In this issue of the Journal (Griffin and Whirham, 1980), we present the results of some experiments in which the durations were all less than the shortest duration (1 min) specified in the ISO Standard. How- ever, the results are highly relevant to the Standard since it is often used to evaluate motions containing short periods of high level vibration. Furthermore both oug. experiment8 and the ISO Standard define time dependencies. Both are averaging procedures and, while they may appear to relate to different exposure periods, they can both be used to average over short periods.

The ISO guidelines include both a time dependency and the requirement to measure root mean square ac- celeration. The latter defines an averaging procedure but without a defined integration time. The ISO time dependency implies little change in effect below about 10 min--which is inconsistent with the use of rms averaging over such a period.

Our experimental results indicate that the ISO time dependency should not be extrapolated to periods less than I min since this will grossly over-estimate the discomfort of short duration motions. Conversely, studies conducted for a few seconds to determine a dis- comfort limit will grossly under-estimate the discom- fort for longer periods. According to our data the rms averaging procedure (i.e.,a2t =constant, where a is acceleration level and ! is time) which is implied in the ISO Standard will under-estimate the effects of short durations and over-estimate the severity of long dura- tions. However the a4t relationship which better ap- proximates our experimental results, provides a con- venient compromise between the two procedures in- herent in the current Standard (see Fig. 1).

Our experiments, conducted to determine the effects

on discomfort of the level and duration of impulses, were obtained with exposure periods up to 32 s. They lead to the definition of the rmq (based on a4t = con- stant) for evaluating such motions in preference to the rm8 (based on a2t =constant). However, our data do not indicate the maximum period over which the rmq should be determined. Figure 1 shows the time depen- dency that is defined if the averaging procedure is ap- plied over 24 h. Again it may be seen that a4t = constant is a reasonable compromise between large effects at long duration in the current Standard and the alterna- tive simple, but suspect, absence of any time depen- dency. The use of the same relationship between level and time for assessing bumps and evaluating longer duration vibrations is attractive.

There is no experimental proof of this new time de- pendency for periods greater than 32 s. However, con- sidering the relatively small change in level with time it seems very doubtful that the factor of two change in discomfort over a sixteenfold change in time from, say, 1-16 h could be measured. It may be tentatively con- cluded that while the rmq procedure has been evolved from considering only short duration motions it defines a time dependency for long periods which is no less valid than available alternative time dependencies. More definitely, it greatly reduces the difficulties and dangers in using the rms and ISO time dependencies with motions containing short periods at high levels.

Griffin, M.J., and Whirlram, E. M. (1980). "Discomfort pro- duced by impulsive whole-body vibration," J. Acoust. Soc. Am. 68, 1277-1284.

international Standards Organization (1978). "Guide for the evaluation of human exposure to whole-body vibration," ISO 2631-1978.

Analytical expressions for critical-band rate and critical bandwidth as a function of frequency

E. Zwicker and E. Terhardt

Institute of Electroacoustics, Technical Uniuersity Munich, D-8000 Munich 2, Germany (Received 14 July 1980; accepted for publication 8 August 1980)

The critical-band rate as well as the critical bandwidth are functions of frequency. Th dependencies have been given in table form. For effective use of these values in computers, relatively simple equations are given to express the dependence of critical-band rate on frequency with an accuracy btter than 0.2 Bark and that of critical bandwidth on frequency with an accuracy btter than 10% over the whole auditory range.

PACS numbers: 43.66.Ba, 43.66.Dc, 43.66.Cb [BSI

INTRODUCTION

In order to make the values of the critical bandwidth available to interested people and to ensure that mea- surements and calculations based on critical bands

would be comparable, an earlier paper (Zwicker, 1961) was published containing the values in table form, as reproduced in Table I. That paper pointed out that the critical bands have a certain width but their position on the frequency scale is not fixed. Thus a critical-band-

1523 J. Acoust. Soc. Am. 68(5), Nov. 1980; 0001-4966/80/111523-03500.80; 1980 Acoust. Soc. Am.; Letters to the Editor 1523 Downloaded 10 May 2010 to 200.39.9.56. Redistribution subject to ASA license or copyright; see http://asadl.org/journals/doc/ASALIB-home/info/terms.jsp

TABLE I. Values of critical band rate and critical bandwidth as functions of frequency.

Critical Frequency Critical Center band rate. Hz bandwidth frequency

Bark Hz Hz

0 0 100 50

1 100

100 150 2 200

100 250 3 300

100 350 4 400

110 450 5 510

120 570 6 630

140 700 7 770

150 840 8 920

160 1000 9 1080

190 1170 10 1270

210 1370 11 1480

240 1600 12 1720

280 1850 13 2000

320 2150 14 2320

380 2500 15 2700

450 2900 16 3150

550 3400 17 3700

700 4000 18 4400

900 4800 19 5300

1100 5800 20 6400

1300 7000 21 7700

1800 8500 22 9500

2500 10 500 23 12000

3500 13500 24 15 5OO

rate function (Tonheitsfunktion) was proposed in graphi- cal form. Meanwhile, critical-band rate and critical bandwidth have been frequently applied in psychoacous- tics as well as in speech research. Accordingly, it is desirable to have the relationships available not only in graphical and tble form, but also as analytical expres- sions for use with calculators and computers. Simple analytical expressions have recently been proposed (Terhardt, 1979). The critical-bandwidth formula given there is in good agreement with the original table (error

represents a different simple approximation which may he used between 0.6 kHz and 7 kHz with an accuracy of 0.25 Bark. The equation is

z/Bark= 8.7 + 14.21ogo(f/kHz). (3) II. CRITICAL BANDWIDTH, CB

The critical bandwidth is approximately proportional to the reciprocal of the first derivative of critical-band rate as a function of frequency. Mathematical deriva- t/on of a critical-bandwidth formula from Eq. (1) does not, however, fulfill the criteria of accuracy and sim- plicity.

Critical bandwidth is constant at low frequencies but increases with the logarithm of frequency at high fre- quencies. No simple equation seems accurately to de- scribe this behavior. The data in Table I have previ- ously been rounded to the nearest ten. They are re- produced in Fig. 2 as dots and show some irregulari- ties, especially near 0.4 and ? kHz. The proposed formula calculates the critical bandwidth cBc in Hz:

CBc/Hz = 25 + 7511 + 1.4(f/kHz)2] 69. (4) It approximates the tabulated data with an accuracy of

10%. The solid Hne in Fig. 2 represents Eq. (4). The departures, mostly produced by the rounding procedure, are shown at the top of the figure. The analytical ex- pression (4) gives values which approach 100 Hz at low frequencies and show small deviations at 0.5, 1, 2, and 4 kHz.

ACKNOWLEDGMENTS

We thank the staff of the Institute of Electroacoustics, especially Dipl.-Ing. W. Aures for fruitful discussions on this subject. This work was carried out in the

5O00

2000

1000

500

200

100 0.02

0.05 0.1 0.2 0.5 1 2 5 10 i

0.05 0.1 0.2 0.5 1 2 5 10 20 frequency (kHz)

FIG. 2. Critical bandwidth. CB. as a function of frequencyf. The relative difference between the values CB calculated with the indicated equation and the values CB T from Table I is given on top.

Sonderforschungsbereich 50, Kybernetik, M{inchen, which is supported by the Deutsche Forschungsgemein- schaft.

Terhardt, E. (1979). "On the perception of spectral informa- tion in speech," in Heari Mechanisms ad Speech, edited by O. Creutzfeld, H. Scheich, and C. Schreiner (Springer, Berlin), pp. 281--291. EBBS-Workshop, G6ttingen, 26-28 April 1979.

Zwicker, E. (1961). 'ubdivisinn of the audible frequency range into critical bands (FrequenZgruppen)," J. Acoust. Soc. Am. 33, 248.

Speed of sound in air as a function of frequency and humidity

C. L. Morley and G. P. Howell Institute of Sound and Vibration Research, University of Southampton, Southampton S09 5NH, England (Received 7 May 1980; accepted for publication 22 luly 1980) The model of atmospheric air given in ANSI SI.26-1978 ("Method for the Calculation of the Absorption of Sound by the Atmosphere") has been used to predict the frequency and humidity dependence of the sound speed in air. Good agreement with existing measurements is demonstrated, apart from an unexpla/ned discrepancy at low relative humidity.

PACS numbers: 43.28.Fp, 43.85.Dj, 43.35.Ae

INTRODUCTION

We have used the model of atmospheric air in Ref. 1 to predict the frequency dependence of sound speed caused by vibrationai relaxation effects. Since the vi- brational relaxation frequencies for oxygen and nitrogen in air are increased by the presence of water vapor molecules, the difference between the sound speed c,

at any given frequency and the equilibrium value c o at zero frequency is a function of humidity. In addition, the presence of water vapor in air reduces the molec- ular weight and (to a lesser extent) the specific-heat ratio, thus causing an increase in co. We have com- bined these effects to yield predictions of sound speed in humid air, for comparison with Harris's measure- ments. ,

1525 J. Acoust. Soc. Am. 68(5), Nov. 1980; 0001-4966/80/111525-03500.80; 1980 Acoust. Soc. Am.; Letters to the Editor 1525

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