental tone, and the rest of the frequencies are called overtones or harmonics. Overtones have frequencies that are multiples of the fundamental frequency: 2ν, 3ν, 4ν, ...

Usually the largest amplitude of the spectrum corresponds to the fundamental tone. It is he who is perceived by the ear as the pitch of the sound (see below). Overtones create the "color" of the sound. Sounds of the same pitch, created by different instruments, are perceived by the ear in different ways precisely because of the different ratios between the amplitudes of the overtones. Figure 3.1 shows the spectra of the same note (ν = 100 Hz) played on a grand piano and clarinet.

Rice. 3.1. Note spectra of piano (a) and clarinet (b)

The acoustic spectrum of noise is solid.

The content of the article

SOUND AND ACOUSTICS. Sound is vibrations, i.e. periodic mechanical disturbance in elastic media - gaseous, liquid and solid. Such a disturbance, 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, which deals with the issues of the origin, propagation, reception and processing of sound waves, is called acoustics. Sound can be inaudible if its frequency is beyond the sensitivity of the human ear, or it propagates in an environment such as a solid that cannot have direct contact with the ear, or its energy is quickly dissipated in the environment. Thus, our usual process of perceiving sound is only one side of acoustics.

SOUND WAVES

Consider a long pipe filled with air. From the left end, a piston that fits snugly against the walls is inserted into it (Fig. 1). If the piston is sharply moved to the right and stopped, then the air in the immediate vicinity of it will momentarily be compressed (Fig. 1, a). Then the compressed air will expand, pushing the air adjacent to it on the right, and the compression region, which originally appeared near the piston, will move along 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, particle displacement and velocity. For sound waves, these deviations from 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 the atmospheric pressure.

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

Wave motion, of course, is not characteristic only of sound: light and radio signals propagate in the form of waves, and everyone knows the waves on the surface of the 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. We denote the period through 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. Displacement of the piston with simple harmonic vibrations for any moment in time t can be written as

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

Suppose the piston vibrates according to the harmonic vibration formula. Then, when it moves to the right, compression arises, as before, and when it moves to the left, the pressure and density will decrease relative to their equilibrium values. There is no compression, but a rarefaction of the gas. In this case, the right will spread, as shown in Fig. 2, a wave of alternating compression and rarefaction. 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 peaks) is called the wavelength. It is customary to denote it with a Greek letter. l(lambda). Wavelength l is the distance traveled by the wave in time T... So l = Tv, or v = l f.

Longitudinal and transverse waves.

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

The most significant difference between these two types of waves is that the shear wave has the property polarization(vibrations 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 displacement of the particles, 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. Thus, the elastic properties of a metal rod are not the same during torsion, compression and bending. And the corresponding wave vibrations propagate at different speeds.

Elastic is a medium in which 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 WITH- modulus of elasticity, depending on the material and the type of deformation.

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

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

Sound speed in a solid rod.

The long rod can be stretched or compressed by the 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... The quantity D L/L we will call the relative deformation, and the force per unit area of ​​the cross-section of the bar, the stress. So the voltage is F/A, where A - cross-sectional area of ​​the bar. When applied to such a rod, Hooke's law has the form

where Y- Young's modulus, i.e. the modulus of elasticity of the bar for tension or compression, characterizing the material of the bar. Young's modulus is small for easily stretchable materials such as rubber and large for tough materials such as steel.

If now, by blowing a hammer on the end of the rod, a compression wave is excited in it, then it will propagate with a speed where r, as before, is the density of the material from which the rod is made. The values ​​of wave velocities for some typical materials are given in table. one.

Table 1. SOUND SPEED 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 compression is associated with the movement of the lateral surface of the rod (Fig. 3, a).

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

The bending wave speed in the bar depends on the wavelength. This wave is called "dispersive".

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

where m- shear modulus, which characterizes the elastic properties of the material in relation to shear. Some typical shear wave velocities are given in Table. one.

Velocity in extended solid media.

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

Longitudinal deformation is plane deformation, i.e. one-dimensional compression (or rarefaction) in the direction of wave propagation. The shear wave deformation is the 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 associated with the bulk modulus V(the definition of which is given below) and the shear modulus m of the material by the ratio C L = B + 4/3m. 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 made of the same material. Therefore, it is given by expression. Its values ​​for common solid materials are given in table. one.

Gas velocity.

In gases, only one type of deformation is possible: compression - rarefaction. Corresponding modulus of elasticity V called the modulus of volumetric deformation. It is determined by the ratio

–D P = B(D V/V).

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

The magnitude V 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 pressure change in the sound wave occurs without heat exchange with the surrounding particles. This change is called adiabatic. It was found 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 compression-rarefaction waves, as in gases. The speed is given by the same formula. However, a liquid is much less compressible than a gas, and therefore the quantity V, more and density r... The speed of sound in liquids is closer to the speed in solids than in gases. It is much less than in gases, it depends on the temperature. For example, the speed in fresh water is 1460 m / s at 15.6 ° C. In seawater of normal salinity, it is 1504 m / s at the same temperature. 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 the boundaries, so-called standing waves are generated. A standing wave is the result of the superposition of two waves, one traveling in the forward direction and the other in the opposite direction. There is a picture of oscillations not moving in space with alternation of antinodes and nodes. At the antinodes, the deviations of the vibrating 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 initial, rectilinear position. When photographing waves in a string, nodes and antinodes of the fundamental tone and overtones are clearly visible.

The standing wave pattern greatly facilitates the analysis of the vibrational motion of a string of a given length. Let there be a string of length L fixed 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 standing waves are possible that have knots at the boundary points. The lowest frequency of vibration of a string corresponds to the maximum possible wavelength. Since the distance between the nodes is l/ 2, the frequency is minimum 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. The corresponding frequency, called the fundamental frequency or fundamental tone, is given by the expression f = v/2L, where v- the speed of wave propagation along the string.

There is a whole series 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. natural frequencies of vibrations of the string. They increase in proportion to the numbers of the natural series: higher harmonics in 2, 3, 4 ... etc. times the frequency of the fundamental vibration. Such a range of sounds is called natural or harmonic scale.

All this is of great importance in musical acoustics, which will be discussed in more detail below. For now, note that all natural frequencies are present in the sound produced by the string. The relative contribution of each of them depends on the point at which the vibrations of the string are excited. If, for example, you pluck a string 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 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 - the tension force of the string, and r L - the mass of a unit of string length. Therefore, the spectrum of natural frequencies of the string is given by the expression

Thus, an increase in string tension leads to an increase in vibration frequencies. Reduce the vibration frequency at a given T you can take a heavier string (large r L) or by increasing its length.

Standing waves in organ pipes.

The theory stated in relation to the string can be applied to the vibrations of air in an organ-type pipe. An organ pipe can be simply 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 appears at the open end, and a knot at the closed end. Consequently, a pipe with two open ends has a fundamental frequency at which half the wavelength fits over the pipe length. A pipe, in which one end is open and the other closed, has a fundamental frequency at which a quarter of the wavelength fits over 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- pipe length). In the first case, the result is the same as for the string: the overtones are equal to double, triple, etc. the value of the fundamental frequency. However, for a pipe open at one end, the overtones will be greater than the fundamental frequency at 3, 5, 7, etc. once.

In fig. 4 and 5 schematically show the picture of standing waves of the fundamental frequency and the first overtone for pipes of the two types considered. The offsets are shown here as transverse for reasons of convenience, but are actually longitudinal.

Resonant vibrations.

Standing waves are closely related to the phenomenon of resonance. The natural frequencies mentioned above are also the resonant frequencies of a string or organ pipe. Suppose a loudspeaker is placed near the open end of an organ pipe, which emits a signal of one specific frequency, which can be changed as desired. Then, if the frequency of the loudspeaker signal coincides with the fundamental frequency of the pipe or with one of its overtones, the pipe will sound very loud. This is because the loudspeaker excites the vibrations of the air column with significant amplitude. They say that the pipe resonates 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 first applied it (in the theory of heat).

The graph of the dependence of the relative energy of sound vibrations on frequency is called the frequency spectrum of sound. There are two main types of such spectra: discrete and continuous. Discrete spectrum consists of separate lines for frequencies, separated by empty gaps. All frequencies are present in the continuous spectrum within its bandwidth.

Periodic sound vibrations.

Sound vibrations are periodic if the vibrational process, no matter how complex it is, repeats 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 vibrations (Fig. 6, a) with a change in amplitude from + A before - A and period T = 1/f... Another simple example is the triangular sawtooth vibration shown in Fig. 6, b... An example of periodic oscillations of a more complex shape with corresponding harmonic components is shown in Fig. 7.

Musical sounds are periodic oscillations and therefore contain harmonics (overtones). We have already seen that in the string, along with the oscillations of the fundamental frequency, other harmonics are excited to one degree or another. The relative contribution of each overtone depends on how 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.

Sound pulse spectrum.

A common type of sound is a sound of short duration: clap of hands, knock on the door, sound of an object falling to the floor, cuckoo crowing. 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 must equally well “pass” all these frequencies.

The main features of a sound pulse can be found out by considering a pulse of a simple form. Suppose the sound is a vibration of duration D t at which the pressure change 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 oscillations that we would have if the same signal was infinite.

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

D f D t" one.

This relationship is valid for all sound impulses. Its meaning is that the shorter the pulse, the more frequencies it contains. Suppose that a sonar is used to locate a submarine, which emits ultrasound in the form of a pulse with a duration of 0.0005 s with a signal frequency of 30 kHz. The bandwidth is 1 / 0.0005 = 2 kHz, and the frequencies actually contained in the radar pulse spectrum are in the range from 29 to 31 kHz.

Noise.

Noise is understood as any sound generated by multiple sources that are not coordinated with each other. An example is the noise of the foliage of trees shaking by the wind. Jet engine noise is caused by the turbulence of the high-speed exhaust stream. Noise as an annoying sound is considered in Art. ACOUSTIC ENVIRONMENTAL POLLUTION.

The intensity of the sound.

The sound volume may vary. It is easy to figure out 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 of wavefront area per unit of time. In other words, if we take a unit 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. Intensity is usually expressed in W / cm 2 (or W / m 2).

Let us give the value of this value for some familiar sounds. The amplitude of the excess pressure that occurs during normal conversation is about one millionth of atmospheric pressure, which corresponds to an acoustic intensity of sound of the order of 10 –9 W / cm 2. The total power of the sound emitted during a normal conversation is about only 0.00001 W. 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 loudest sound that the ear can tolerate is about 10-14 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 quantity and measure it in decibels. The logarithmic value of the intensity is the logarithm of the ratio of the value of the value under consideration to its value taken as the initial one. Intensity level J in relation to some conditionally selected intensity J 0 equals

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

Thus, one sound exceeding the other in intensity by 20 dB is 100 times greater in intensity.

In the practice of acoustic measurements, it is customary to express the sound intensity in terms of the corresponding amplitude of the excess pressure R e... When pressure is measured in decibels relative to some conditionally selected pressure R 0, the so-called sound pressure level is obtained. Since the sound intensity is proportional to the value 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 = 2Ч 10 –5 Pa corresponds to the standard hearing threshold for sound with a frequency of 1 kHz. Table 2 lists the 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 department
Office
Living quarters
The territory of the residential area at night
Broadcast studio

Volume.

Sound pressure level is not simply related to 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 loudness of sounds that are not tied to the comparison conditions cannot be compared. Still, the comparison of pure tones is interesting. To do this, determine the sound pressure level at which the given tone is perceived as equal to the standard tone with a frequency of 1000 Hz. In fig. 9 shows curves of equal loudness obtained in the experiments of Fletcher and Manson. For each curve, the corresponding standard tone sound pressure level of 1000 Hz is indicated. For example, at a tone frequency of 200 Hz, a sound level of 60 dB is required for it to be perceived as equal in volume to a 1000 Hz tone with a sound pressure level of 50 dB.

These curves are used to determine the background - a unit of loudness level, which is also measured in decibels. Background is the sound volume level for which the sound pressure level of an equal 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 loudness level of 50 backgrounds.

The lower curve in Fig. 9 is the hearing threshold curve for the good ear. The audible frequency range is approximately 20 to 20,000 Hz.

Sound wave propagation.

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

Plane waves.

The simplest wave front 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 tube can be considered approximately flat, as well as a spherical wave at a great distance from the source.

Spherical waves.

A wave with a spherical front, emanating from a point and propagating in all directions, can also be attributed to simple types of waves. Such a wave can be generated 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, since the energy is distributed over a sphere of increasing radius.

If a point source generating a spherical wave emits a power of 4 p Q, then, since the surface area of ​​a sphere with a radius r equals 4 p r 2, the intensity of sound in a spherical wave is

J = Q/r 2 ,

where r- distance from the source. Thus, the intensity of the spherical wave decreases in inverse proportion to 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 wavefront propagation. To clarify it, let us consider the known wavefront shape at any moment in time. It can be found after time D t, if each point of the initial wavefront is considered as a source of an elementary spherical wave propagating over this interval at a distance v D t... The envelope of all these elementary spherical wavefronts will be the new wavefront. The Huygens principle allows one to determine the shape of the wavefront throughout the entire 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 around an obstacle. Diffraction is analyzed using 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 sound is many times longer than that of light, diffraction of sound waves is less surprising than diffraction of light. So, you can talk to someone standing around the corner of the building, although he is not visible. A sound wave easily bends around an angle, while light, due to its small wavelength, produces harsh shadows.

Consider the diffraction of a plane sound wave incident on a solid flat screen with an aperture. 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 full diffraction is obtained: the wavefront of the outgoing wave will be spherical, and the wave will reach all points behind the screen. If l slightly less D, then the outgoing wave will propagate mainly in the forward direction. Finally, if l much less D, then all its energy will propagate in a straight line. These cases are shown in Fig. 10.

Diffraction is also observed when there is an obstacle in the path of the sound. If the size of the obstacle is 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 the wavelength or less, the sound diffracts to some extent in all directions. This is taken into account in architectural acoustics. For example, sometimes the walls of a building are covered with protrusions with dimensions on 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 moving in one medium falls on the 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 change direction at the interface, i.e. refract. In fig. 11 shows the simplest case when a plane wave is incident at right angles to a flat surface separating two different substances. If the intensity reflectance, which determines the fraction of reflected energy, is R, then the transmission coefficient will be T = 1 – R.

For a sound wave, the ratio of overpressure to vibrational space 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 characteristic impedance of gases is much lower than that of liquids and solids. Therefore, if a wave in the air hits a thick solid object or the surface of deep water, then the sound is almost completely reflected. For example, for the boundary between air and water, the ratio of characteristic impedances is 0.0003. Accordingly, the energy of sound passing from air to 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, the 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 much higher 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 or comparable to the sound wavelength. If the wall thickness is a multiple of half the wavelength of sound in the wall, then the coefficient of wave transmission at perpendicular incidence is very large. The baffle would be absolutely transparent for the sound of this frequency, if not for the absorption, which we neglect here. If the wall thickness is much less than the sound wavelength in it, then the reflection is always small, and the transmission is large, unless special measures are taken to increase the sound absorption.

Refraction of sound.

When a plane sound wave is incident at an angle on the interface between the media, 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. Refractive geometry at a flat boundary is shown in Fig. 12. The angles between the direction of the waves and the normal to the surface are designated q 1 for the incident wave and q 2 - for refracted past. The relationship between these two angles includes only the ratio of the speeds of sound for the two media. As with light waves, these angles are related by Snell's (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 motion. In the atmosphere and ocean, vertical temperature gradients are usually observed due to horizontal stratification. 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 in some place near the surface of the Earth the air is warmer than in higher layers. Then, with an increase in altitude, the air temperature here decreases, and with it the speed of sound decreases. The sound emitted by the source near the Earth's surface will go up due to refraction. This is shown in Fig. 13, which depicts sound "beams".

The deflection of the beams of sound, shown in Fig. 13 is generally 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 the equality sin q/v= const, referring to any point on the ray. Thus, if the beam enters 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 the speed of sound.

From fig. 13 shows that there is a region located at some distance from the source, where the 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 deflected 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 distant 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 adequate sound refraction, cannon fire 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 beams are deflected downward, resulting in a zone of silence similar to that shown in Fig. 13 for the atmosphere. For the ocean, the corresponding picture will turn out if this picture is simply turned over.

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

Interference of sound waves.

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

Standing waves as a result of interference.

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

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

Beats.

Interference of harmonic waves of different frequencies is also possible. When the two frequencies differ little, what are known as beats occurs. Beats are changes in the amplitude of a sound that occur at a frequency equal to the difference between the original frequencies. In fig. 14 shows an oscillogram of beats.

Note that the beat frequency is the amplitude modulation frequency of the sound. Also, the beat should not be confused with the difference frequency resulting from harmonic distortion.

Beats are often used when tuning two tones in unison. The frequency is adjusted until the beats cease to be heard. Even if the beat frequency is very low, the human ear is able to detect periodic increases and decreases in sound volume. Therefore, beating is a very sensitive method of tuning in the audio range. If the tuning is not accurate, then the frequency difference can be determined by ear by counting the number of beats per second. In music, beats of higher harmonic components are also perceived by ear, which is used when tuning a 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 dissipated. Due to the processes of heat transfer, intermolecular interaction and internal friction, sound waves are absorbed in any medium. The absorption rate 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 overpressure decreases as a function of the distance traveled by the propagating wave. Decrease in the amplitude of overpressure -D R e when passing the distance D X proportional to the amplitude of the initial overpressure R e and distance D X... In this way,

–D P e = a P e D x.

For example, when the absorption loss is said to be 1 dB / m, this means that at a distance of 50 m, the sound pressure level decreases by 50 dB.

Absorption due to internal friction and thermal conductivity.

When particles move, associated with the propagation of a sound wave, friction between different particles of the medium is inevitable. In liquids and gases, this friction is called viscosity. Viscosity, which causes the irreversible transformation of acoustic wave energy into heat, is the main reason for the absorption of sound in gases and liquids.

In addition, absorption in gases and liquids is due to heat loss during compression in a wave. We have already said that during the passage of a wave, the gas in the compression phase heats up. In this fast-moving process, heat usually does not have time to be transferred to other areas 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. This is related to the absorption of sound due to thermal conductivity. Such absorption occurs in compression waves in gases, liquids and solids.

Sound absorption, due to both viscosity and thermal conductivity, usually increases with the square of the frequency. Thus, high-frequency sounds are absorbed much more strongly than low-frequency sounds. For example, at 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 structural defects in solids. The point is that polycrystalline solid materials consist of small crystallites; when sound passes, deformations occur in them, leading to the absorption of sound energy. Sound is also scattered at the boundaries of the crystallites. In addition, even single crystals contain dislocation-type defects that contribute to the absorption of sound. Dislocations are violations of the alignment of atomic planes. When a sound wave causes atoms to vibrate, dislocations move and then return to their original position, dissipating energy due to internal friction.

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

MUSICAL ACOUSTICS

Musical sounds.

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

A 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 a periodic sound can be represented as a sum of oscillations with frequencies that are multiples of the fundamental frequency f: 2f, 3f, 4f etc. It was also noted that the vibrating strings and pillars of air emit musical sounds.

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

Sound pitch.

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

So, for example, it was found that the assessment of the height of a single-frequency sound depends somewhat on the level of its loudness. With a significant increase in the volume level, say 40 dB, the apparent frequency can decrease by 10%. In practice, this dependence on loudness is irrelevant, since musical sounds are much more complex than single-frequency sound.

In the question of the relationship between pitch and frequency, something else is more significant: if musical sounds consist 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. 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 sound spectrum. True, most often the fundamental frequency is present in one way or another 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 distortion (not to mention the fact that there are psychological and subjective aspects of hearing). The ear is capable of producing certain frequencies, in addition, the sound wave undergoes non-linear distortion in it. Frequency selectivity is due to the difference between the loudness of the sound and its intensity (Fig. 9). Harder to explain is harmonic distortion, which is the appearance of frequencies that are not present 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 total and difference frequencies are excited in it f 1 + f 2 and f 1 - f 2. The greater the amplitude of the initial vibrations, the greater the contribution of the "extra" frequencies.

Thus, due to the nonlinearity of the acoustic characteristics of the ear, frequencies that are absent in the sound may appear. These frequencies are called subjective tones. Let us assume that the sound consists of pure tones of frequencies of 200 and 250 Hz. Due to the nonlinearity of the response, frequencies of 250 - 200 = 50, 250 + 200 = 450, 2ґ 200 = 400, 2ґ 250 = 500 Hz, etc. will additionally appear. The listener will think that there is a whole set of combination frequencies in the sound, but their appearance is actually due to the nonlinear 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, as studies have shown, subjective frequencies arise only with 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 measurement of musical pitch.

In the history of music, sounds of different frequencies have been taken as the fundamental tone that determines the entire musical structure. The currently accepted frequency for the first octave A is 440 Hz. But in the past it has varied from 400 to 462 Hz.

The traditional way of determining 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 there are now more convenient devices for determining the pitch, such as a stable frequency reference generator (with a quartz resonator), which can be smoothly tuned over the entire sound range. True, accurate calibration of such a device is rather difficult.

A widespread stroboscopic method of measuring the pitch, in which the sound of a musical instrument sets the frequency of stroboscopic lamp flashes. The lamp illuminates the 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 changes in pitch, but its sensitivity is frequency dependent. It is maximal near the lower threshold of hearing. Even the untrained ear can detect a frequency difference of as little as 0.3% in the 500 to 5000 Hz range. The 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 is understood as those features of musical sounds that give musical instruments and voices their unique specificity, even if we compare sounds of the same pitch and volume. This is the sound quality, so to speak.

Timbre depends on the frequency spectrum of the sound and its changes over time. It is determined by several factors: the distribution of energy over tones, the frequencies that occur at the moment of the appearance or termination of sound (the so-called transition tones) and their decay, as well as the slow amplitude and frequency modulation of the sound ("vibrato").

Overtones intensity.

Consider a stretched string, which is excited by a pluck 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- its length. Thus, only frequencies will be present f 1 , 3f 1 , 5f 1, etc. The relative amplitudes of these harmonics are shown in Fig. 15, 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 harmonic energy distribution depends on the excitation method. When the string is excited, the fundamental frequency dominates in its middle and even harmonics are completely suppressed. If the string is fixed in its middle part and plucked somewhere else, 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 stated above, the sound quality of musical instruments depends on the harmonic distribution of energy. When the pitch of many instruments, and especially the human voice, changes, the harmonic distribution changes so that the main overtones are always located approximately in 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 a soundboard and an air resonator. The width of natural resonances is usually large, due to which the radiation efficiency at the corresponding frequencies is higher. In brass instruments, the formants are determined by the bell from which the sound comes out. Overtones in the formant range are always strongly emphasized, since 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 significantly related to this. Even when the instrument sustains a long note, there is a slight periodic modulation of frequency and amplitude that enriches the sound - "vibrato". This is especially true for stringed instruments such as the violin and for the human voice.

In many instruments, for example, the piano, the duration of the sound is such that a constant tone does not have time to form - the excited sound rapidly increases, and then its rapid decay follows. Since the decay of overtones is usually caused by frequency-dependent effects (such as acoustic radiation), it is obvious that the overtone distribution changes over the course of the 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 keyboard) have practically no constant tone. In such cases, it is possible to speak about the spectrum of overtones only conditionally, since the sound changes rapidly over time. The rise and fall characteristics are also an important component of the timbre of such instruments.

Transitional tones.

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

SOUND OF MUSICAL INSTRUMENTS

Musical sounds can be excited and modified in many ways, and therefore musical instruments come in a variety of forms. Most of the instruments were created and improved by the musicians themselves and skilled craftsmen, who did not resort to scientific theory. Therefore, acoustic science cannot explain, for example, why the violin has such a shape. However, it is quite possible to describe the properties of the sound of a violin based on the general principles of playing it and its construction.

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 the musical instrument covers at least three (the large organ is ten). In most cases, overtones extend to the very edge of the audible sound range.

Musical instruments have three main parts: an oscillating element, a mechanism for exciting it, 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 vibrations of strings and air pillars of a fixed length are harmoniously related to each other, in many instruments the main vibrating elements are strings and air pillars. With a few exceptions (flute is one of them), you cannot take a single-frequency sound on instruments. When the main vibrator is excited, a sound is produced that contains overtones. Some vibrators have resonant frequencies that are not harmonic. Instruments of this kind (such as drums and cymbals) are used in orchestral music for expressiveness and emphasis on rhythm, but not for melodic development.

Stringed instruments.

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

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

where T Is the tension force of the string, and r L Is the mass of a unit of string length. 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 dictated by the proper selection of tension and mass. Other frequencies are obtained by shortening the length of the string with your fingers.

In other instruments, such as the piano, one of many pre-tuned strings is provided for each note. Tuning a piano where there is a wide range of frequencies is not an easy task, especially in the low frequency region. The tension of all piano strings is almost the same (approximately 2 kN), and the variety of frequencies is achieved by varying the length and thickness of the strings.

A stringed instrument can be excited by plucking (for example, on a harp or banjo), percussion (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 string is struck is the piano. The large body 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 the 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 blow on all but the shortest strings falls on a point located at a distance of 1/7 of the string's length 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 hammer, other harmonics located near the seventh are suppressed as well.

Violin family.

In the violin family of instruments, long sounds are produced with a bow, with the help of which an alternating driving force is applied to the string to keep the string vibrating. Under the action of a moving bow, the string is pulled to the side due to friction until it breaks off due to an increase in the tension force. Returning to the starting position, she 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 the 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 with 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 the low frequencies. Therefore, the same note, played on two instruments of the violin family, acquires a different timbre color due to the difference in the structure of overtones.

The violin has a pronounced resonance near 500 Hz, due to the shape of its body. When a note that is close to this frequency is played, an unwanted vibrating sound called a "wolf tone" may 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 stand out very much in the general spectrum of the sound being extracted.

Wind instruments.

Woodwind instruments.

The natural vibrations of air in a cylindrical pipe of finite length were discussed earlier. The 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 a column of air.

Air vibrations are excited either by vibrations in the air stream falling on the sharp edge of the resonator wall, or by vibrations of the flexible surface of the tongue in the air stream. In both cases, periodic pressure changes occur in the 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 - now on one side of the wedge, then on the other side of the wedge. The frequency of their formation is the greater, the higher the speed of the air flow. If such a device is acoustically connected to a resonating air column, then the frequency of the edge tone 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 fundamental frequency of the air column is excited only when the air flow rate exceeds a certain minimum value. In a certain interval of speeds exceeding this value, the frequency of the edge tone is equal to this fundamental frequency. At an even higher air flow velocity (close to that at which the edge frequency, in the absence of coupling 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 overdoing.

Air pillars in instruments such as organ, flute and piccolo flute are excited by edge tones. When playing the flute, the performer excites the edge tones by blowing from the side into the side opening near one of the ends. Notes of one octave, starting from "D" and higher, are obtained by changing the effective length of the trunk, opening the side holes, at a normal edge tone. Higher octaves are received by blow.

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

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

Unlike a pipe that is open at both ends, which has a full set of overtones, a pipe that is open only at one end has only odd harmonics ( cm. above). This is the configuration of the clarinet, and therefore the even harmonics are poorly expressed. The overdrive 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 the bore of its barrel is tapered, while in the clarinet the section of the channel is constant over most of its length. The vibration frequencies in a conical barrel are more difficult to calculate than in a cylindrical tube, but still there is a full set of overtones. In this case, the vibration frequencies of a tapered tube with a closed narrow end are the same as for a cylindrical tube open at both ends.

Brass instruments.

Copper ones, including French horn, trumpet, cornet-a-piston, trombone, horn and tuba, are stimulated by the lips, which in combination with a specially shaped mouthpiece are similar to the action of a double reed. The air pressure when sound is excited is much higher here than in woodwind. Brass horns, as a rule, are a metal barrel with cylindrical and conical sections, ending with a bell. The sections are selected to provide a full range of harmonics. The total barrel length ranges from 1.8 m for a pipe to 5.5 m for a tuba. The tube is coiled for ease of handling and not for acoustic reasons.

With a fixed barrel length, the performer has only notes determined by the natural frequencies of the barrel (and the fundamental frequency is usually "unbeatable"), and higher harmonics are excited by an increase in air pressure in the mouthpiece. So, on a horn of a fixed length, you can only play a few notes (second, third, fourth, fifth and sixth harmonics). On other brass instruments, the frequencies between the harmonics are taken with a change in the length of the barrel. 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 stage. 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 tools, this is achieved by effectively extending the full length of the barrel with three side 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 picked up by exciting different series of overtones corresponding to these seven barrel 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 be played over a much wider range of harmonics than the horn. The part of the solo trumpet in the works of I. Bach contains many passages in the fourth octave of the row, reaching the 21st harmonic of this instrument.

Percussion instruments.

Percussion instruments are made to sound by striking the body of the instrument and thereby exciting its free vibrations. Such instruments differ from a piano, in which vibrations are also excited by a shock, in two respects: an oscillating body does not give harmonic overtones and it itself can emit sound without an additional resonator. Percussion instruments include drums, cymbals, a xylophone, and a triangle.

The vibrations of solids are much more complicated than that of an air resonator of the same shape, since there are more modes of vibration in solids. Thus, compression, bending and torsion waves can propagate along a metal rod. Therefore, a cylindrical rod has much more vibration modes and, therefore, resonant frequencies than a cylindrical air column. Moreover, these resonant frequencies do not form a harmonic series. The xylophone uses bending vibrations of solid bars. The ratios of the overtones of the oscillating xylophone bar to the fundamental frequency are as follows: 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 does not have a 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 a vibrating solid that makes musical sounds is a bell. Bells vary in size, from a small bell to a multi-tone church bell. 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 original overtone row of the bell is not harmonic, and the overtone ratios are not the same for different bells. For example, for one large bell, the measured ratios of the frequencies of the overtones to the fundamental frequency were 1.65, 2.10, 3.00, 3.54, 4.97 and 5.33. But the distribution of energy over tones changes rapidly immediately after the bell is struck, and, apparently, the shape of the bell is chosen in such a way that the dominant frequencies are connected approximately harmoniously. The pitch of a bell is not determined by the fundamental frequency, but by the note that dominates immediately after being struck. It corresponds to about the fifth overtone of the bell. After some time, the lower overtones begin to predominate in the sound of the bell.

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

Sound is elastic waves in a medium (often in air) that are invisible, but perceived by the human ear (the wave affects the eardrum of the ear). A sound wave is a longitudinal compression and rarefaction wave.

If we create a vacuum, will we distinguish between sounds? Robert Boyle placed the clock in a glass vessel in 1660. After evacuating the air, he did not hear a sound. Experience proves that medium is necessary for sound propagation.

Sound can also spread in liquid and solid media. The impact of stones is clearly audible under the water. Place the clock on one end of the wooden board. By putting your ear to the other end, you can clearly hear the ticking of the clock.

The sound wave propagates through the tree

The source of sound is necessarily oscillating bodies. For example, a guitar string does not sound in its normal state, but as soon as we force it to oscillate, a sound wave arises.

However, experience shows that not every oscillating body is a source of sound. For example, a weight suspended on a thread does not make a sound. The fact is that the human ear does not perceive all the waves, but only those that create bodies that vibrate at a frequency from 16Hz to 20,000Hz. Such waves are called sound... Oscillations with a frequency less than 16Hz are called infrasound... Oscillations with a frequency greater than 20,000 Hz are called ultrasound.

Sound speed

Sound waves do not propagate instantly, but with a certain finite speed (similar to the speed of uniform motion).

That is why during a thunderstorm we first see lightning, that is, light (the speed of light is much greater than the speed of sound), and then sound comes.

The speed of sound depends on the environment: in solids and liquids, the speed of sound is much higher than in air. These are tabular measured constants. With an increase in the temperature of the medium, the speed of sound increases, with a decrease, it decreases.

Sounds are different. To characterize the sound, special values ​​are introduced: loudness, pitch and timbre of the sound.

The sound volume depends on the vibration amplitude: the larger the vibration amplitude, the louder the sound. In addition, our ear's perception of the loudness of sound depends on the frequency of vibration in the sound wave. Higher frequency waves are perceived as louder.

The frequency of the sound wave determines the pitch. The higher the vibration frequency of the sound source, the higher the sound it emits. Human voices are divided into several ranges in pitch.

Sounds from different sources are a collection of harmonic vibrations of different frequencies. The component of the longest period (lowest frequency) is called the pitch. The rest of the sound components are in overtones. The set of these components creates the color, the timbre of the sound. The set of overtones in the voices of different people is at least slightly, but different, and this determines the timbre of a particular voice.

Echo... An echo is formed as a result of the reflection of sound from various obstacles - mountains, forests, walls, large buildings, etc. An echo occurs only when the reflected sound is perceived separately from the originally pronounced sound. If there are many reflective surfaces and they are at different distances from a person, then the reflected sound waves will reach him at different times. In this case, the echo will be multiple. The obstacle must be at a distance of 11m from the person so that the echo can be heard.

Reflection of sound. Sound reflects off smooth surfaces. Therefore, when using a horn, sound waves are not scattered in all directions, but form a narrowly directed beam, due to which the sound power increases, and it spreads over a greater distance.

Some animals (for example, bat, dolphin) emit ultrasonic vibrations, then perceive the reflected wave from obstacles. So they determine the location and distance to surrounding objects.

Echolocation... This is a method for determining the location of bodies by the ultrasonic signals reflected from them. It is widely used in navigation. On ships establish sonars- instruments for recognizing underwater objects and determining the depth and topography of the bottom. A sound emitter and receiver are placed at the bottom of the vessel. The emitter gives short signals. By analyzing the delay time and direction of the returning signals, the computer determines the position and size of the object that reflected the sound.

Ultrasound is used to detect and identify various types of damage in machine parts (voids, cracks, etc.). The device used for this purpose is called ultrasonic flaw detector... A stream of short ultrasonic signals is sent to the part under study, which are reflected from the inhomogeneities inside it and, returning, enter the receiver. In places where there are no defects, the signals pass through the part without significant reflection and are not recorded by the receiver.

Ultrasound is widely used in medicine to diagnose and treat certain diseases. Unlike X-rays, its waves do not have a harmful effect on tissues. Diagnostic ultrasound examinations (ultrasound) allow, without surgical intervention, to recognize pathological changes in organs and tissues. A special device directs ultrasonic waves with a frequency of 0.5 to 15 MHz to a specific part of the body, they are reflected from the examined organ and the computer displays its image on the screen.

Infrasound is characterized by low absorption in various media, as a result of which infrasonic waves in air, water and the earth's crust can propagate over very long distances. This phenomenon finds practical application in locating violent explosions or the position of firing weapons. The propagation of infrasound over long distances in the sea makes it possible natural disaster predictions- tsunami. Jellyfish, crustaceans, etc. are able to perceive infrasound and long before the onset of a storm they feel its approach.

This lesson covers the topic "Sound Waves". In this lesson, we will continue our study of acoustics. First, we will repeat the definition of sound waves, then we will consider their frequency ranges and get acquainted with the concept of ultrasonic and infrasonic waves. We will also discuss the properties inherent in sound waves in various environments, and find out what characteristics they have. .

Sound waves - these are mechanical vibrations, which, propagating and interacting with the organ of hearing, are perceived by a person (Fig. 1).

Rice. 1. Sound wave

The section that deals with these waves in physics is called acoustics. The profession of people who are called "rumors" in the common people is acoustics. A sound wave is a wave propagating in an elastic medium, it is a longitudinal wave, and when it propagates in an elastic medium, compression and relaxation alternate. It is transmitted over time over a distance (Fig. 2).

Rice. 2. Propagation of a sound wave

Sound waves include those vibrations that are carried out with a frequency of 20 to 20,000 Hz. For these frequencies, the corresponding wavelengths are 17 m (for 20 Hz) and 17 mm (for 20,000 Hz). This range will be referred to as audible sound. These wavelengths are given for air, in which the speed of sound propagation is.

There are also such ranges that acoustics deal with - infrasonic and ultrasonic. Infrasound are those that have a frequency less than 20 Hz. And ultrasonic ones are those that have a frequency of more than 20,000 Hz (Fig. 3).

Rice. 3. Ranges of sound waves

Every educated person should navigate in the frequency range of sound waves and know that if he goes to an ultrasound scan, the picture on the computer screen will be built with a frequency of more than 20,000 Hz.

Ultrasound - these are mechanical waves, similar to sound waves, but with a frequency from 20 kHz to a billion hertz.

Waves with a frequency of more than a billion hertz are called hypersound.

Ultrasound is used to detect defects in cast parts. A stream of short ultrasonic signals is directed to the part to be examined. In those places where there are no defects, the signals pass through the part without being registered by the receiver.

If there is a crack, air cavity or other inhomogeneity in the part, then the ultrasonic signal is reflected from it and, returning, enters the receiver. This method is called ultrasonic flaw detection.

Other examples of ultrasound applications are ultrasound machines, ultrasound machines, and ultrasound therapy.

Infrasound - mechanical waves, similar to sound waves, but having a frequency of less than 20 Hz. They are not perceived by the human ear.

Natural sources of infrasonic waves are storms, tsunamis, earthquakes, hurricanes, volcanic eruptions, and thunderstorms.

Infrasound is also an important wave that is used to vibrate the surface (for example, to destroy some large objects). We launch infrasound into the soil - and the soil is crushed. Where is this used? For example, in diamond mines, where ore is taken in which there are diamond components, and crushed into small particles to find these diamond inclusions (Fig. 4).

Rice. 4. Application of infrasound

The speed of sound depends on environmental conditions and temperature (Fig. 5).

Rice. 5. The speed of propagation of a sound wave in various media

Note: in air, the speed of sound at is, at, the speed increases by. If you are a researcher, then this knowledge may be useful to you. You may even come up with some kind of temperature sensor that will record temperature differences by changing the speed of sound in the environment. We already know that the denser the medium, the more serious the interaction between the particles of the medium, the faster the wave propagates. We discussed this in the last paragraph using the example of dry air and humid air. For water, the speed of sound propagation. If you create a sound wave (knock on a tuning fork), then the speed of its propagation in water will be 4 times greater than in air. Information will travel 4 times faster by water than by air. And even faster in steel: (fig. 6).

Rice. 6. The speed of propagation of a sound wave

You know from the epics that Ilya Muromets used (and all the heroes and ordinary Russian people and boys from Gaidar's RVS), used a very interesting method of detecting an object that is approaching, but is still far away. The sound it makes when driving is not yet heard. Ilya Muromets, leaning his ear to the ground, can hear it. Why? Because sound is transmitted at a higher speed on solid ground, which means it will reach the ear of Ilya Muromets faster, and he will be able to prepare to meet the enemy.

The most interesting sound waves are musical sounds and noises. What objects can create sound waves? If we take a wave source and an elastic medium, if we make the sound source vibrate harmoniously, then we will have a wonderful sound wave, which will be called a musical sound. These sources of sound waves can be, for example, the strings of a guitar or grand piano. This can be a sound wave that is created in the gap of an air pipe (organ or pipe). From music lessons, you know the notes: do, re, mi, fa, sol, la, si. In acoustics, they are called tones (Fig. 7).

Rice. 7. Musical tones

All objects that can emit tones will have special features. How do they differ? They differ in wavelength and frequency. If these sound waves are created by non-harmonious sounding bodies or are not connected into a common orchestral piece, then such a number of sounds will be called noise.

Noise- random vibrations of various physical nature, characterized by the complexity of the temporal and spectral structure. The concept of noise is everyday and there is physical, they are very similar, and therefore we introduce it as a separate important object of consideration.

Let's move on to quantitative estimates of sound waves. What are the characteristics of musical sound waves? These characteristics apply exclusively to harmonic sound vibrations. So, sound volume... What determines the volume of a sound? Consider the propagation of a sound wave in time or the oscillation of a sound wave source (Fig. 8).

Rice. 8. Sound volume

At the same time, if we added not very much sound to the system (for example, tapped a piano key softly), then there will be a quiet sound. If we raise our hand loudly, we call this sound by hitting the key, we will get a loud sound. What does it depend on? A quiet sound has a lower vibration amplitude than a loud sound.

The next important characteristic of musical sound and any other is height... What does the pitch of the sound depend on? The pitch depends on the frequency. We can make the source oscillate often, or we can make it oscillate not very quickly (that is, make fewer oscillations per unit time). Consider the time sweep of high and low sound of the same amplitude (Fig. 9).

Rice. 9. Sound pitch

An interesting conclusion can be drawn. If a person sings in bass, then his source of sound (these are the vocal cords) oscillates several times slower than that of a person who sings soprano. In the second case, the vocal cords vibrate more often, therefore, more often they cause foci of compression and vacuum in the propagation of the wave.

There is another interesting characteristic of sound waves that physicists do not study. This timbre... You know and easily distinguish the same piece of music, which is performed on the balalaika or on the cello. What is the difference between these sounds or is this performance? At the beginning of the experiment, we asked people who extract sounds to make them of approximately the same amplitude, so that the sound volume was the same. It's like in the case of an orchestra: if you don't need to select an instrument, everyone plays about the same, with the same strength. So the timbre of balalaika and cello is different. If we were to draw the sound that is extracted from one instrument, from another, using diagrams, they would be the same. But you can easily distinguish these instruments by their sound.

Another example of the importance of timbre. Imagine two singers who graduate from the same music college with the same teachers. They studied equally well for grades. For some reason, one becomes an outstanding performer, while the other is dissatisfied with his career all his life. In fact, this is determined exclusively by their instrument, which causes just vocal vibrations in the environment, that is, their voices differ in timbre.

Bibliography

1. Sokolovich Yu.A., Bogdanova G.S. Physics: a handbook with examples of problem solving. - 2nd edition redistribution. - X .: Vesta: Ranok publishing house, 2005. - 464 p.
2. Peryshkin A.V., Gutnik E.M., Physics. 9th grade: textbook for general education. institutions / A.V. Peryshkin, E.M. Gutnik. - 14th ed., Stereotype. - M .: Bustard, 2009 .-- 300 p.

1. Internet portal "eduspb.com" ()
2. Internet portal "msk.edu.ua" ()
3. Internet portal "class-fizika.narod.ru" ()

Homework

1. How does sound propagate? What could be the source of the sound?
2. Can sound propagate in space?
3. Is every wave that reaches a human hearing organ is perceived by it?

A specific sensation, perceived by us as a sound, is the result of the vibrational motion of an elastic medium - most often air - on the human hearing system. Oscillations of the medium are excited by a sound source and, propagating in the medium, reach the receiving apparatus - our ear. Thus, the infinite variety of sounds we hear is caused by oscillatory processes that differ from each other in frequency and amplitude. Two sides of the same phenomenon should not be confused: sound as a physical process is a special case of oscillatory motion; as a psycho-physiological phenomenon, sound is some specific sensation, the mechanism of the occurrence of which has been studied in detail at the present time.

Speaking about the physical side of the phenomenon, we characterize sound by its intensity (strength), its composition and the frequency of oscillatory processes associated with it; with sound sensations in mind, we are talking about loudness, timbre, and pitch.

In solids, sound can propagate both in the form of longitudinal and transverse vibrations. Since liquids and gases do not have shear elasticity, it is obvious that sound can propagate in gaseous and liquid media only in the form of longitudinal vibrations. In gases and liquids, sound waves are alternating thickening and rarefaction of the medium, moving away from the sound source at a certain speed characteristic of each medium. The surface of the sound wave is the geometrical position of the particles of the medium, which have the same phase of oscillation. The surfaces of sound waves can be drawn, for example, so that between the surfaces of adjacent waves there is a thickening layer and a rarefaction layer. The direction perpendicular to the surface of the wave is called the ray.

Sound waves in a gaseous environment can be photographed. For this purpose, behind the sound source, place

a photographic plate, onto which a beam of light from an electric spark is directed from the front so that these rays from an instant flash of light fall on the photographic plate, passing through the air surrounding the sound source. In fig. 158-160 shows the photographs of sound waves obtained by this method. The sound source was separated from the photographic plate by a small screen on a stand.

In fig. 158, but it can be seen that the sound wave has just come out from behind the screen; in fig. 158, b, the same wave was filmed a second time after a few thousandths of a second. In this case, the surface of the wave is a sphere. In the photograph, the image of the wave is obtained in the form of a circle, the radius of which increases over time.

Rice. 158. Photo of a sound wave at two times (a and b). Reflection of a sound wave (c).

In fig. 158, c shows a photograph of a spherical sound wave reflected from a flat wall. Here you should pay attention to the fact that the reflected part of the wave, as it were, comes from a point located behind the reflecting surface at the same distance from the reflecting surface as the sound source. It is well known that the phenomenon of the reflection of sound waves is explained by the echo.

In fig. 159 shows the change in the wave surface when a sound wave passes through a lens-shaped bag filled with hydrogen. This change in the surface of the sound wave is a consequence of the refraction (refraction) of sound rays: at the interface between two media, where the wave velocity is different, the direction of wave propagation changes.

Rice. 160 reproduces a photograph of sound waves with a four-slit screen in the path of propagation. Passing through the slots, the waves bend around the screen. This phenomenon of bending around by waves of the encountered obstacles is called diffraction.

The laws of propagation, reflection, refraction and diffraction of sound waves can be deduced from the Huygens principle, according to which each particle brought into vibration

the environment can be considered as a new center (source) of waves; the interference of all these waves gives the actually observable wave (ways of applying Huygens' principle will be explained in the third volume on the example of light waves).

Sound waves carry with them a certain amount of movement and, as a result, put pressure on the obstacles they encounter.

Rice. 159. Refraction of a sound wave.

Rice. 160. Diffraction of sound waves.

To clarify this fact, let us refer to Fig. 161. In this figure, the dotted line shows the sinusoid of displacements of the particles of the medium at a certain moment of time during the propagation of longitudinal waves in the medium. The velocities of these particles at the considered moment of time are depicted by the cosine, or, what is the same, by the sinusoid ahead of the sinusoid of displacements by a quarter of the period (in Fig. 161 - the solid line). It is easy to figure out that thickening of the medium will be observed where at a given moment the displacement of particles is equal to zero or close to zero and where the velocity is directed in the direction of wave propagation. On the contrary, rarefaction of the medium will be observed where the displacement of particles is also equal to zero or close to zero, but where the velocity of the particles is directed in the direction opposite to the propagation of waves. So, in condensations, particles move forward, in rarefactions - backward. But in

Rice. 161. In the thickening of a passing sound wave, particles move forward,

the thickened layers contain a greater number of particles than in the rarefions. Thus, at any moment of time in traveling longitudinal sound waves, the number of particles moving forward slightly exceeds the number of particles moving backward. As a result, the sound wave carries with it a certain amount of movement, which is manifested in the pressure that the sound waves exert on the obstacles they meet.

The sound pressure was experimentally investigated by Rayleigh and Petr Nikolaevich Lebedev.

Theoretically, the speed of sound is determined by the Laplace formula [§ 65, formula (5)]:

where K is the modulus of all-round elasticity (when compression is performed without the influx and release of heat), density.

If the body is compressed while maintaining the body temperature constant, then the values ​​for the elastic modulus are lower than in the case when the compression is performed without the influx and release of heat. These two values ​​of the modulus of all-round elasticity, as proved in thermodynamics, relate in the same way as the heat capacity of a body at constant pressure to the heat capacity of a body at constant volume.

For gases (not too compressed) the isothermal modulus of elasticity is simply equal to the gas pressure. If, without changing the gas temperature, we compress the gas (increase its density) by times, then the gas pressure will increase by times. Consequently, according to the Laplace formula, it turns out that the speed of sound in a gas does not depend on the density of the gas.

From gas laws and Laplace's formula, it can be deduced (§ 134) that the speed of sound in gases is proportional to the square root of the absolute temperature of the gas:

where is the acceleration of gravity, the ratio of the temperature capacities is the universal gas constant.

At C, the speed of sound in dry air is equal at average temperatures and average humidity, the speed of sound in air is considered equal to The speed of sound in hydrogen at is

In water, the speed of sound in glass in iron is

It should be noted that sound shock waves caused by a shot or explosion, at the beginning of their path, have a speed

significantly exceeding the normal speed of sound in a given environment. A shock sound wave in air, caused by a strong explosion, can have a speed near the sound source that is several times higher than the normal speed of sound in air, but already at a distance of tens of meters from the explosion site, the wave propagation speed decreases to a normal value.

As already mentioned in § 65, sound waves of different lengths have practically the same speed. Exceptions are those frequency ranges for which a particularly rapid damping of elastic waves during their propagation in the medium under consideration is characteristic. Typically, these frequencies lie far beyond the hearing range (for gases at atmospheric pressure, these are frequencies on the order of vibrations per second). Theoretical analysis shows that the dispersion and absorption of sound waves are associated with the fact that it takes some, albeit small, time to redistribute energy between the translational and vibrational motions of molecules. This leads to the fact that long waves (waves of the sound range) move somewhat slower than very short "inaudible" waves. So, in carbon dioxide vapor at and atmospheric pressure, sound has a speed, while very short, "inaudible" waves propagate with a speed

A sound wave propagating in a medium can have a different shape, depending on the size and shape of the sound source. In the most technically interesting cases, the sound source (emitter) is some kind of vibrating surface, such as, for example, a telephone membrane or a loudspeaker diffuser. If such a sound source emits sound waves into an open space, then the waveform essentially depends on the relative dimensions of the emitter; the emitter, the dimensions of which are large in comparison with the length of the sound wave, emits sound energy in only one direction, namely in the direction of its oscillatory motion. On the contrary, a radiator of small size in comparison with the wavelength emits sound energy in all directions. The shape of the wave front in both cases will obviously be different.

Let us first consider the first case. Imagine a rigid flat surface of a sufficiently large (compared to the wavelength) size, performing oscillatory movements in the direction of its normal. Moving forward, such a surface creates a thickening in front of it, which, due to the elasticity of the medium, will propagate in the direction of the displacement of the emitter). Moving back, the emitter creates a rarefaction, which will move in the medium following the initial thickening. For a short-term oscillation of the emitter, we will observe on both sides of it a sound wave, characterized by the fact that all particles of the medium that are at an equal distance from the emitting surface of the average density of the medium and the speed of sound with:

The product of the average density of the medium and the speed of sound is called the acoustic resistance of the medium.

Acoustic resistance at 20 ° С

(see scan)

Let us now consider the case of spherical waves. When the size of the emitting surface becomes small compared to the wavelength, the wavefront is noticeably curved. This is because the vibrational energy propagates in all directions from the emitter.

The phenomenon can be best understood with the following simple example. Let's imagine that a long log has fallen on the surface of the water. The resulting waves travel in parallel rows on either side of the log. The situation is different when a small stone is thrown into the water, and the waves propagate in concentric circles. The log is relatively large

with a wavelength at the surface of the water; the parallel rows of waves coming from it represent a visual model of plane waves. The stone is small in size; the circles diverging from the place of its fall give us a model of spherical waves. When a spherical wave propagates, the wavefront surface increases in proportion to the square of its radius. At a constant power of the sound source, the energy flowing through each square centimeter of the spherical surface of the radius is inversely proportional.Since the vibration energy is proportional to the square of the amplitude, it is clear that the amplitude of oscillations in a spherical wave should decrease as the reciprocal of the first power of the distance from the sound source. The equation of a spherical wave has, therefore, the following form: