The emission, propagation and perception of sound

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Soft notes of music, the hum of an engine, the stridence of a jackhammer… How are sounds produced and how do they get to our ears? Sound is composed of oscillations that propagate in compressible media, especially in fluids and solids, with characteristics that are specific to each of them. In this article, the concepts of frequency, wavelength and sound power are discussed. We will even go a little further by identifying the concepts of proper modes and resonance, illustrated by the examples of the swing and the music produced by a violin.

1. What is sound? What is it?

son - propagation - onde - Encyclopedie de l'environnement - ondes acoustiques émises par les instruments de l’orchestre - festival - acoustic waves - perception of sound
Figure 1. The acoustic waves emitted by the orchestra’s instruments reach the sensors in the listener’s ears. [Source: Pixabay]
Sounds are alternating compression and relaxation, which propagate in the air, or in any other compressible medium, from a transmitter to a receiver; they are also called acoustic waves. Often, the transmitter is a solid material, more or less flexible, mechanically excited. It can be an anvil vibrated by the shock of the hammer, a guitar string excited by the musician’s finger, or the membrane of a loudspeaker set in motion by an electromagnet. Similarly, pressure fluctuations in fluids such as air or water, caused by propellers, jets from nozzles, or turbulence produced by the movement of a vehicle, create sounds that propagate in this environment.

The ambient fluid transmits its own vibration to its surroundings, forcing the nearest layers of molecules to follow its movements back and forth. In turn, they transmit to the adjacent layers these alternating compression and relaxation, and so on (see the animation of the cover image). The air, apparently at rest, but which we know well that it consists of molecules in perpetual agitation and that it is compressible [1], is very sensitive to these tremors which, in turn, tighten and then remove the layers of molecules. Water, much denser than air (about 800 times) and much less compressible (about 100,000 times), is also made up of molecules in agitation, but these repel each other with Coulomb forces [2] that are very difficult to overcome. Like water, liquids are therefore also capable of carrying sounds.

son - propagation - onde - Encyclopedie de l'environnement - schema de l'oreille humaine - schema human ear - human ear
Figure 2. Schematic illustration of the human ear, showing the membrane of the eardrum, on which are fixed the ossicles whose movements are transmitted to the brain by the auditory nerves. [Source: DR]
These pressure variations can reach a receiver, which is often a flexible membrane such as the eardrum of an ear (Figure 1) or the sensor of a microphone. This receptor in turn vibrates with the layers of molecules closest to it. In the case of the human or animal ear (Figure 2), the auditory nerves transmit the information received to the brain, which is used to recognizing and interpret it. Microphones, on the other hand, transform the oscillations of their diaphragm into an electrical signal for recording or amplification and retransmission by an audio speaker.

2. Speed, frequency and wavelength

In a gas like air, sound propagation results from an essential property: the mobility of molecules, with an average speed of about 480 m/s under normal conditions (see the article Pressure, temperature, heat). But this agitation does not have a privileged orientation: it diffuses the energy of the tremor in all directions. However, the speed of sound propagation in the air, called celerity, implies that all molecules located in a very small volume (the fluid particle) undergo the same ordered and collective displacement. This explains why the speed of sound, although linked to the average speed of molecules, is only a fraction of this average speed, around 340 m/s.

This sound velocity is very low compared to the speed of light (300,000 km/s in vacuum and almost as much in air). This explains why the spectator of a football match, sitting in the stands at a distance of about 170 m from the centre circle, only hears the player’s shot half a second after seeing his foot hit the ball. On the other hand, over relatively short distances such as the dimensions of a concert hall, sound waves are able to accurately convey very subtle information that music lovers appreciate. A well-known exercise: in stormy weather, knowing that the perception of lightning is almost instantaneous, how can we assess how far away the storm has just broken?

son - propagation - onde - Encyclopedie de l'environnement - variation vibrations memrane microphone - time variation of the amplitude microphone membrane - perception of sound - human ear
Figure 3. Time variation of the amplitude of the vibration amplitude of a microphone membrane. We note their oscillatory nature and the presence of various frequencies and amplitudes. [Source: pixabay]
One of the main properties of waves of any kind is their period T; in the case of sound, T is the time between two successive compressions or releases at a given point. Instead of the period, it is common and quite equivalent to use its inverse, the frequency f = 1/T which therefore represents the number of oscillations per second. The frequency is counted in Hertz, noted Hz (1 Hz = 1 s-1). With the sound velocity c and its period T or its frequency f, it is elementary to construct the wavelength of the sound vibration λ = cT = c/f, distance between two successive maxima or minima. A frequency of 1000 Hz, relatively well centred in the audible range, therefore corresponds to a wavelength of 34 cm (see Figure 3 and following paragraph).

In our environment, everything is constantly moving and therefore emitting sounds. In our own bodies, blood circulates, the lungs swell and deflate, the digestive system is also animated. But our relatively soft flesh absorbs these pressure variations well; this requires the doctor to use a stethoscope to hear the beats of our hearts or the fluctuations in air velocity in our bronchi. In addition, human ears are only sensitive to a relatively narrow frequency band, between 15 Hz and 15 000 Hz. That is why we do not perceive our inner noises, except for our own voice. This allows us to focus our attention on the noises coming from the outside, often more pleasant, and above all more useful by all the information they allow us to capture. Other animal species perceive sounds in very different frequency bands. Elephants emit and hear infrasound frequencies of less than 10 Hz, which they use to communicate with each other. At the other end, bats emit and hear ultrasounds with a frequency greater than 30,000 Hz, whose echoes reflected by their environment allow them to read them in complete darkness.

Generally speaking, the speed of sound is practically independent of frequency. However, it depends on the density of the medium and is therefore quite sensitive to temperature and pressure variations. This speed is all the greater as the density of the medium and its compressibility are smaller. Thus, in helium, whose compressibility is close to that of air but whose density is much lower, the speed of sound is almost 3 times greater than in air.

3. The sound power

To characterize the power of a sound, it is common to use the sound pressure p, the maximum amplitude of the local fluctuation of atmospheric pressure. Usually, this fluctuation is very small. For example, a normally speaking person produces an acoustic pressure of about 0.01 pascal (Pa) at a distance of one metre, 10 million times less than the normal atmospheric pressure, close to 100000 Pa.

This sound pressure is proportional to the maximum amplitude of the vibrations shown in Figure 2; in reality, the sensation perceived by the human ear is more proportional to the logarithm [3] of the pressure than to the pressure itself. This means that a sound pressure 10 times higher is felt as a sound twice as intense, a sound pressure 100 times higher as a sound 4 times more intense. The power carried by a sound wave being proportional to the square of the sound pressure, this led to characterize the power of a sound by the quantity 10 log10 (p2/p2ref)=20 log10 (p/pref), named the decibel. The decibel (dB), or tenth of the Bel, is the commonly used sound power unit, so named in tribute to Graham Bell (1847-1922), a Scottish scientist who invented the telephone. It is equal to 20 log10 (p/pref) where p denotes the acoustic pressure and pref a reference pressure arbitrarily set at 20 μPa (20×10-6 Pascal, or 0.2 billionth of atmospheric pressure), which represents the hearing threshold in most humans. [4] and noted dB, where pref  a reference pressure arbitrarily set at 20 μPa (20 × 10-6 Pascal, or 0.2 billionth of atmospheric pressure). This reference corresponds to the audibility threshold for most people, i.e. 0 dB. Thus, a 20-decibel increase in the number of decibels represents a 10-fold increase in the amplitude of sound pressure, or a 100-fold increase in sound power, also called sound intensity.

Here are some orders of magnitude that give practical meaning to the decibel:

  • Quiet country atmosphere: 40 dB or p=2 × 10-8 times atmospheric pressure.
  • Bustling street with traffic: 80 dB or p=2 × 10-6 times atmospheric pressure.
  • Airport with aircraft at take-off: 120 dB or p=2 × 10-4 times atmospheric pressure.
  • Legal safety limit for short exposure: 135 dB or one thousandth of atmospheric pressure.

Everyone knows that it is necessary to raise your voice more and more to call someone who is moving away. As it propagates, the sound emitted by any source loses its intensity quite quickly, for two reasons. First, since its energy is distributed over an increasingly large hemispheric front as the wavefront moves away from the emission site, its intensity decreases due to the opposite of the area of this front, i.e. as the inverse of the square of the distance to the source. In addition, this initial energy, made up of gas compression and expansion, is gradually dissipated into heat by viscosity (see the article Pressure, Temperature and Heat).

4. Simple oscillations: the pendulum and the swing

son - propagation - onde - Encyclopedie de l'environnement - Pendule simple - pendulum
Figure 4. Simple pendulum. Almost all the mass is concentrated at the lower end in the conical shape that oscillates by sweeping an arc of a circle centred on the attachment point at the top of the chain. [Source: pixabay]
Let’s start our discovery of oscillating phenomena with the case of the simple pendulum. It has only one degree of freedomThis refers to each parameter, or coordinate, used to characterize the position of an object or its center of gravity on its trajectory. : the position of its center of gravity on an arc-shaped trajectory (Figure 4). Very quickly after the excitation has ceased, any disordered initial disturbances cease and the oscillations of this pendulum continue with a well-defined frequency, called its natural frequency, equal, for small oscillations, to (1/2π)(l/g)1/2, where l denotes the length of the pendulum and g the gravity. The fact that this natural frequency depends on gravity reflects the exchange between potential energy (maximum at high points) and kinetic energy (maximum at low point). This regular periodic movement is the proper mode of oscillation of the pendulum.

Now suppose that we bring additional energy to the pendulum by choosing the frequency of this input carefully. Example: each time the pendulum passes through its low point, it is pushed to provide it with an additional speed; this energy will be added to the previous energy of the pendulum and the amplitude of its oscillations will increase. It is then said that the frequency of this new excitation is in phase with the oscillation of the pendulum and that it leads to a resonance of the pendulum. This resonance results in large amplitudes, such that, at each oscillation, the energy supply is exactly equal to the energy dissipated.

son - propagation - onde - Encyclopedie de l'environnement - Balançoire à son point bas
Figure 5. Swing at its lowest point. The energy supply, or momentum, is achieved by projecting the legs forward and the chest backward, which also reduces friction. [Source: Fragonard, Les hasards heureux de l’escarpolette]
On the other hand, if no energy is supplied to the system, the amplitude of oscillations will gradually decrease since the energy will be regularly reduced by friction.

The case of the swing (Figure 5) shows another form of resonance. With a little practice, the children can launch themselves into the low point and thus increase the amplitude of the oscillation. It should also be noted that they provide additional energy while reducing the resistance of the air by lying as far as possible along their trajectory when the swing passes at its low point.

5. Stationary waves: vibrating string and music

Neither the pendulum nor the swing produce sounds (only the squeaks of their fasteners can be heard). Beyond this simple case, a new step must be taken to illustrate how an oscillating body produces sounds. The prototype of such a system is the rope stretched between two fixed points, subjected to excitation, which vibrates and emits sounds. This is the process used to produce music with violins, guitars, harps and other stringed instruments. These mechanical systems are very complex, because they have an infinite number of degrees of freedom – all possible positions – and as many different modes. Among these, we distinguish the fundamental mode, such that only one half wavelength is present between the two fixed points, and its harmonics whose frequencies are multiples of the fundamental frequency. The animation in Figure 6 illustrates the first three modes, the imagination makes it possible to guess the infinite sequence of harmonics not represented.

son - propagation - onde - Encyclopedie de l'environnement - Oscillation d’une corde vibrante entre deux points - oscillation of vibrating string - acoustic waves - perception of sound - human ear
Figure 6. Oscillation of a vibrating string between two fixed points: fundamental mode and two first harmonics [Source: By Christophe Dang Ngoc Chan (cdang) (Own work)[GFDL (http://www.gnu.org/copyleft/fdl.html) or CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0/)], via Wikimedia Commons]
One of the most interesting properties of this oscillating system is its ability to form standing waves, as illustrated on the animation of Figure 6. This oscillation is remarkable in that it highlights a succession of knots (points where the amplitude of the movement of the string is zero) and bellies (points where it is maximum). The number of knots and bellies depends on the distance between the two fixed points and the tension of the rope. The fact that the propagation along the rope has disappeared justifies the name standing wave. On the other hand, the propagation of this very particular excitation in the surrounding air continues to the point of transporting the sound emitted by the string to a listener’s ear or microphone. The length of the note depends on how long the excitation is maintained, for example, how long the bow rubs against a violin string.

It is also possible to create standing waves between a fixed point and a free end. The fundamental mode then includes only a quarter wavelength between these two ends. Although it consists of two metal branches instead of a vibrating string, the tuning fork illustrates this system very well, to such an extent that its ability to emit an “A” of quite high purity can become the reference for an entire orchestra.

The oscillation of the vibrating string is audible but not visible. On the other hand, the one in the fire tube of the Video below shows that it is possible to visualize the periodic fluctuations in pressure within the propane gas present in this appliance. The flow of gas through the orifices distributed along the tube and the height of each small flame are proportional to this pressure: zero at knots and maximum at bellies.

Video. Ruben’s fire tube. A loudspeaker sends acoustic waves into a long tube where standing waves are installed for a range of frequencies that resonate with the tube. At the left end of the propane is pumped into the tube; it exits through a series of holes on the upper generator, creating a longitudinal flame. The local height of this flame, proportional to the local pressure in the tube, highlights the bellies and knots within it.

son - propagation - onde - Encyclopedie de l'environnement - caisse de résonance d'un violon - sound box violin
Figure 7. Cross-section view of the sound box of an unvarnished violin. 1. 2. Table. 3. Front cleat. 4. Button hole. 5. Soul. 6. Hearing. 7. Splice. 8. Corner. 9. Rear cleat. 10. Sound bar.
Source: By English: Unknown photograph. Improved and (a little) colorised by Dake. Numbering added by SuperManu / French: The original photographer is unknown. The photo was cleaned and a little bit colored by Dake. The figures have been added by SuperManu. (L’Art du luthier (Auguste Tolbecque))[Source : GFDL (http://www.gnu.org/copyleft/fdl.html), CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0/), CC BY-SA 2.5-2.0-1.0 (http://creativecommons.org/licenses/by-sa/2.5-2.0-1.0), GFDL (http://www.gnu.org/copyleft/fdl.html), CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0/) or CC BY-SA 2.5-2.0-1.0 (http://creativecommons.org/licenses/by-sa/2.5-2.0-1.0)], via Wikimedia Commons]
In all musical instruments, especially stringed instruments, sound production comes from the resonance between the excitation and a set of specific modes, including harmonics (Figures 6 and Video). It is by seeking this resonance that the musician selects the note he wants to make heard. The purity of the sound emitted is due to the precise selection of the excited eigenmodes. In general, the vibration of a string alone is barely audible. This is why, in violins, cellos and guitars, the strings are coupled to a soundboard; located near the strings, it is constructed in such a way as to form a sound box with other pieces (see Figure 7). In turn, this box has its own modes of vibration that can resonate with those of the string. Thanks to its significant dimensions, this soundboard amplifies sounds and transmits them to the air, where they spread to listeners. It is finally the area of the soundboard that determines the range [5] of the sound emitted. It varies from about 500 cm2 for a violin intended for an adult (the length AB in Figure 7 is about 35 to 36 cm), to 1000 cm2 for a cello and 2000 cm2 for a double bass, instruments characteristic of symphony orchestras.

Rocky cliffs, an ancient theatre enclosure, a noise barrier… Everyone knows that natural or artificial walls can have an influence on the signal perceived by the listener. This influence is due both to their shape, more or less favourable to resonances, and to their surface condition. Smooth, firm and elastic walls reflect sound well; on the other hand, rough walls, or walls coated with soft and absorbent materials, do not reflect it well. This concerns the phenomenon of echo, the analysis of which leads to important applications such as the design of concert halls of high acoustic quality, or protection against noise pollution.

6. Sound propagation in water and solids

Because it results from their compressibility, sound spreads in all media, especially in liquids such as water. In the seas, this phenomenon is of considerable interest because, since light does not penetrate to great depths, it provides one of the preferred means of diagnosis. Fishermen use sound or ultrasound to detect schools of fish, geographers survey underwater landforms and national navies around the world identify ships and submersible friends or enemies in their vicinity. It is also with ultrasound that marine mammals communicate. The range of frequencies that can be used in seawater ranges from 30 Hz to 1.5 MHz, a value 100 times higher than the human audible limit of about 15,000 Hz. The speed of sound in the water is about 1450 to 1550 m s-1. As shown in Figure 8, it varies mainly with temperature and depth, i.e. with pressure, but is not very sensitive to changes in salinity (see the article The Marine Environment).

son - propagation - onde - Encyclopedie de l'environnement - propagation du son c en fonction de la profondeur dans l’atlantique
Figure 8: Typical variations in temperature T, salinity S, and sound velocity c as a function of depth in the Atlantic Ocean; the influence of salinity variations is much smaller than the influence of temperature, especially near the surface, and pressure, especially at depth. [Source: http://lecalve.univ-tln.fr/oceano/fiches/fiche3F.htm]
In the seas, sound rays are reflected by the free surface, so that underwater noise cannot be heard from the outside and requires special equipment to be picked up: sonars. They are also reflected from the bottom, and they are partially reflected by intermediate interfaces such as the thermocline (see The marine environment) or as more or less diffuse interfaces separating waters of different densities, particularly off the estuaries of major rivers. Even in the absence of strong interfaces, sound paths are rarely straight. They are generally diverted to the area where the speed of sound is lower, where they concentrate. This area then acquires a waveguide function [6]. This is particularly the case for the layer between the free surface and the depth where the velocity is the lowest (around 1000 m in Figure 8). The deeper layers then constitute “sound shadow” zones that can only be examined by plunging the transmitting and receiving systems below this particular depth.

son - propagation - onde - Encyclopedie de l'environnement - Trajets des ondes acoustiques - tracks acoustic waves - perception of sounds
Figure 9. Tracks of acoustic waves emitted from A or B to the receiving ears OG and OD located respectively on the left and right of the human or animal. The phase shift between the perceived sounds is zero when they come from A ; it reaches a wavelength fraction equal to 2πd/λ when they come from B. For man this value is therefore close to λ/2 in air, λ/10 in water. For a dolphin that emits sounds with a frequency of 10000 Hz, d/λ is of the order of unity.

The human body, largely composed of water, does not reflect underwater sound waves. Because of the distance between their ears (about 17 cm), humans perceive two sounds in the air whose phase shift makes it possible to detect their origin. We have seen that for a sound with a frequency equal to 1000 Hz, the wavelength is equal to 34 cm. If the transmitter is located in front of this human (point A in Figure 9), at equal distance from both ears, they perceive this sound at the same time, without any phase shift. But if the transmitter is located in the ear line (point B in Figure 9), the sounds perceived by the ears have a time shift d/c of 1/2000 s and a value d/λ of 17/34, or half a wavelength [7]. This significant phase shift allows the brain to feel a sound relief, i.e. to know where the sound comes from. It is this phase shift that allows the conductor to detect the sound emitted by each instrument within a musical ensemble.

On the contrary, in water, the time difference between the sounds perceived by the two ears (d/c) is about 1/10,000 s (when the transmitter is located on the side), and their maximum phase shift. The value of d is then reduced to one-tenth of the wavelength at a frequency of 1000 Hz and becomes too low to allow this detection. Immersed in water, humans are therefore unable to feel the sound relief. On the other hand, marine mammals such as dolphins emit sounds 10 to 100 times higher than the highest frequency perceived by the human ear. Their wavelength), between 1.5 and 15 cm, is then less than the distance between their ears (about 15 cm). The phase shift between these sounds is then reduced to a significant fraction of the wavelength even when the transmitter is not in line with their ears. It allows them not only to communicate with each other, but also to situate themselves in relation to each other and in relation to any possible obstacles in the darkness of the seabed.

In solid media, sound spreads even faster than in liquids. This is because solids are even less compressible than liquids. Thus, in the anvil steel mentioned at the beginning of this article, the speed is about 5000 m/s. If its length is close to 50 cm, it means that the entire anvil has felt the shock of the hammer in one ten-thousandth of a second, while the sound waves in the neighbouring air take almost 2 milliseconds to travel the same distance. In other words, approximately, it is the entire anvil, and not just the area impacted by the hammer, that emits the sound heard by the blacksmith.

 


References and notes

Cover image. The acoustic waves emitted by a radially vibrating sphere propagate in all directions in the form of alternating compression and relaxation of the gas layers adjacent to each other. Source: By Thierry Dugnolle (Own work)[CC0], via Wikimedia Commons]

[1] The compressibility of air is both large enough for this fluid medium to carry sound and small enough to justify the use of incompressible approximations to describe aerodynamics at speeds well below the velocity of sound.

[2] The negative electrical charges of the electrons surrounding the nuclei of the hydrogen and oxygen atoms that form the water molecule are subjected to the Coulomb force that strongly pushes them away from each other when their distance becomes very small. In turn, these electrons transmit this repulsive force to atoms and molecules.

[3] The logarithm is the mathematical operation that substitutes an addition for a multiplication: log (ab) =log(a)+log(b). Multiply the sound pressure by 10, the number of which the decimal logarithm is the unit, then results in the addition of a unit to its logarithm. In the usual notations, log10 refers to the decimal logarithm, while Log refers to the naperian logarithm, whose base is the irrational number e = 2.71828..

[4] Name in honour of Graham Bell (1847-1922), Scottish scientist and inventor of the telephone. The decibel (dB) is one-tenth of the Bel, very rarely used.

[5] The range of a sound characterizes its pitch, i.e. its frequency, whether it is a sound produced by an instrument or a human voice. The range of the violin is between 300 and 1400 Hz, that of the cello between 70 and 750 Hz and that of the double bass between 60 and 350 Hz.

[6] A waveguide is a physical system used to confine waves in a particular environment, at least over a certain distance. It is widely used with light and electromagnetic waves in general, for example in optical fibres. In the case of acoustic waves, particularly in seawater, its effectiveness is less clear-cut due to the more fuzzy nature of the interfaces, which do not channel the different wavelengths in the same way.

[7] With a 90-degree orientation difference of emitters A and B in Figure 9, we can therefore associate a 360/20 = 18-degree phase shift with respect to the axis of the human body, for a frequency of 100 Hz for which d/λ=1/20. The ratio 18/90 = 1/5 characterizes the sensitivity of humans to sound relief in the air.


The Encyclopedia of the Environment by the Association des Encyclopédies de l'Environnement et de l'Énergie (www.a3e.fr), contractually linked to the University of Grenoble Alpes and Grenoble INP, and sponsored by the French Academy of Sciences.

To cite this article: MOREAU René (February 7, 2019), The emission, propagation and perception of sound, Encyclopedia of the Environment, Accessed December 9, 2024 [online ISSN 2555-0950] url : https://www.encyclopedie-environnement.org/en/physics/emission-propagation-and-perception-of-sound/.

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声音的产生、传播和感知

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son - propagation - onde - perception of sound - human ear

  轻柔的乐符、发动机的嗡鸣、电钻的尖响……这些声音是如何产生,又是如何传到我们耳中的呢?声音由在可压缩介质(特别是液体和固体)中传播的振动组成。在不同的介质中,声音的特性也各有不同。本文不仅讨论频率、波长和声强的概念,还会以秋千和小提琴的乐音为例,来进一步说明模态和共振的概念。

1. 声音究竟是什么?

环境百科全书-声音的发射、传播和感知-声波传播到达听众的耳朵里
图1. 管弦乐队演奏乐器时,产生的声波传播到听众耳中的听觉感受器。[图片来源:Pixabay]

  声音又名声波(acoustic waves),是一种疏密相间的波,能够通过空气或任何其他可压缩的介质,从声源传播到接收者;声源通常为受到机械激励的具有一定弹性的固体材料,如:一块受到锤子敲击而振动的铁砧,一根被音乐家手指拨动的吉他琴弦,或一片因电磁铁作用而鼓动的扬声器膜片。同样地,空气或水等流体在受到扰动(如螺旋桨、喷嘴喷出的气流、车辆运动产生的湍流等)而出现压力波动时,也会发出声音,并在环境中传播。

  流体中的分子可以向周围传递自身振动,带动相邻的分子层随之运动,这些分子层又将这种疏密相间的振动继续传递给下一分子层,如此循环(如封面动画所示)。众所周知,空气看似静止,却包含大量不断无规则运动的分子,且可以压缩[1]。空气对改变其分子层密度和位置的振动非常敏感,因而能够传播声波。水同样由无规则运动的分子组成,但密度要比空气大得多(约为空气的800倍),而可压缩性则弱得多(约为空气的1/100000),分子之间存在着难以克服的库仑斥力[2]。水和其他液体也具备传导声音的能力。

环境百科全书-声音的发射、传播和感知-人耳
图2. 人耳的解剖示意图。其中听小骨固定于鼓膜上,其振动经由听觉神经传递给大脑。
[图片来源:DR]
(图2.Oreille externe 外耳,Oreille moyenne 中耳,Oreille interne 内耳,Pavillon 耳廓,Conduit auditif 耳道,Osselets 听小骨,Nerf auditif 听觉神经,Tympan 鼓膜,Cochlée 耳蜗)

  当分子间的压力变化传播到接收器,通常是鼓膜(图1)或麦克风膜片等柔性膜时,接收器将随最近的分子层一同振动。对于人耳或动物耳(图2)而言,听觉神经会将接收到的信息传递给大脑,进行识别和解读;而对于麦克风而言,膜片的振动将被转换为电信号记录下来,或由音频扬声器放大后重新传输。

2. 速度、频率和波长

  在空气等气体中,声音的传播得益于一大基本性质:分子的运动性。在标准状态下,分子运动的平均速度约为480 m/s(详见《压强、温度和热量》)。但是这种运动并不是定向的,而是向四面八方扩散能量。然而,声音在空气中的传播速度,或者说波速(celerity),反映的则是一个极小体积内的所有分子(即一个流体质点)在同一方向上的集体有序位移。这就解释了为什么声速虽然与分子的平均速度有关,但却只能达到该平均速度的一个分数,约为340 m/s。

  真空中的光速约为300000 km/s,空气中的光速与之相差无几。相比之下,声速就缓慢得多。正因如此,在足球比赛中,坐在距离中圈约170米的看台上的观众,在看到球员的脚触球半秒钟后,才能听到踢球的声音。尽管如此,但在相对较短的距离内,如音乐厅一类的小空间,声波就能够原原本本、近乎同步地将乐符信息传递到乐迷耳中供其欣赏。关于声速与光速,有一道经典的练习题:在暴风雨中,我们几乎是瞬间就能看到闪电,那么该如何估算出风暴距离我们有多远呢?

环境百科全书-声音的发射、传播和感知-麦克风膜片的振幅随时间的变化
图3. 麦克风膜片的振幅随时间的变化图象。由此可见声音的振动性质,以及不同的频率和振幅。[来源:Pixabay]

  周期(period)T是所有波的主要性质之一;对于声波而言,T是以给定时间点为始,两个相邻密部或疏部的间隔时间。相比之下,周期的倒数即频率(frequency)ff = 1/T)更为常见,且可与周期起到相同的描述效果。频率表示每秒钟振动的次数,单位为赫兹(Hz,1 Hz = 1s-1)。在声速c和周期T或频率f已知的情况下,可以很容易地计算出声音的振动波长(wavelength of sound vibration)λ = cT = c/f,即两个相邻密部或疏部之间的距离。1000 Hz的频率刚好位于可听声波的频段中点,而与该频率相对应的波长为34 cm(见图3及后文段落)。

  在我们所处的环境中,一切物体都处于运动当中,也因此会不可避免地发出声音。在我们体内,血液在循环,肺部在扩张与收缩,消化系统也在持续运转,但我们相对柔软的肉体可以很好地吸收这些压力变化,因此,医生需要用听诊器才能听到我们的心跳或支气管内气流的速度变化。此外,人耳只对15 Hz到15000 Hz这段相对较窄的频谱内的声音敏感。因此,除了自己的嗓音,我们感知不到体内的其他响声,这也使得我们能够将注意力更好地集中在外界的声音上,这些声音通常更悦耳,最重要的是,它们能让我们捕捉到更为有用的信息。不同种类的动物能感知到的声音频段也大相径庭。例如,大象能够发出并听到低于10 Hz的次声波,彼此之间也用该频段的声波进行交流。与之相反,蝙蝠却能发出并听到频率超过30000 Hz的超声波,这些超声波在周围环境中反射产生回声,帮助它们探测完全黑暗的环境。

  通常来讲,声速与频率无关,但取决于介质密度,因而对温度和压力的变化非常敏感。介质的密度越小、可压缩性越弱,声速就越大。所以,在可压缩性接近空气、但密度却比空气低得多的氦气中,声速约为其在空气中的3倍。

3. 声音的强度

  我们一般用声压(sound pressure)p来表示声音的强度,这一物理量指的是声波造成大气压力局部波动的最大振幅。通常来说,这种波动是非常微弱的。例如,一个正常说话的人会在一米外产生约0.01帕斯卡(Pa)的声压,是正常大气压(约为105 Pa)的千万分之一。

  声压与图2所示的最大振幅成正比;而相较于声压本身,人耳听觉实际感知到的声音强度显然更与其对数[3]成正比关系。这意味着当声压高出10倍,人耳感受到的声音强度是原来的2倍;声压高出100倍,则人耳感受到的强度是原来的4倍。声波的强度与声压的平方成正比,故取10 log10p2/p2ref=20 log10p/pref来表征声音的强度,并称之为分贝。分贝(dB)或1/10贝尔是常用的声强单位,是为了纪念发明电话的苏格兰科学家格雷厄姆·贝尔(Graham Bell,1847-1922)而命名的。它等于20 log10p/pref),其中p表示声压,pref为参考压力,随机设定为20 μPa(20×10-6帕斯卡,或十亿分之零点二个标准大气压),这一参考值能够代表大多数人的最低听力阈值[4],即0 dB。因此,声音强度每增加20分贝,则表示声压或声音振幅扩大了10倍,或声音的强度(又称声强,sound intensity)扩大了100倍。

  以下给出了一些情况下的声音强度数值,它们赋予分贝实际意义:

  • 宁静的乡村:40 dB或p=2×10-8个标准大气压。
  • 车水马龙的繁华街道:80 dB或p=2×10-6个标准大气压。
  • 飞机起飞时的机场:120 dB或p=2×10-4个标准大气压。
  • 短时间内暴露的法定安全限值:135 dB或1/1000个标准大气压。

  众所周知,当呼喊一个正在走远的人时,我们需要不断地提高嗓门。任何声源发出的声音在传播过程中,其强度都会迅速降低,原因有二:第一,声波离开声源后,其能量会呈半球面状不断扩散,因此,随着半球面积的增加,声波强度也逐渐降低,这一趋势可用与声源距离平方的倒数来表示。第二,这种由气体压缩和膨胀而形成的初始能量,会经由粘性耗散转化为热量(详见《压强、温度和热量》)。

4. 简谐运动:摆锤和秋千

环境百科全书-声音的发射、传播和感知-单摆
图4. 单摆。几乎所有的质量都集中在下端的圆锥体上,圆锥体的摆动会扫过以链条顶部连接点为圆心的圆弧。[图片来源:Pixabay]

  让我们以单摆为例,开启对振动现象的探索与发现。单摆只有一个自由度[即用于表示物体在运动轨迹上的位置或重心的参数或坐标],重心始终保持在一段弧形的轨迹上(图4)。当外力刺激刚刚停止,起始扰动结束的情况下,摆锤将以一个明确的频率继续摆动,该频率被称为固有频率(natural frequency)。在小幅摆动的情况下,固有频率等于(1/2π)(l/g)1/2,其中l表示单摆的长度,g表示重力加速度。固有频率取决于重力,这一事实反映了势能(高点处的最大值)和动能(低点处的最大值)之间的转换。上述这种有规律的周期性运动正是摆锤的振动模式。

  现在,假设我们以给定的频率为摆锤增加额外的能量。例如,每当摆锤通过最低点时,都推动它一次,来提升其速度;这一额外的能量将与摆锤先前的能量累加,使摆锤的振幅增大。我们称这种推动与摆锤的振动同相(in phase),并且引发了摆锤的共振(resonance)。共振会增大振幅,每次推动增加的能量都刚好可以与振动过程中耗散的能量相抵。相反地,如果系统得不到能量供给,每振动一次,摩擦力就会耗损一部分能量,振幅也因此会逐渐减小。

环境百科全书-声音的发射、传播和感知-处于最低位置的秋千
图5. 处于最低点的秋千。向前伸腿和挺胸可以给秋千增加额外的动力,同时也能减少摩擦力。
[来源:《少女心中的爱情升华》,弗拉戈纳尔(Fragonard)的布面油画]

  秋千的例子(图5)展现了另一种形式的共振。孩子们稍加练习就能加速冲向最低点,从而增加摆动的幅度。此外,值得注意的是,当秋千经过最低点时,他们会尽可能地沿着运动轨迹切线方向平躺,从而减少空气阻力,同时提供额外的能量。

5. 驻波:弦振动与音乐

  摆锤和秋千都不会发出声音(安全扣的吱吱声除外)。因此,除了上述的简单案例,我们也有必要借助新例子,进一步说明振动的物体是如何发出声音的。这种发声系统的原型是一根在两个定点之间拉紧的弦,在受到激励后开始振动,同时发出声音。这便是用小提琴、吉他、竖琴或其他弦乐器演奏乐曲的过程。这种机械系统非常复杂,因为在弦上存在无数个分布于任意位置的自由度,以及许多不同的模态。我们将其分波长是两定点间距两倍的基波(fundamental mode),和频率为基波整数倍的谐波(harmonics)。图6中的动画展示了最简单的三种模态,由此我们可以推测出无限多的可能的谐波。

环境百科全书-声音的发射、传播和感知-一根弦在两个定点之间的振动
图6. 两个定点之间的弦振动:基波和两种一次谐波。
[来源:克里斯托夫·当国·陈(Christophe Dang Ngoc Chan,cdang)(个人作品)[美国国家地球物理流体动力学实验室(GFDL),http://www.gnu.org/copyleft/fdl.html或知识共享许可协议(CC-BY-SA-3.0),http://creativecommons.org/licenses/by-sa/3.0/],维基百科共享资源]

  这种振动系统最有趣的特性之一是其能够形成驻波(standing waves),如图6的动画所示。驻波的鲜明特点在于它是由一系列的波节(knots),即振幅为零的点,和波腹(bellies),即最大振幅所在的点组成的。波节和波腹的数量取决于两个固定点之间的距离和弦的张力。弦上的波似乎停在原地,并未沿着某一方向传播,这一现象正是驻波这一名字的由来。这种非常特殊的波也能够在周围空气中传播,不断地将弦发出的声音传递到听众的耳朵或麦克风中。音符的时长则取决于施加的激励(如,用琴弓摩擦小提琴琴弦)的时长。

  此外,在固定端和自由端之间也可以产生驻波。但这种驻波的基波只包含1/4波长。音叉就是这一现象的最佳写照,它由两根金属棒而非振动的弦组成,却能够发出音准极高的“A”(音名,对应唱名La,译者注),可作为整个管弦乐队的参考音。

  通常,弦的振动可闻却不可见,但下面视频中的驻波火焰管却是一个特例。沿管道分布的一系列小孔处可以显示管中丙烷气压的周期性波动。无论是通过小孔的气体流量,还是每个小火焰的高度,都与该处的压力成正比,在波节处为零,在波腹处最大。

  在所有乐器,尤其是弦乐器中,声音的产生源自物理激励带来的振动与一组特定模态(包括谐波)之间的共振(如图6与上文视频所示)。音乐家正是通过找寻这种共振,来选择自己想要演奏的音符。所发出声音的音准取决于对激励方式的精确选择。一般来说,单根弦的振动几乎是听不到的。这就解释了为什么小提琴、大提琴和吉他的琴弦要与共鸣板(soundboard)相连,二者与其他部件又共同组成共鸣箱(sound box)(图7)。共鸣箱本身具有振动波,可以与琴弦的振动形成共振。

环境百科全书-声音的发射、传播和感知-未经修饰的小提琴音箱的横截面图
图7. 未上漆的小提琴共鸣箱的横截面图。1.2.背板和面板,3.尾木,4.尾钮,5.音柱,6.f孔,7.侧板,8.角木,9.衬条,10.低音梁。
[图片来源:英语:未知照片。由德克(Dake)改进和(略微)补色。编号由SuperManu添加;法语:原始摄影师未知。由德克(Dake)改进和(略微)补色。编号由SuperManu添加。(路德艺术(奥古斯特·托尔贝克(Auguste Tolbecque)))
[来源:美国国家地球物理流体动力学实验室,GFDL(http://www.gnu.org/copyleft/fdl.html),知识共享许可协议(CC-BY-SA-3.0,http://creativecommons.org/licenses/by-sa/3.0/,CC BY-SA 2.5-2.0-1.0,http://creativecommons.org/licenses/by-sa/2.5-2.0-1.0),维基百科共享资源]

  得益于其尺寸,共鸣板可以将声音放大并传播到空气中,进而传播给听众。最后,需要说明的是,共鸣板的面积决定了乐音的音域[5]。从500 cm2的成人小提琴(图7中AB点之间的长度约为35-36 cm)到1000 cm2的大提琴,再到2000 cm2的低音提琴,不同音域的乐器共同组成了交响乐团。

  悬崖峭壁、古剧院的围墙、隔音屏障……众所周知,天然或人造的壁面会对感知到的声音信号产生显著影响。这种影响是由两个因素共同决定的:一是其形状是否利于共振,二是其表面条件。光滑、坚固、富有弹性的壁面能很好地反射声音;相反地,粗糙或者涂有柔性吸波材料的墙壁则不利于声音的反射,这与回声(echo)现象有关。目前,对回声现象的研究已有诸多重要的应用,如设计高音质音乐厅,减轻噪声污染等。

6. 声音在水和固体中的传播

  因为声音的产生源自介质的可压缩性,所以所有的介质都可以传导声音,特别是水等液体。人们对声音在海洋中的传播尤其感兴趣,因为光不能穿透很深的海域,这种情况下,声音就成为了首选方案之一。渔民探测鱼群,地理学家考察水下的地形地貌,世界各国海军识别附近敌友舰艇,都离不开声波或超声波。海洋中的哺乳动物也是通过超声波来进行交流的。在海水中可以使用的音频范围从30 Hz到1.5 MHz,这个值比15000 Hz的人类听觉极限高出约100倍。水中的声速约为1450至1550 m/s,如图8所示,它主要随温度和深度(即压力)变化而变化,但对盐度变化并不敏感(见《海洋环境》)。

环境百科全书-声音的发射、传播和感知-大西洋的温度T、盐度S和声速c随深度而产生的典型变化
图8. 大西洋的温度T、盐度S和声速c随深度的变化趋势;可见盐度与温度变化对声速的影响主要体现在接近海面的浅水区,其中,盐度变化的影响比温度变化的影响小得多;而压力变化的影响在深海区域更为明显。
[来源:http://lecalve.univ-tln.fr/oceano/fiches/fiche3F.htm]

  在海洋中,声波会在海水表面发生反射,因此在水上是无法直接听见水下的噪声的,需要使用声纳(sonar)这种特殊装备进行采集。此外,在海底、海洋中的分层处,如温跃层(见《海洋环境》),以及不同密度水体的交汇处,特别是主要的河流入海口附近,声波也会发生反射。即使没有穿过清晰的分层,声音的路径也很少是一条直线。声波通常会向声速较低的区域转折并聚集,并在该区域内发生波导(waveguide)作用[6]。对于海洋表层和声速最低的中间层(图8中约1000 m处)而言,这一现象尤为明显。1000 m以下的各层因此形成了“声音盲区”,只有将发射和接收系统置于这一深度以下才能对其进行探测。

环境百科全书-声音的发射、传播和感知-声波轨迹
图9. 从声源A和B分别传播至人或动物的左耳OG和右耳OD的声波轨迹。当声音来自A处时,两耳感知到的声音相移为零;当声音来自B处时,相移则达到2πd/λ。对于人类来说,这个值在空气中接近λ/2,在水中则接近λ/10。对于发出频率为10000 Hz声音的海豚来说,d/λ的数量级一般为1。

  人体主要是由水组成的,因而不能在水下反射声波。由于双耳之间存在一定距离(约17cm),人类在空气中可以感知到两种不同的声音,而它们的相移(phase shift)使得追溯声源成为可能。前文提到,频率为1000 Hz的声音,其波长等于34 cm。如果声源位于听者前方(图9中的点A),且与左右耳的距离相等,那么两只耳朵会同时感知到这个声音,而不存在任何相移。但如果声源位于两耳的连线上(图9中的B点),耳朵感知到的声音会产生一个d/c为1/2000 s的时移,d/λ值为17/34,即波长的一半[7]。这种显著的相移能够使大脑感知到声源方向。指挥家可以根据相移识别出合奏中每种乐器发出的声音。

  相比之下,在水中,当声源位于侧面,即相移最大时,人耳感知到的声音时移(d/c)约为1/10000 s。当频率为1000 Hz时,d的值仅相当于波长的十分之一,无法被人耳察觉。因此,人类在水下无法辨别声源。然而,对海豚等海洋哺乳动物来说,其发出的声音比人耳能感知到的最高频率还要高10到100倍,波长(λ)一般在1.5-15 cm之间,小于它们两耳之间的距离(约15 cm)。因此,即使声源与它们的耳朵不在一条直线上,与波长相比,这些声音的相移也已经足够显著,不仅能让它们相互交流,还能让它们在黑暗的海底定位彼此或可能的障碍物。

  在固体介质中,声音的传播速度甚至比在液体中还要快。这是因为固体比液体的可压缩性更弱。在本文开头提到的铁砧中,声音的传播速度约为5000 m/s,如果其长度约为50 cm,则意味着整个铁砧仅在1/10000 s内就能感受到锤子的冲击,而空气中的声波需要几乎2 ms才能传播相同的距离。换言之,铁匠听到的打铁声,是整个铁砧,而不仅仅是被锤子敲击的区域发出的。


参考资料及说明

封面图片:由径向振动的球体发出的声波,通过相邻气体层疏密相间的振动,向四面八方传播。[来源:蒂埃里·杜格诺尔(Thierry Dugnolle)(个人作品)[CC0],维基百科共享资源]

[1] 空气的可压缩性既大到足以使这种流体介质传导声音,又小到可以在远低于声速的速度下用不可压缩的流体力学方程来近似描述空气动力学。

[2] 构成水分子的氢原子和氧原子的原子核周围,带负电荷的电子受到库仑力的作用,在距离非常小时,彼此之间会发生强烈排斥。这种排斥力也会影响原子和分子。

[3] 对数是一种数学运算,两个数乘积的对数等于这两个数各自的对数之和:log (ab) =log(a)+log(b)。例如,声压增加10倍,则其以10为底的对数只增加1倍。在通常的记法中,log10 指的是以10为底的对数,而Log指的是以e为底的纳皮尔(Naperian)对数,其底数为无理数e = 2.71828…

[4] 分贝(dB)是以苏格兰科学家、电话的发明者格雷厄姆贝尔(Graham Bell,1847-1922)的名字命名的。分贝是贝尔的十分之一,但我们很少直接使用贝尔做单位。

[5] 音域是描述音高,即乐声或人声频率的特征,小提琴的音域在300至1400 Hz之间,大提琴的音域在70至750 Hz之间,低音提琴的音域在60至350 Hz之间。

[6] 波导是一种物理概念,指的是在一定距离内,将波限制在特定的区域中。波导现象通常广泛应用于光波和电磁波,一个典型例子就是光纤。而对于声波,特别是海水中的声波,波导的效果并不明显,因为尽管不同波长的声波在不同分层中的传播方式存在差异,但水下各层之间的分界却较为模糊。

[7] 图9中声源A和B之间的角度为90°,当声音频率为100 Hz时,d/λ=1/20,以人体中轴线计算,相移约为360/20=18°。18/90=1/5,这一比率体现了人类对空气中声音源头的敏感度。


The Encyclopedia of the Environment by the Association des Encyclopédies de l'Environnement et de l'Énergie (www.a3e.fr), contractually linked to the University of Grenoble Alpes and Grenoble INP, and sponsored by the French Academy of Sciences.

To cite this article: MOREAU René (April 12, 2024), 声音的产生、传播和感知, Encyclopedia of the Environment, Accessed December 9, 2024 [online ISSN 2555-0950] url : https://www.encyclopedie-environnement.org/zh/physique-zh/emission-propagation-and-perception-of-sound/.

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