Campbell Scientific CSAT3 3-D Sonic Anemometer - Manual (Page 66)

Appendix C. CSAT3 Measurement Theory
The speed of sound in moist air is a function of temperature and humidity and
is given by:
C 2 = γP ρ ⁄ = γR
d
T
v
= γR
d
T(1 + 0.61q) (6)
where γ is the ratio of specific heat of moist air at constant pressure to that at
constant volume, P is pressure, ρ is air density, R
d
is the gas constant for dry
air, T
v
is virtual temperature, T is the air temperature, and q is the specific
humidity defined as the ratio of the mass of water vapor to the total mass of air
(Kaimal and Gaynor, 1991; Wallace and Hobbs, 1977).
Note that γ is a function of specific humidity. It would be convenient if the
effects of humidity could be consolidated into one term.
The specific heats for moist air at constant pressure and volume are given by:
C
p
= qC
pw
+ (1 − q)C
pd
= C
pd
(1 + 0.84q) (7a)
C
v
= qC
vw
+ (1 − q)C
vd
= C
vd
(1 + 0.93q) (7b)
where C
p
and C
v
are the specific heats of moist air at constant volume and
pressure, C
pw
and C
vw
is the specific heat of water vapor, and C
pd
and C
vd
is the
specific heat of dry air, respectively (Fleagle and Businger, 1980).
Substitute Eq. (7a) and (7b) into (6) and ignore the higher order terms. This
yields
c 2 = γ
d
R
d
T
s
= γ
d
R
d
T(1 + 0.51q) (8)
where T
s
is sonic virtual temperature and γ
d
is the ratio of specific heat of dry
air at constant pressure to that at constant volume (Fleagle and Businger, 1980;
Kaimal and Gaynor, 1991; Kaimal and Businger, 1963; Schotanus et al., 1983).
With Eq. (8), the effect of humidity, on the speed of sound, is included in the
sonic virtual temperature.
The sonic virtual temperature, in degrees Celsius, is given by Eq. (9), where γ
d
= 1.4 and R
d
= 287.04 JK –1 kg –1 .
T
s
=
c 2
γ
d
R
d
− 273.15 (9)
REFERENCES
Kaimal, J. C. and Businger, J. A.: 1963, “A Continuous Wave Sonic
Anemometer-Thermometer”, J. Applied Meteorol., 2, 156-164.
Kaimal, J. C. and Gaynor, J. E.: 1991, “Another Look at Sonic Thermometry”,
Boundary-Layer Meteorol., 56, 401-410.
Fleagle, R. G. and Businger, J. A.: 1980, An Introduction to Atmospheric
Physics, Academic Press, Inc., New York.
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