Natural temporal variability of atmospheric acoustic absorption coefficients

Applied Acoustics 34 ( 199l ) 1t I- [ 2 l

Natural Temporal Variability of Atmospheric Acoustic Absorption Coefficients

D. K e i t h W i l s o n a & D e n n i s W. T h o m s o n b aGraduate Program in Acoustics and Department of Meteorology, Department of Meteorology, 503 Walker Building, The Pennsylvania State University, University Park, Pennsylvania 16802, USA (Received 25 January 1991; accepted 8 March 1991)

A BSTRA CT Since acoustic absorption coefficients depend upon environmental temperature and humidity, they vary in response to changing atmospheric conditions on a variety of time scales. This natural temporal variability is often neglected in outdoor noise assessment work, where averaged meteorological data are used to estimate absorption. In this stud)" the authors assess the validio' of calculating absorption coefficients from meteorological data which have been averaged on various time scales. The results indicate that the use of annual mean meteorological data can result in frequency-dependent r.m.s. errors ranging from 20 to 50%. Seasonal averaging can result in r.m.s, errors of 20 to 30%. Even daily averages may be unsatisfactory: for some of the cases presented absorption changes by 200% during the course of a single day.

1 INTRODUCTION A t m o s p h e r i c acoustic absorption coefficients depend strongly u p o n environmental t e m p e r a t u r e and humidity. Because temperature and humidity vary substantially on turbulent, diurnal, synoptic (weather system) and seasonal time scales, so too does acoustic absorption. Nonetheless, as pointed out by Thomson, ~ temporal variability o f absorption coefficients is often neglected in o u t d o o r noise assessment work, where monthly, seasonal or annual m e a n meteorological data are often used to estimate absorption. 111 Applied Acoustics 0003-682X/91/$03.50 © 1991 Elsevier Science Publishers Ltd, England. Printed in Great Britain

112

D. Keith Wilson, Dennis W. Thomson

The temporal variability of absorption coefficients may also be important in the modification of signal and noise spectra, and in the propagation of shock waves. This paper is an extension of an earlier one by Thomson.~ For this study the authors utilized a considerably larger database of temperature and humidity measurements. The measurements consist of half-hour averages recorded during 1987 at the Larson Agricultural Research Center near Rock Springs in central Pennsylvania. All sensors, which were research quality, were installed two meters above the ground. Using the formulae provided in the American National Standard (ANSI S1.26-1978), 2 absorption coefficients were calculated from the meteorological data. Note that absorption coefficients were not actually measured as a part of this study. According to the standard, the formulae should be accurate to within _+10% between 0 and 40°C. 2 Some of the values presented in this study were calculated for air colder than this range, down to -20~C. Although the calculations are probably less accurate for these sub-zero temperatures, the ANSI formulae still appear to be consistent with existing data) The authors also do not consider time scales less than one half hour in this study. Such time scales are normally associated with turbulence in the atmospheric boundary and surface layers. For quantification of variability on time scales ranging from 10-1 to 103 s, T h o m s o n has suggested defining a structure function for the acoustic absorption coefficient) This approach has already been shown to be useful in the interpretation of turbulence fluctuations in the acoustic and optical indices of refraction [see, for example, Brown & Hall 3 and Wyngaard et al.4).

2 REVIEW OF A B S O R P T I O N BY AIR Before discussing the evaluation of temporally varying absorption coefficients, it will be helpful to review the physical processes by which sound energy is absorbed by air. More detailed discussions can be found in Kinsler et al. 5 and Bass et al. 6 Absorption in a molecular gas results from viscosity, thermal conduction and molecular relaxation. For audible frequencies, relaxation of the vibrational modes ofdiatomic nitrogen and oxygen molecules is the primary contribution to absorption. This process involves energy transfer between the translational (external) degrees of freedom and the vibrational (an internal) degree of freedom of the molecules. When the temperature associated with the translational motion fluctuates, energy is transferred to

Atmospheric acoustic absorption coefficients

113

the vibrational mode. The relaxation time, r~, is used to describe the rate of this reaction. The frequency dependence of vibrational absorption is givert by

fZ/fr

(1)

~ b ~ 1 + (S'fr) 2 where f is the frequency of the sound wave and f~= 1/~r is called the relaxation frequency. Hence, for a fixed acoustic frequency S attenuation is maximized when J~ =fl The relaxation frequencies for the translational modes of nitrogen and oxygen depend on both temperature and humidity. Figures 1 and 2 show the dependence of absorption coefficients on temperature and molar concentration of water vapor at two different frequencies, 250 and 4000Hz, respectively. The absorption peak at low humidities occurs when the relaxation frequency of oxygen coincides with the acoustic frequency. As

(

,q

¢/ ,el,<~ c

Fig. 1. The absorption coefficient at 250 Hz, in dB/100 m, as a function of temperature and humidity. Temperature is expressed in °C and humidity as moles of water vapor per kilomole of air.

114

D. Keith Wilson, Dennis W. Thomson

M.

t.

z--

Fig. 2.

The absorption coefficient at 4000 Hz as a function of temperature and humidity. Units are the same as for Fig. 1.

will be seen in the next section, this 'absorption resonance' has interesting consequences when the air is particularly dry. In this paper the humidity will always be expressed as a molar concentration of water vapor, i.e. the ratio of the number of moles of water vapor to the total number of moles in an air sample. More commonly, meteorologists use the relative humidity or mixing ratio. The molar concentration of water vapor can be converted to relative humidity by multiplying it by the ratio of the atmospheric pressure to the saturation vapor pressure. The reason why the molar concentration is used here, instead of relative humidity or mixing ratio, is that it is the physically relevant measure of humidity for absorption calculations. Figure 3 is a scatter plot of the daily mean molar concentration of water vapor versus temperature, as recorded for the entire year of 1987 at the authors' micrometeorological field site near Rock Springs, PA. By examining these data one can determine which sections of the absorption surfaces (such as those shown in Figs 1 and 2) may be important in analyzing a given atmospheric sound propagation problem.

115

A t m o s p h e r i c acoustic absorption coefficients

0

E o

m

0 "1"

u~ oo

0

g

=8% ==q.°

E- I t-

ii i

Ig IIl~ I m l

0 -20

-10

0

10

20

30

temperature (C)

Fig. 3.

Scatter plot of mean daily molar concentration of water vapor (%) versus temperature (°C), at Rock Springs, PA, for the year 1987.

3 ABSORPTION CALCULATIONS Calculations of time-dependent absorption coefficients from the 1987 Rock Springs micrometeorological data show that the variability characteristics depend strongly upon both the sonic frequency and season. The daily mean data, shown in Fig. 4 is examined first. The two uppermost curves show the temperature and molar concentration of water vapor, respectively. As expected, the air is relatively cold and dry in the winter, and warm and moist in the summer. Superimposed on the annual cycle are changes lasting approximately 3-5 days. This variability (known meteorologically as synoptic scale) is essentially the consequence of the passage of relatively dryer and moister air masses, which are associated with high and low pressure systems, respectively. In the eastern US these air masses are normally classified as being of continental (Canadian) or modified maritime (Gulf of Mexico) origin. The next four curves in Fig. 4 show the calculated absorption coefficients at 63,250, 1000 and 4000 Hz, respectively. At all frequencies absorption has greater variability in the winter than in the summer. This behavior is particularly pronounced at 4000 Hz. It is also interesting that absorption at 63 and 4000Hz is typically greater in the winter than in the summer, although little seasonal dependence is evident at 250 Hz.

116

D. Keith Wilson, Dennis W. Thomson

2

sz

, 0

0.05

=

o.8J 'il ~' I,,

1

!t, ttll

2, l 0

,I .

0

.

40

.

.

.

80

.

.

120

.

.

.

160 Julian

.

.

200

.

.

.

240

.

.

280

320

360

day

Fig. 4. Daily mean values of temperature and humidity at Rock Springs, PA. for the year 1987, and the resulting calculations of acoustic absorption coefficients. Temperature is in °C and humidity is moles water vapor per kilomole of air. The absorption coefficients, expressed in dB/100m, are calculated at four frequencies: 63, 250, 1000 and 4000 Hz.

Figure 5 shows, with half-hour resolution, the micrometeorological data and absorption calculations for the month of January. The variability of the absorption coefficients is particularly dramatic between 24 and 29 January, the dryest days which occurred that year. During this five-day period the water vapor content of the air was less than 0-002 moles per mole of air. Consequently, during this period absorption was near the 'resonance' peak throughout most of the audible range. For example, at 250 Hz absorption rapidly increased and then underwent dramatic fluctuations. At 4000 Hz absorption was actually on the dry side of the resonance peak, so that while absorption was increasing at 250 Hz it was decreasing at 4000 Hz. Figure 6 shows the micrometeorological data and absorption calculations

Atmospheric acoustic absorption coefficients

0

0

8.

-~o

E

-20

_z,

~o- i

5

o~

1-

117

~- o.~s i 0.05 -!

-?8 Q

0.6

1

7

13 Julian

19

25

31

day

Fig. 5. Half-hourmeasurementsof temperatureand humidityat Rock Springs, PA, for the month of January 1987, and the resulting calculations of acoustic absorption coefficients. Units and frequenciesare the same as for Fig.4.

for the month of July. From a synoptic-scale point of view, absorption coefficients are apparently not so highly variable in the summer as compared to the winter. However, the diurnal variability is much stronger in the summer. The reason for this is the seasonal difference in daytime solar heating of the ground. During the winter the sun is lower in the sky, so that the intensity of solar radiation is less. Furthermore, in the winter, when the ground is snow covered, 75-95% of the solar energy is reflected or absorbed in radiative processes; only 5-20% is reflected by a grassy field. Thus, in the summertime, acoustic absorption coefficients will essentially track the diurnal temperature changes. They will peak during the warmest part of the day, which is normally mid to late afternoon.

0.02

.~

0

182

2

0.6

o.,;1

188

194 200 Julian day

206

'

'

'

212

'

Fig. 6. Same as Fig. 5, but for the month of July 1987.

g

o

o,I

20

E¢.-

o.oi

30

,_~

20

3o

Fig. 7.

0

2

s

0.2

9 ;~

0.6

0.05

-

)

i

~

)

97

,

i

,

i

!

v

,

,

i

,

103 109 Julian day

,

115

!i

121

t

S:tmc as Fig. 5, but f o r the m o n t h o f A p r i l 1987.

91

0.15-1

0.0"~

0.02

0

lO

~ o o

¢'

~

E

20

q~

4.

Atmospheric acoustic absorption coefficients

119

Figure 7 shows the behavior of absorption coefficients for the month of April. During this transitional month, increases in absorption correlate well with decreases in the humidity, as was characteristic of January. However, some periods of pronounced diurnal oscillations, as were characteristic of July, can also be seen. On some days the absorption increased by as much as 200% above the nighttime levels.

4 STATISTICAL ANALYSIS When mean meteorological data are used to calculate absorption, there are two potential sources of error. First, significant variability on time scales less than the averaging period may be neglected. Secondly, because the dependence of absorption on temperature and humidity is nonlinear, there is the potential for serious bias error. The actual mean absorption is thus not normally the same as absorption calculated from the mean temperature and humidity values. Figure 8 shows the r.m.s, errors (square root of the variances) resulting from use of meteorological data averaged over the year, at octave-band center frequencies from 63 to 8000 Hz. The contribution to the error from the various averaging periods and the bias error are also indicated. The total variance can be written 2 2 + az_y + b z ah-h-, = a2h--a + af--w + a;¢_,, + tr;_,

(2)

where b is the bias error and the subscript hh stands for half hour, d for day, w for week, m for month, s for season and y for year. For example, a;Ly is the variance of the seasonal absorption coefficients about the year's mean. In Fig. 8 the contributions to the error are computed by dividing the standard deviation for each averaging period by the total of standard deviation. For example, the bar sections indicated with cross-hatching for 'month' have size cr,,_,,/crhh_y x 100%. This statistical analysis can be considered representative of conditions prevalent in much of the northeastern Atlantic seaboard and mid-western regions of the United States. The illustrated r.m.s, errors probably underestimate variability characteristics of dryer areas such as the high plains and southwestern deserts. It is interesting to note in Fig. 8 that contributions to the error by averaging on time scales seasonal or less are about the same for all frequencies shown. In fact, although it is not directly evident from the graph, r.m.s, errors of 20-30% typically result from averaging on the monthly or seasonal time scales. When the seasonal data are averaged over the course of the year, however, the error is strongly frequency-dependent. F r o m 125 to

D. Keith Wilson, Dennis W. Thomson

120 60

DAY WEEK 50

MONTH

SEASON YEAR

4O

l

BIAS

i

m o re =r. uu

i i

i Z

m

!

m

2O

ot I i

63

N i 125

250

i

i

i

500

1000

2000

4000

8000

FREQUENCY (Hz)

Fig. 8. The r.m.s, errors resulting from the use of mean meteorological data to estimate absorption coefficients, at octave-band center frequencies from 63 to 8000Hz. The contributions to the errors from averaging over various time scales are also shown (see text for details).

500 Hz there is little additional error. At both lower and higher frequencies, however, annual averaging is the most significant source of error. Note also that the bias error caused by using annual mean meteorological data (as opposed to directly averaging the absorption coefficients over the course of the year) is most significant from 500 to 4000 Hz, precisely the frequency range which is usually important in environmental noise impact assessments.

5 CONCLUDING REMARKS The calculations presented suggest that, in a wintertime dry air mass, small changes in the humidity may cause dramatic fluctuations in absorption. In fact, absorption may increase at some frequencies while it decreases at others. Under such conditions the spectrum of a received broadband signal propagating in the atmosphere may vary dramatically with both time of day and distance. Absorption depends less strongly on humidity during the summertime, and does not exhibit the dramatic fluctuations characteristic of dry wintertime air. During the summer all frequencies exhibit consistent diurnal cycling: absorption tends to be greatest during the daytime. In much noise assessment work it may be impractical to assemble a database with temporal resolution as fine as the one used for this study. How,

Atmospheric acoustic absorption coefficients

121

then, can a consultant apply the results of this study? At a minimum, it appears that mean seasonal or monthly data, rather than yearly data, should be used to calculate absorption. Such a procedure should be practical using most existing climatological data bases. A better approach would be to make calculations from a large ensemble of meteorological measurements averaged over time scales less than a day, and then statistically analyze the resulting predictions. Only in this manner can one properly account for the natural variability in absorption resulting from diurnally and synopticaliy changing weather conditions.

A C K N O W L E D G E M ENTS Support for the micrometeorological measurement program during which these measurements were made was provided by N O A A Grant NA85-W-CC-06145. Analysis of the data was performed in conjunction with studies of atmospheric acoustic propagation supported by Atmospheric Sciences Laboratory, White Sands Missile Range (Grant N00039-88-C-0051) and O N R (Grant N000014-86-K-06880). The authors also gratefully acknowledge the assistance of Richard T h o m p s o n in performing the field measurements.

REFERENCES 1. Thomson, D. W., Regarding the temporal variability of frequency-dependent atmospheric acoustic absorption coefficients. Proceedings of Noise-Con 88, State College, PA, 1987, pp. 251-6. 2. Method for calculation of the absorption of sound by the atmosphere. American National Standards Institute, ANSI S1.26-1978. American Institute of Physics, New York, 1978. 3. Brown, E. H. & Hall, E E Jr, Advances in atmospheric acoustics. Rev. Geophys. Space Phys., 16 (1978) 47-110. 4. Wyngaard, J. C., Izumi, Y. & Collins, S. A. Jr, Behavior of the refractive-indexstructure parameter near the ground. J. Opt. Soc. Am., 61 (1971) 1646-50. 5. Kinsler, L. E., Frey, A. R., Coppens, A. B. & Sanders, J. V., Fundamentals of Aco,stics (3rd edn). John Wiley, New York, 1982, pp. 141-62. 6. Bass. H. E., Sutherland, L. C., Piercy, J. & Evans, L. E., Absorption of sound by the atmosphere. In Physical Acoustics, Vol. XVII, ed. W. E Mason. Academic Press, New York, 1984, pp. 145-232.