Introduction to Acoustics

1030 Part H Engineering Acoustics
Part H 24.4
Table 24.1 Tabulated values of Ev for the heat conduction correction (ANSI S1.15: 2005, and IEC 61094-2:1992-03)
Real part of Ev
X ′
Imaginary part of Ev
R ′ = 0.2 R ′ = 0.5 R ′ = 1 R ′ = 0.2 R ′ = 0.5 R ′ = 1
0.72127 0.71996 0.72003 1.0 0.24038 0.22323 0.22146
0.80092 0.80122 0.80128 2.0 0.17722 0.16986 0.16885
0.83727 0.83751 0.83754 3.0 0.14818 0.14304 0.14236
0.85907 0.85920 0.85922 4.0 0.13003 0.12614 0.12563
0.87393 0.87402 0.87403 5.0 0.11732 0.11421 0.11380
0.89343 0.89348 0.89349 7.0 0.10030 0.09807 0.09777
0.91082 0.91086 0.91086 10.0 0.08477 0.08321 0.08300
0.93693 0.93694 0.93694 20.0 0.06086 0.06007 0.05997
0.94850 0.94851 0.94851 30.0 0.05002 0.04950 0.04942
0.95540 0.95541 0.95541 40.0 0.04349 0.04310 0.04304
0.96358 0.96359 0.96359 60.0 0.03568 0.03541 0.03538
0.96846 0.96846 0.96846 80.0 0.03098 0.03078 0.03076
0.97179 0.97179 0.97179 100.0 0.02776 0.02761 0.02758
0.98005 0.98005 0.98005 200.0 0.01972 0.01964 0.01963
0.98590 0.98590 0.98590 400.0 0.01399 0.01395 0.01395
0.99003 0.99003 0.99003 800.0 0.00992 0.00990 0.00989
The numerical values given are considered accurate to 0.00001
ber [24.35]. Ballagh [24.36] took into consideration heat
loss effects due to the microphone diaphragms at the
ends of the cavity. Jarvis [24.37] pointed out the need
to include the impedance of the driver microphone during
computation of the heat conduction correction. It
is difficult to decide accurately at which middle frequencies
the isothermal–adiabatic transition occurs, and
which portions of the cavity to include in the impedance
of the microphone. The exact nature of this transition
depends upon the frequency of the calibration and the
dimensions of the coupler. The sound pressure generated
by the transmitter microphone, i. e. a constant-volume
displacement source, will change accordingly. The effect
can be considered as an apparent increase in the
coupler volume expressed by a complex correction factor
∆H to the geometrical volume V in Sect. 24.A or
to the cross-sectional area S0 defined in Z = ρcS −1
0 in
Sect. 24.B.
The heat conduction correction factor is given by:
γ
∆H =
, (24.29)
1 + (γ − 1)Ev
where Ev is the complex temperature transfer function
defined as the ratio of the space average of the sinusoidal
temperature variation associated with the sound pressure
to the sinusoidal temperature variation that would be
generated if the walls of the coupler were perfectly nonconducting
(24.30), and γ is the ratio of specific heats of
the gas inside the coupler. Tabulated values for Ev can
be found in Table 24.1 [24.35] as a function of the parameters
R ′ and X ′ ,whereR ′ is the length-to-diameter
ratio of the coupler, X ′ = fℓ 2 /(γαt), f is the frequency,
ℓ is the volume-to-surface ratio of the coupler and αt is
the thermal diffusivity of the gas enclosed.
For finite cylindrical couplers as described in
[24.38], annex C, the approximation described below
for the complex quantity Ev will be satisfactory
where
Ev = 1 − S + D1S 2 + (3/4) √ π D2S 3 , (24.30)
�
S = −i 1
2π X ′
�1/2 = 1 − i
D1 = π R′2 + 8R ′
π(2R ′ ,
+ 1) 2
D2 = R′3 − 6R ′2
3 √ π(2R ′ .
+ 1) 3
2 √ ,
π X ′
The modulus of Ev, as calculated from (24.30), is
accurate to 0.01% within the range 0.125 < R ′ < 8and
for X ′ > 5. The first two terms in (24.30) constitute an
approximation that may be used for couplers that are not
right circular cylinders.