Introduction to Acoustics

56 Part A Propagation of Sound
Part A 3.7
first excited vibrational states is a function of temperature
when the gas is in thermodynamic equilibrium,
but during an acoustic disturbance the redistribution of
molecules to what is appropriate to the concurrent gas
temperature is not instantaneous. The knowledge that
the relaxation processes are vibrational relaxation processes
and that only the ground and first excited states
are appreciably involved allows the sound speed increments
to be determined from first principles. One need
only determine the difference in sound speeds resulting
from the two assumptions: (i) that the distribution
of vibrational energy is frozen, and (ii) that the vibrational
energy is always distributed as for a gas in total
thermodynamic equilibrium. The resulting sound speed
increments [3.14]are
(∆c)ν
c
= (γ − 1)2
2γ
nν
n
�
T ∗ �2 ν
e
T
−T ∗ ν /T , (3.261)
where n is the total number of molecules per unit volume
and nν is the number of molecules of the type corresponding
to the designation parameter ν. The quantity
T is the absolute temperature, and T ∗ ν
is a characteristic
temperature, equal to the energy jump ∆E between the
two vibrational states divided by Boltzmann’s constant.
The value of T ∗ 1 (corresponding to O2 relaxation) is
2239 K, and the value of T ∗ 2 (corresponding to N2 relaxation)
is 3352 K. For air the fraction n1/n of molecules
that are O2 is 0.21, while the fraction n2/n of molecules
that are N2 is 0.78. At a representative temperature of
20 ◦ C, the calculated value of (∆c)1 is 0.11 m/s, while
that for (∆c)2 is 0.023 m/s.
Because relaxation in a gas is caused by two-body
collisions, the relaxation times at a given absolute temperature
must vary inversely with the absolute pressure.
The relatively small (and highly variable) number of
water-vapor molecules in the air has a significant effect
on the relaxation times because collisions of diatomic
molecules with H2O molecules are much more likely
to cause a transition between one internal vibrational
quantum state and another. Semi-empirical expressions
for the two relaxation times for air are given [3.59, 60]
by
pref
p
pref
p
1
= 24 + 4.04 × 10
2πτ1
6 � �
0.02 + 100h
h
,
0.391 + 100h
(3.262)
� �1/2 1 Tref
= (9 + 2.8×10
2πτ2 T
4 h e −F )
(3.263)
F = 4.17
�� � �
1/3
Tref
− 1 . (3.264)
T
The subscript 1 corresponds to O2 relaxation, and the
subscript 2 corresponds to N2 relaxation. The quantity
h here is the fraction of the air molecules that are H2O
molecules; the reference temperature is 293.16 K; and
the reference pressure is 105 Pa. The value of h can be
determined from the commonly reported relative humidity
(RH, expressed as a percentage) and the vapor
pressure of water at the local temperature according to
the defining relation
h = 10 −2 (RH) pvp(T)
. (3.265)
p
However, as indicated in (3.262)and(3.263), the physics
of the relaxation process depends on the absolute humidity
and has no direct involvement with the value of the
vapor pressure of water. A table of the vapor pressure
of water may be found in various references; some representative
values in pascals are 872, 1228, 1705, 2338,
10 6
10 5
10 4
10 3
10 2
10
Relaxation frequency (Hz)
10 –5
0.5 1 2.5 5 10 25 50 100
Relative humidity RH at 20°C
0.5 1 2.5 5 10 25 50 100
Relative humidity RH at 5°C
10 –4
(O2)
1/2πτ1
(N2, 20°)
1/2 πτ2
10 –3
10 –2
10 –1
Fraction of air molecules that are H2O, h
Fig. 3.14 Relaxation frequencies of air as a function of
absolute humidity. The pressure is 1.0 atmosphere