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2.4 Acoustics

2.4.1 The speed and attenuation of sound

Gases and vapours

The attenuation of plane sound waves in neper/unit length is α = (l/2d)ln(I₀/I_d), where the initial intensity I₀, has decreased to I_d after traversing distance d. When expressed in decibels per unit length, convenient for practical application, the value is 8.686α. In normal monatomic gases α varies classically as f ², f being the frequency, but in polyatomic gases relaxation phenomena may dominate over classical absorption. (See Herzfeld and Litovitz, 1959.) For gases at moderate pressure the speed of sound c = √(γp/ρ) or alternatively c = √/(γRT/M), p being the ambient gas pressure, γ the ratio of the specific heat at constant pressure to that at constant volume, R the gas constant, T the absolute temperature and M the molecular weight. Thus, to first order, c is proportional to √T and is independent of gas pressure; at higher pressures, however, the value is changed due to contributions from the second and higher virial coefficients. The speed of sound waves bounded by walls or tubes is less than the free-space value. Dispersion is observed for sound propagation in many polyatomic gases due to molecular relaxation: above the relaxation frequency the rotational or vibrational degrees of freedom are not excited, the specific heats are modified and the speed of sound is increased. The table below gives experimentally determined values of sound speed in gases and vapours, selected from the literature. Where known, both the low-frequency and high-frequency limiting values are given, the latter being in brackets.

Speed of sound in gases and vapours

Attenuation of sound in air

The attenuation of sound in air due to viscous, thermal and rotational loss mechanisms is simply proportional to f ². However, losses due to vibrational relaxation of oxygen molecules are generally much greater than those due to the classical processes, and the attenuation of sound varies significantly with temperature, water-vapour content and frequency. A method for calculating the absorption at a given temperature, humidity, and pressure can be found in ISO 9613-1 (1993). The table gives values of attenuation in dB km⁻¹ for a temperature of 20°C and a pressure of 101.325 kPa. The uncertainty is estimated to be ± 10%.

Attenuation of sound in air (dB km⁻¹)

Speed of sound in air

The speed of free progressive sound waves in standard dry air containing 0.03% CO₂ by volume is 331.46 ± 0.10 m s⁻¹ at a temperature of 0°C and a pressure of 101.325 kPa (see Cramer, 1993). The speed of sound in air changes with temperature, water vapour content, and CO₂ content. The table gives values of the speed in m s⁻¹ for a range of temperatures and humidities at 0.03% CO₂ by volume. The uncertainty in the values in the table is estimated to be 0.1 m s⁻¹.

Speed of sound in air (m s⁻¹)

Speed, attenuation and non-linearity of sound in liquids

The speed of dilatational waves in unbounded fluids is c = (β_aρ)^−1/2, β_a being the adiabatic compressibility and ρ the density. In practice the propagation velocity of each part of a sound wave depends on the local amplitude and the importance of this effect is determined by the ratio B/A = (ρ/c²)∂²p/∂ρ² (where p is the pressure and all quantities are at ambient pressures and constant entropy). For plane waves the sound speed is then c + βu, where u is the particle velocity and β = 1 + B/2A (see Hamilton and Blackstock, 1988). In normal fluids the classical attenuation is proportional to f ², f being the frequency, but relaxation phenomena may dominate; the attenuation generally decreases with increasing temperature. Values for various liquids are given below, selected from numerous sources; f is in Hz and α is in neper m⁻¹. Where known, the frequency at which the attenuation was measured is given in brackets, in MHz. The temperature for the B/A values is given in brackets, in °C.

Properties of sound in liquids

    * The acoustic properties of helium are highly complex and dependent on temperature; see Rudnick (1980).

Acoustic properties of distilled water

Air-free distilled water is non-dispersive but the temperature dependence of sound speed is anomalous, mainly because of the temperature dependence of the adiabatic compressibility of the water molecule itself; the maximum speed occurs at a temperature of approximately 74.16 °C. The pressure coefficient is approximately + 0.156 m s⁻¹ atm⁻¹at 20 °C (see Del Grosso and Mader, 1972, Chen and Millero, 1976, Bilaniuk and Wong, 1993). The estimated uncertainty in the tabulated values is less than ± 0.02 m s⁻¹. The attenuation of sound in distilled water is proportional to f ², at least for the range 3–70 MHz, but is considerably in excess of that expected from the shear viscosity alone. The excess is attributed to structural relaxation between an open type of molecular structure, where each water molecule has four nearest neighbours tetrahedrally arranged about it, and a closer type of packing in which there are twelve nearest neighbours (see Litovitz and Davis, 1965).

Properties of sound in distilled water

Speed and attenuation in sea water

The speed of sound in sea water is a function of the temperature, the excess pressure, and the salinity, their relative importance being in that order (see Lovett, 1978, and Spiesberger, 1993); values in the table relate to surface depth and a salinity of 3.5%. The attenuation of sound in sea water is much higher than in distilled water owing to complex relaxation phenomena involving MgSO₄, MgCO₃ and B(OH)₃ (see Mellen et al., 1980); values in the table, relating to surface depth, a pH of 8.0 and a salinity of 3.5%, are based on Schulkin and Marsh (1962, 1978).

Velocity of sound in sea water

Attenuation of sound in sea water (dB km⁻¹)

Speed and attenuation in solids

In isotropic solids, both shear (transverse) and longitudinal waves can be propagated. The velocity of shear waves in an extensive medium is c_S = √(Gρ) = √{E/2ρ(1 + σ)}, E being Young’s modulus, G the rigidity modulus and σ Poisson’s ratio, and this is also the velocity of torsional waves in thin cylindrical bars. The speed of longitudinal or irrotational waves in an extensive medium is c_L = √{(K + G)/ρ} = √{E(1 − σ)/ρ(1 − 2σ)(1 + σ)}, K being the bulk modulus. In straight uniform bars and in tubes thin compared with a wavelength, the speed of longitudinal waves is c_E = √(E/ρ). Surface waves propagating along the surface of an extensive solid are generally known as Rayleigh waves and propagate with speed c_SR = ac_S, where a is the least positive root of the equation

In plates, the propagating modes are dispersive and for thin isotropic plates these Lamb waves reduce to two types: flexural (tending to zero speed) and antisymmetric (P₀) waves (Achenbach, 1984).

Variation of a with Poisson’s ratio

σ	0.20	0.25	0.30	0.35	0.40
a	0.9110	0.9194	0.9274	0.9350	0.9422

In anisotropic solids, which may have as many as 21 independent elastic constants, there may exist, for a given direction of the wave normal, three distinct displacement vectors each associated with a distinct plane wave velocity. Of the three waves, one is analogous to the longitudinal and the others to transverse waves in the isotropic case. The directions of the respective displacement vectors are mutually orthogonal and are in general oblique to the wave normal. A generalized Rayleigh-type wave may be propagated, in a limited number of directions, along the surface of an extensive anisotropic medium. See Achenbach (1984).

Velocities and attenuation constants for various solids in the region of 20 °C are given below. In the case of metals, factors such as texture, cold work, stress, hardening, tempering and aging can cause significant departures from the values given in the table. Properties of plastics vary considerably with molecular weight, with additives and with temperature. Rocks or building materials can be equally variable. In view of this, many values are approximate and relate to materials of variable composition; the velocities given should therefore only be regarded as typical. The bracketed figures following the attenuation are frequencies in MHz; for plastics and many other solids, the variation of attenuation with frequency is often approximately linear. Composition of materials is given as percentage by weight.

Speed and attenuation of waves in solids

References

J. D. Achenbach (1984) Wave propagation in elastic solids, North-Holland, Amsterdam.
N. Bilaniuk and G. S. K. Wong (1993) J. Acoust. Soc. Am., 93, 1609–12.
C.-T. Chen and F. J. Millero (1976) J. Acoust. Soc. Am., 60, 1270–3.
O. Cramer (1993) J. Acoust. Soc. Am., 93, 2510–2513.
V. A. Del Grosso and C. W. Mader (1972) J. Acoust. Soc. Am., 52, 1442–6.
C. M. Davis and J. Jarzynski (1972) Liquid water–acoustic properties: Absorption and relaxation, in Water–a comprehensive treatise (ed. F. Franks) vol. 1, Plenum, New York, 443–61.
ISO 9613-1 (1993) Acoustics–Attenuation of sound during propagation outdoors, Part 1: Calculation of the absorption of sound by the atmosphere.
F. E. Fox and G. D. Rock (1946) Phys. Rev., 70, 68–73.
M. F. Hamilton and D. T. Blackstock (1988) J. Acoust. Soc. Am., 83, 74–7.
K. F. Herzfeld and T. A. Litovitz (1959) Absorption and Dispersion of Ultrasonic Waves, Academic Press.
T. A. Litovitz and C. M. Davis (1965) Structural and shear relaxation in liquids, in Physical Acoustics (ed. W. P. Mason), vol. II Part A: Properties of Gases, Liquids and Solutions, Academic Press, 281–349.
J. R. Lovett (1978) J. Acoust. Soc. Am., 63, 1713–18.
R. H. Mellen et al. (1980) J. Acoust. Soc. Am., 68, 248–57.
J. M. M. Pinkerton (1947) Proc. Phys. Soc., 62B, 129–41.
I. Rudnick (1980) J. Acoust. Soc. Am. 68, 36–45.
M. Schulkin and H. W. Marsh (1962) J. Acoust. Soc. Am., 34, 864–5 (errata in (1963) J. Acoust. Soc. Am., 35, 739).
M. Schulkin and H. W. Marsh (1978) J. Acoust. Soc. Am., 63, 43–8.
J. L. Spiesberger (1993) J. Acoust. Soc. Am., 93, 2235–7.
V. Uhlendorf et al. (1985) J. Phys. E: Sci. Instrum., 18, 151–7.

D.R.Bacon, D.R.Jarvis

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