Introduction to Acoustics

Method 2
Alternatively, over the temperature range 0–30 ◦ C, with
a computation [24.72–74] that is based on the derivation
of real-gas sound speed from virial coefficients, the
dry-air sound speed c0 calculated with 16 coefficients
similar to those shown in Table 24.8, is 331.46 m/s.
The uncertainty is approximately 545 ppm [24.73, 74],
which encompasses sound speeds c0 from 331.279 to
331.641 m/s. It should be noted that the sound speed of
331.29 m/s obtained with the first method is within the
uncertainty range of the second method [24.50].
Within the normal range of environmental conditions
during calibration, the following simplified
equation [24.38, 40, 50] for the sound speed in humid
air may be used:
� �1/2 T
c = c0 (1 + 0.165hs)∆, (24.B4)
T0
where hs is the fractional molar concentration of moisture,
and ∆ is a factor to compensate for dispersion.
Values of ∆ = 0.99935 and ∆ = 0.99965 are found
in [24.81] and [24.82], respectively. A value of 1.0001
has also been obtained [24.72].
Over a frequency range of 10 Hz to 10 kHz, by ignoring
the dispersion factor ∆, the uncertainty in the
sound speeds obtained with methods 1 and 2 increases
by approximately 300 ppm.
By ignoring dispersion, at the reference conditions
of 23 ◦ C, 101.325 kPa and 50% relative humidity, the
sound speed obtained with (24.B4) is 196 ppm higher
than that obtained with (24.B3).
24.B.3 Ratio of Specific Heats of Air
There have been several experimental [24.83, 84] and
theoretical [24.71–74,85,86] investigations into the variation
of the ratio of specific heats of air with temperature,
humidity, pressure and carbon dioxide content [24.75]
([24.86], pp. 36–39). A general empirical equation has
been obtained [24.71] for the calculation of the variation
of γ/γ0 with relative humidity h, temperature t and
carbon dioxide content hc
γ/γ0 = a0 + a1t + a2t 2 + a3hc + a4hct + a5hct 2
+ a6h + a7ht + a8ht 2 + a9ht 3 + a10(hc) 2
+ a11h 2 + a12hthc , (24.B5)
where a0–a12 are coefficient constants listed in Table
24.8. With (24.B5), the above normalized ratio of
specific heats can be deduced by multiplying γ/γ0 by
the corresponding reference dry-air ratio of specific
Microphones and Their Calibration 24.B Physical Properties of Air 1047
heats γ0. ForCO2 content hc values from 0% to 1%,
and for temperatures of 0–30 ◦ C, humidity from 0 to
1 (relative humidity 0% to 100%), and at a barometric
pressure of 101.325 kPa, the ratio of specific heats computed
using the numerical coefficients in Table 24.8,
fit the theoretical data with a standard uncertainty of
±17 ppm.
Uncertainties in γ0
Based on method 1 above [24.71], the dry-air specific
heat ratio γ0 is 1.3998. The standard uncertainty in γ0 is
approximately 400 ppm.
Similarly, based on method 2 [24.72–74], the dry-air
specific heat ratio γ0 is 1.4029; the standard uncertainty
in γ0 is over 760 ppm.
24.B.4 Viscosity and Thermal Diffusivity
of Air for Capillary Correction
The viscosity η of air, in the parameters in (24.39)
of Sect. 24.4.3, is a function of temperature t. Anempirical
equation which is based on a least-squares fit to
published data [24.87]is
η =[17.26797 + (5.0756 × 10 −2 )t
− (4.4028 × 10 −5 )t 2 + (5.0000 × 10 −8 )t 3 ]
×10 −6 Pa s (24.B6)
The equation for the thermal diffusivity αt of
air [24.38, 40]is
αt = η(9κ − 5)/(4κρ)m 2 /s . (24.B7)
Uncertainties in η
Over the temperature range from −80 ◦ C to 100 ◦ C,
when compared with the published data [24.87], the
maximum deviation in η obtained with (24.B6) is
±1 ppm from the data points. However, the data point
uncertainty may be several percent [24.87].
For standard dry air at a pressure of 101.325 kPa,
the numerical values calculated with the above equations
for the viscosity are 1.7268 × 10 −5 Pa s and
1.8413 × 10 −5 Pa s, at 0 and 23 ◦ C, respectively; and
the corresponding values for the thermal diffusivity
are 1.812 34 × 10 −5 m 2 /sand2.095 49 × 10 −5 m 2 /s,
respectively.
Examples. Aiming at the highest accuracies (method 1),
Table 24.9 gives the recommended values of the
quantities given in (24.B2)–(24.B4) for the reference
Part H 24.B