Sound

The sound is defined as a perturbation wavelike that typically propagates as an audible wave of pressure in an elastic medium (such as a gas, liquid, or solid) and which generates an auditory sensation.

The wave phenomenon, associated with the sound, causes that the numerous particles of the medium in which it is transmitted to vibrate, thus propagating the disturbance to neighboring particles. While this perturbation is propagated, which carries both information and energy, the individual particles, even in the case of fluids (gases and liquids), always remain in the proximity of their original position. In other words, there are local vibrations (compression and rarefaction) of particles:

  • in the case of gases or liquids, which cannot transmit shear stresses, these vibrations are always parallel to the direction of the propagating wave. Therefore we speak of longitudinal waves;
  • in the case of solids, which can transmit shear stresses, there are also vibrations perpendicular to the direction of the wave, which therefore corresponds to transverse waves.

The displacement characteristics of the particles around the equilibrium positions depend on the characteristics of the source that produced the perturbation.

In acoustics, in addition to the speed of propagation (which measures the speed with which the signal moves from one point to another of the transmission medium), other characteristic properties of the waves must be considered, such as the frequency, the period, and the wavelength.

The frequency, related to the rapidity with which the particles oscillate in every single point, is the number of oscillations per unit of time: is measured in cycles per second, Hertz [Hz]. In the case of normal-hearing adult individuals, the audible frequency range extends approximately from 20 Hz to 16000 Hz. The inverse of the frequency is called period (measured in seconds): it is the necessary time for the particles to make a complete oscillation.

Sound intensity

Sound intensity is defined as the sound power carried by sound waves per unit area. The usual context is the measurement of sound intensity in the air at a listener’s location. Sound intensity is not the same physical quantity as sound pressure. The basic units are W/m2 or W/cm2. Many sound intensity measurements are made relative to a standard threshold of hearing intensity I0:

\[I_0=10^{-12}\;\dfrac{\textrm{W}}{\textrm{m}^2}=10^{-16}\;\dfrac{\textrm{W}}{\textrm{cm}^2}\]

The most common approach to sound intensity measurement is to use the decibel (dB) scale:

\[I_{dB}=10\log_{10}\left[\dfrac{I}{I_0}\right]\]

Decibels measure the ratio of a given intensity \(I\) to the threshold of hearing intensity so that this threshold takes the value 0 decibels (0 dB). To assess sound loudness, as distinct from an objective intensity measurement, the sensitivity of the ear must be factored in.

Sound pressure (acoustic pressure)

Sound pressure or acoustic pressure is the local pressure deviation from the ambient (average or equilibrium) atmospheric pressure, caused by a sound wave. In the air, sound pressure can be measured using a microphone, and in water with a hydrophone. Since audible sound consists of pressure waves, one of the ways to quantify the sound is to state the amount of pressure variation relative to atmospheric pressure caused by the sound.

Infrasound

Infrasound sometimes referred to as low-frequency sound, lower in frequency than 20 Hz or cycles per second of the “normal” limit of human hearing (20 kHz).

The study of such sound waves is sometimes referred to as infrasonics, covering sounds beneath 20 Hz down to 0.1 Hz and rarely to 0.001 Hz. This frequency range can be used for monitoring earthquakes, charting rock and petroleum formations below the earth, and also in ballistocardiography and seismocardiography to study the mechanics of the heart.

Infrasound is characterized by an ability to get around obstacles with little dissipation.

Ultrasound (ultrasonics)

Ultrasounds are mechanical sound waves. Unlike acoustic phenomena, the frequencies that characterize ultrasounds are higher than those normally heard by a human ear.

The frequency conventionally used to discriminate acoustic waves from ultrasonic waves is set at 20 kHz up to several gigahertz, higher than the upper audible limit of human hearing (below 20 kHz in healthy young adults).

Ultrasounds are generated using piezoelectric materials, which have particular mechanical-electrical characteristics. These particular materials such as quartz or barium titanate have the characteristic of generating an electrical potential difference if compressed or stretched in the transverse direction; vice versa, if a potential difference is applied to their extremes, these are compressed or dilated in a transverse direction.

This last feature is used to generate these mechanical waves above the audibility range (ultrasounds). Depending on the type of material: different frequencies of ultrasound, different propagations in the materials, and therefore different power characteristics of the generating machines are obtained.

A second system for generating ultrasounds is based on magnetostriction: a ferromagnetic core subjected to an alternating magnetic field (maximum 200 kHz) is put into vibration at ultrasonic frequencies. This system is, for example, used for the production of industrial ultrasonic washing machines.

Like any other type of wave phenomenon, ultrasounds are subject to reflection, refraction and diffraction phenomena and can be defined by parameters such as frequency, wavelength, propagation speed, intensity (measured in decibels), the attenuation (due to the acoustic impedance of the medium crossed). Few people can perceive them clearly and can become disturbing listening to them.

Ultrasound applications

Ultrasounds are mostly used in the medical and industrial fields as they are widely used in ultrasound scans, in non-destructive tests and many appliances used for surface cleaning of small objects. Ultrasounds are also used to nebulize water in some types of humidifiers.

They are often prescribed as physiotherapy following trauma or fractures, although there are conflicting studies on their effectiveness. Even sonar employs frequency ranges that often border on the ultrasound range.

The main applications beyond those mentioned above also relate to the mechanical field, especially in the welding of plastic materials and non-destructive testing of welds. The welding of plastic materials employing ultrasound is often used when a certain aesthetic quality is required but above all speed of execution. Two plastic objects (preferably of the same material so that the molecular friction is high) are put in contact with each other, and a metal parallelepiped (sonotrode) leans on one of them emitting ultrasounds and then putting it in vibration.

The friction generated will melt the plastic parts in contact by joining them. The shape and frequency at which the sonotrode will vibrate depend on the geometry of the object to be welded. The aesthetic quality is excellent even if water tightness is not ensured, so if it is a fundamental requirement, it is preferable to consider another type of welding (eg, hot-blade welding).

Sound propagation speed

Sound waves propagate with a characteristic speed of the transmission medium: while the frequency of local vibrations depends on the source, the propagation speed depends exclusively on the transmission medium.

Sound propagation in gases

In the case of ideal gases (which can also be considered air in standard temperature conditions, 25 °C, and pressure, 1 atm), the sound propagation speed, which will be denoted by c, can be expressed by the following relationship:

\[c=\sqrt{\dfrac{kp_0}{\rho_0}}\]

where k = cP/cV (the so-called adiabatic index) is the ratio between the specific heat at constant pressure and the specific heat at constant volume; p0 [Pa] is the gas pressure and ρ the density (mass per unit of volume) of the gas itself.

Considering adiabatic transformations (without heat exchanges) derives from the fact that the sound propagation speed in the medium is so high, compared to the speed with which heat exchange processes take place, that these processes can be considered null.

Demonstration: having to do with a perfect gas, we can use the equation of state of ideal gases:

\[p_0V_0=nR_0T_0=\dfrac{m}{m_M}R_0T_0\]

where, with reference to the considered gas, V0 is the volume of the gas itself, n [kmol] the amount of gas, T0 [K] is the absolute temperature (measured in K), R0 = 8314 [J/kmol⋅K] the universal constant of gases, m the mass, mM [kg / kmol] the molar mass. Taking into account that ρ0 is the mass per unit of volume (the density), we can use the equation of state to write that:

\[\rho_0=\dfrac{m}{V_0}=\dfrac{p_0V_0m_M}{R_0T_0V_0}=\dfrac{p_0m_M}{R_0T_0}\]

Substituting this expression in that the sound propagation speed, we obtain that:

\[c=\sqrt{\dfrac{kT_0R_0}{m_M}}\]

Based on this last relation (known as Laplace’s law), we can say that the sound propagation speed is independent of the gas pressure, while it is directly proportional to the square root of the absolute temperature.

In the particular case of air, knowing that k = 1.4 and that the molar mass is mM = 29 [kg/kmol], that relationship leads to obtaining \(c = 20.04\sqrt{T_0}\) [m/s]. Finally, if we refer to the temperature expressed in °C, which we indicate with \(\xi\), we can use, with good approximation, the following relation: \(c = 331.2 + 0.6\xi\) which shows, in practice, that the speed of sound increases by 0.6 m/s for every 1 °C increase in temperature.

Sound propagation in liquids

In the case of liquids the sound propagation speed can be calculated using the following equation:

\[c=\sqrt{\dfrac{1}{K\rho}}\]

where \(K\) is the compressibility coefficient of the liquid under adiabatic conditions and ρ the density (the mass per unit volume). Based on this relationship, the speed with which the sound propagates in a liquid grows with decreasing density.

In most cases the speed of propagation in liquids is greater than in gases.

Sound propagation in solids

In solids, we can have both longitudinal waves, for which the displacement of particles takes place in the same direction of wave propagation, and transverse waves, for which the displacement occurs instead in the orthogonal direction to the direction of propagation.

Considering the longitudinal waves, for which the speed of sound, which we indicate with cl (where the l is for longitudinal), is different according to the geometric shape:

  • for a solid whose shape is mainly longitudinal, we have:

\[c_l=\sqrt{\dfrac{E}{\rho}}\]

  • for a solid in the form of an indefinite plate (extended surface prevalent than the thickness), we have instead that:

\[c_l=\sqrt{\dfrac{E}{\rho(1-\nu^2)}}\]

where E [Pa] is the Young’s modulus, n is the Poisson’s coefficient and ρ the density of the material of which the solid is made.

Finally, as regards transverse waves in solids, their speed ct can be estimated by the following relation:

\[c_t=\sqrt{\dfrac{E}{2\rho(1+\nu)}}\]

In most cases, the speed of sound in solids is higher than that in the air.

Soundproofing

For sound insulation or soundproofing (or acoustic shielding) means all those actions aimed at limiting the unwanted sound and noise transmission, usually by introducing sound-absorbing materials in the path of the sound waves. The application fields where insulation or soundproofing measures are necessary are many, including recording studios, rehearsal rooms, cinemas, and environments dedicated to audio-video, residential, offices, and public places, industry, motoring, boating, hobby, etc.

Sound enters a room in three ways: through the air, through the structure, or by the diaphragm action of floors, walls, and ceilings. Airborne sounds are absorbed by using thick or absorbent walls in which there are no cracks or ducts; doors and windows should fit tightly and be seated in rubber or felt liners. Windows should be double-glazed. Structure-borne vibrations and diaphragm effects are reduced by using double walls with insulating material but as few ties as possible between them.

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