Introduction to Acoustics

1046 Part H Engineering Acoustics
Part H 24.B
For psv the numerical coefficients are:
A ′ 1 = 1.237 8847 × 10−5 ,
B ′ 1 =−1.912 1316 × 10−2 ,
C ′ 1 = 33.93711047 ,
D ′ 1 =−6.343 1645 × 103 .
Within the normal range of environmental conditions
during calibrations, the following simplified equation for
the density of humid air may be used
ρ = ρ0
ps
p0
T0
T (1 − 0.378hs) , (24.B2)
where ρ and ρ0 are the densities of air, p0 and ps are the
reference and static pressure, respectively; T0 and T are
the temperatures in Kelvin and hs is the fractional molar
concentration of moisture.
Uncertainties in ρ
The uncertainty in ρ obtained with (24.B1) has been
estimated [24.69, 70] at approximated 100 ppm. At the
reference conditions of 101.325 kPa and 50% relative
humidity, ρ obtained with (24.B2) are 239 ppm and
293 ppm larger than those obtained with (24.B1)at23 ◦ C
and 30 ◦ C, respectively.
24.B.2 Computation of the Speed
of Sound in Air
Method 1
The speed of sound in air varies with temperature, carbon
dioxide content, relative humidity and barometric
pressure. The sequence of the above parameters is listed
roughly in decreasing order of their influence on the
speed of sound with respect to the day-to-day encounter
of environmental conditions. A large number of investigations
related to the speed of sound [24.50, 71–80]
had been published, detailed references have been
given [24.75], and an updated and comprehensive bibliography
([24.80], Chap. 17, pp. 265–284 and references
therein) on publications relating to sound speed in air is
included at the end of this appendix. It can be shown that
the speed of sound remains relatively constant with barometric
pressure [24.72]: at 23 ◦ C and 50% RH, over the
barometric pressure range 60–110 kPa, the sound speed
varies by less than 76 ppm. In view of this, the following
equation has a relatively small overall uncertainty,
is based on a steady barometric pressure of 101.325 kPa
and may be used to compute the variation of the speed
of sound with temperature, carbon dioxide content and
relative humidity.
A general empirical equation has been obtained
[24.71] for calculation of the variation of c/c0
with relative humidity h, temperature t and carbon
dioxide content hc:
c/c0 = a0 + a1t + a2t 2 + a3hc + a4hct + a5hct 2
+ a6h + a7ht + a8ht 2 + a9ht 3 + a10(hc) 2
+ a11h 2 + a12hthc , (24.B3)
where c and c0 are the sound speed and the reference
dry-air sound speed, respectively; a0–a12 are coefficient
constants listed in Table 24.8. With Table 24.8, the sound
speed can be deduced by multiplying c/c0 with the
corresponding reference dry-air sound speed c0.
For hc values from 0% to 1%, and for t from 0 ◦ C
to 30 ◦ C, for h from 0 to 1 (relative humidity 0% to
100%), and at a barometric pressure of 101.325 kPa, the
sound speed computed using the numerical coefficients
in Table 24.8 fits the theoretical data with a standard
uncertainty of ±48 ppm.
Uncertainties in c0
Over the temperature range 0–30 ◦ C, (24.B3) is fitted to
a computation [24.71] for a real gas at 101.325 kPa at
which the value Cp −Cv is not greatly different from the
universal constant R [24.75]. Based on this approximate
assumption, the dry-air sound speed c0 is 331.29 m/s,
with an uncertainty of approximately 200 ppm [24.75],
which encompasses sound speeds from 331.224 to
331.356 m/s [24.50].
Table 24.8 Coefficient constants for the computation of
c/c0 and γ/γ0 (after [24.71])
Coefficient
constants
c/c0 (24.B3) γ/γ0 (24.B5)
a0 1.000100 1.000034
a1 1.8286 × 10−3 –2.8100 × 10−6 a2 –1.6925 × 10 −6 –2.1210 × 10 −7
a3 –3.1066 × 10 −3 –1.012 23 × 10 −3
a4 –7.9762 × 10 −6 –5.2500 × 10 −6
a5 3.4000 × 10 −9 1.1290 × 10 −8
a6 8.9180 × 10 −4 –3.4920 × 10 −4
a7 7.7893 × 10 −5 –2.8560 × 10 −5
a8 1.3795 × 10 −6 –5.9000 × 10 −7
a9 9.5330 × 10 −8 –2.9710 × 10 −8
a10 1.2990 × 10 −5 4.234 27 × 10 −6
a11 4.8016 × 10 −5 8.0000 × 10 −7
a12 –1.4660 × 10 −6 5.1000 × 10 −7