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 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√T0 [m/s]. Finally, if we refer to the temperature expressed in °C, which we indicate with ξ, we can use, with good approximation, the following relation: c = 331,2 + 0.6ξ 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.

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