The phenomenon of sound

We can say that the phenomenon of sound is "what we hear" (or even better, "what we listen to"). This definition may seem simplistic (Does the phenomenon of sound exist even when no one is there to perceive it?) but, in its generality, it offers the advantage of highlighting the four phases that we believe sound is always made up of:

1. production of mechanical waves by a vibrating source called sound source. Some examples of sound sources are:
   • musical instruments, of which the vibrating part can be a string that is struck (like the piano) or drawn with a bow (like the violin), a membrane, a bar, a plate (like percussion instruments) or a column of air where the vibration is controlled by the musician's breath (like wind instruments);
   • our vocal cords, which are made to vibrate by air coming from our lungs giving rise to the human voice;
   • all phenomena that provoke a "movement of air" (the beating of a hummingbird's wings, an aeroplane breaking the supersonic barrier, an exploding bomb, a hammer banging on an anvil, etc.) having the appropriate physical characteristics (we do not "hear" all movements of air as a sound event; for more details, see the page on audible frequencies).
2. propagation of waves through an elastic medium (usually air); this propagation is the true wave phenomenon and will be called a sound wave from now on.
3. reception and perception of a sound wave by an appropriate apparatus that can transform (and eventually process) sound energy into other forms of energy (e.g. the human auditory system).
4. processing of transformed signals (usually into electrochemical impulses) by the brain.

What makes the study of the world of sound particularly interesting and, at the same time complex, is the fact that it is tightly intertwined with both objective and subjective parameters.

• Objective parameters are the physical quantities that describe the vibrations of sound sources and that characterise sound waves and their propagation. These are sound wave properties that are independent of listeners. Some examples of which are frequency, wavelength, propagation velocity, etc.
• Subjective parameters are all the properties of sound that are perceived and depend on the sensory and cerebral processing of the sound "stimulus" by the subject that is listening. Some examples are pitch, loudness and timbre, as well as more subtle qualities such as pleasantness, consonance and harmony.

Despite the subjective nature of individual perceptions, we study the sound phenomena in all four of its phases to better define both its physical and perceptive parameters and to possibly link them to each other. Even if the physicist's most reductionist dream is to "explain" all the qualities of the sound phenomenon in terms of only objective and measurable quantities, this field holds many partial answers and open problems that are still being studied, which makes it all the more fascinating.

On this page, we will study the second phase, which involves the identifying of what we call sound waves.

What is a sound wave?

In physics, a wave is a perturbation that propagates in space and can transport energy from one point to another through the variation of a physical quantity (see the page on What is a wave?). A sound wave is a particular type of wave in which the perturbation is the variation of pressure created by a vibrating body in its surrounding medium (usually air). This pressure variation can propagate in the medium as a succession of rarefactions and condensations (or rather, as a variation of density). The following animation shows what happens when a vibrating body, in this case a moving wall, is made to oscillate by a motor and piston.

The areas of rarefaction and increment in molecular density are obvious and we can see that what is "advancing" is the wave front, in other words the compression of the medium and not the air molecules, which are only subject to small movements around points of fixed equilibrium, as is shown by the reference molecule highlighted in red.

• Observations

1. We can see that the air density has changed locally but the particles oscillate around a position of equilibrium and are not carried away as if they were in a current.
2. However, the areas of compression-decompression move in the medium with a velocity that depends on the characteristics of the medium itself, particularly its compressibility.
3. As we can see, the molecules oscillate in the same direction as the wave propagation. Technically, we say that a sound wave is a longitudinal wave.
4. If the medium were a solid, the propagation of the elastic wave could also have transverse components because, in solids, these components are favoured by the presence of intermolecular forces that are different from those present in liquids. However, the human ear is insensitive to transverse elastic waves and, therefore, the name "sound wave" is reserved for longitudinal elastic waves in any medium.

Physical properties of sound waves

If we want to quantitatively analyse the sound wave, we must identify the measurable characteristics of the phenomenon.

Medium properties

If we focus on the medium, we can, both point by point in space and moment by moment in time, measure:

• the pressure difference of air with respect to normal atmospheric pressure. This difference is called acoustic pressure. The set of all the local acoustic pressures in the whole space forms a "pressure field". We must point out right from the start that acoustic pressure is a small "ripple" with respect to standard air pressure. Even in the case of extremely intense sounds, its value is about a thousand times less than that of air pressure.
• the density difference of air with respect to air density at equilibrium.
• the displacement of air molecules from their position of equilibrium and their velocity. In this case, we have a "displacement field" and a "velocity field".

These properties are not independent of each other and are in fact connected to each other by, for example, the gas laws (for an example, see the page on Acoustical impedance).

Wave properties

If, while observing the animation, we focus our attention on the wave instead of the medium in which it is propagating, we could measure:

• the period T of a perturbation, which is the time that elapses between the moment in which maximum pressure is reached at a prefixed point and the moment in which this situation occurs again at the same point. Put more simply, in the case of the reference molecule, it is the time that it takes to complete one oscillation around its position of equilibrium;
• the frequency ${\displaystyle f}$ of a wave, in other words, the number of times the molecule oscillates within a unit of time;
• the wavelength ${\displaystyle \lambda }$, which, at any given moment, is the distance between two consecutive areas of greater density increment (dark areas), areas where the acoustic pressure is at its maximum;
• the amplitude of the oscillation, which is the maximum displacement of molecules with respect to their condition of rest; we will see that this is directly linked to the maximum value that acoustic pressure can reach;
• the velocity with which a perturbation advances in a medium, as a ratio between ${\displaystyle \lambda }$ and T. Observe that this velocity does not coincide with the velocity of the individual molecules. It always points to the direction of the propagation of pressure fronts, while that of the molecules changes direction with every time period (see Velocity of mechanical waves);

Not even these properties are completely independent of each other. For an example, see the table on the Fundamental wave variables page, as well as the entire page on How is a wave described?.

More complex waves

Obviously, the animation shows a particularly simple sound wave, where the oscillation is periodic and described by a sine function. There are also oscillations that can be much more complex in two ways:

1. Even if they are periodic, they can have a non-sine shape. In particular, they can be made by the superposition of many sine waves with the appropriate frequency (as explained on the page dedicated to Fourier's Theorem).
2. They can be non-periodic. Actually, to a certain extent, all waves in nature are non-periodic, as they have a beginning and an end both for their duration in time and for their extension in space. However, a wave is usually considered periodic when its time duration is much greater then the time period. For example, a period for a La₃ at 440 Hz lasts only two milliseconds. Therefore, a La₃ that remains unaltered for at least two seconds can effectively be considered as an ideal periodic signal. However, instruments such as the guitar and piano emit sounds that change their loudness and timbre very rapidly in a two-second period and, therefore, should not be considered as periodic during that time.

In both cases, a spectral analysis of the wave can be done, which measures the frequency and amplitude of each sine making up the complete periodic wave. What is "measured" is called the frequency spectrum of the wave. The difference is that, in the case of periodic sounds, the spectrum does not change with time, while for non-periodic sounds the spectrum does change with time, forcing us to use an extra variable in its mathematical description.

From wave to sound

Now that we have defined several measurable properties of the sound phenomenon, we presumably wish to extrapolate indications for the quantitative study of sounds and even of music. However, several difficulties now arise because sound as we perceive it is not simply a faithful copy of sound waves:

1. the auditory system transforms the pressure variations hitting it in a rather complex way that is not always "faithful" to the original;
2. the brain carries out an enormous amount of signal processing, a task in which training or habit, and therefore cultural factors, as well as physical and physiological factors play a very important role.

As a consequence, connecting the subjective properties of sound sensation to measurable and objective physical properties is not always possible. At other times, it appears to be deceptively simple. For example, periodicity and spectral composition (see previous paragraph) were sometimes used to establish the first distinction between sound and noise. However, that distinction can become particularly difficult in some cases or even suffer due to cultural aspects (how would "heavy metal" have been classified by listeners in the 1800s?). If you are interested in further reading on this topic, see the section Sound or noise? and the section Questions and answers, where you will find questions and answers regarding these aspects.

Nevertheless, considering the enormous importance that music has played and continues to play in our civilisation, it is worth trying to discover what might be the physical or physiological basis of its various aspects. Therefore, it is legitimate to ask ourselves such questions as:

• Are there objective properties that allow for the distinction between sound and noise?
• Does the perception of intervals as either consonant or dissonant have a physical or physiological basis or is it only a question of cultural norms?
• Why is modern music in the west based on scales and intervals that are so different from those used in the music of the east?
• Can we associate a specific objective property to each property of perceived sound?

For other relevant questions, see the section on Questions and answers.

In-depth study and links

Several of these questions are dealt with in the following sections:

• Sound or noise?
• Perception of sound
• Physiology of the auditory system
• From sound to music
• Consonance and dissonance

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