Standard Octaves and Sound Pressure. The superposition of several independent sound sources produces multifrequency noise: i=1

Transcription

1 Appendix C Standard Octaves and Sound Pressure C.1 Time History and Overall Sound Pressure The superposition of several independent sound sources produces multifrequency noise: p(t) = N N p i (t) = P i cos(2πf i t + φ i ) i=1 i=1 (C.1) Here, t = time, f = frequency in Hertz, P i = pressure amplitude of the ith frequency and φ i is phase. Noise is usually modeled as a stationary random process, which is valid if time-averaged statistical measures of noise, including root-mean-square (rms) pressure, spectral density,andprobability distribution, are independent of the sample length. The fundamental measure of noise amplitude is rms pressure, the square root of the average over time period T of the square of pressure: ( ) T 1 p rms = p 2 (t)dt (C.2) T 0 The rms of a pure tone (P cos[2πft + φ]) is its amplitude divided by the square root of 2: p rms = P Overall rms sound pressure is the sum of the component mean square pressures [1 3]: p 2 overall rms = 1 T 0 T p 2 (t)dt = N i=1 p 2 ith-independent component,rms = S p (f )df 0 (C.3) provided 1) the components are independent (randomly phased) with respect to each other, so their cross products, p i (t) p j (t) with i j, average to zero over many samples Formulas for Dynamics, Acoustics and Vibration, First Edition. Robert D. Blevins John Wiley & Sons, Ltd. Published 2016 by John Wiley & Sons, Ltd.

2 Appendix C: Standard Octaves and Sound Pressure 423 Table C.1 Decibel scales of sound Notation: p rms = root mean square pressure; ρ=density; c = speed of sound, T = time period [4 6]. (a) (a) In air reference pressure is 20 μpa. In water reference pressure is 1 μpa [7]. (ergodic average), or 2) their periods are nonequal submultiples of the sample time T, as in a Fourier series. The overall sound pressure level (OASPL) is generally expressed in decibels (Table C.1), and the summing can be done in decibels (see Beranek [3], Pierce ([1], pp ) and Example C.1). C.2 Peaks and Crest Peaks in time dominate acoustic damage accumulation. The peak (maximum) of a pure sinusoidal tone is its amplitude. The maximum possible peak value of the sum of N sine waves (Eq. C.1) is the sum of their amplitudes P i. A measure of randomness is its peak-to-rms ratio (also called crest factor), which is the peak of a sample divided by the rms of the sample (Eq. C.3): Peak rms = Pi, p overall rms (2N) 1 2, equal peaks, P 1 = P i, i = 1, 2,, N (C.4) The peak-to-rms ratio of a single sine wave is Itis(2N) 1 2 for random noise that is the sum of N = 1, 2, 3, sine waves with random phases and equal amplitudes (Eq. C.1 with P i = P 1 = P, i = 1, 2,, N) [8]. The peak-to-rms ratio approaches infinity for a Gaussian random process. Normally operating machines usually have peak-to-rms ratios between and 4.

3 424 Appendix C: Standard Octaves and Sound Pressure C.3 Spectra and Spectral Density A plot of sound level against its frequencies is called a noise spectrum. The spectrum of the time history is a plot of the component amplitudes P i versus their frequencies f i.the single-sided acoustic pressure spectral density is defined as the mean square of oscillating pressures at frequencies between f 1 and f 2 divided by bandwidth Δf = f 2 f 1 in Hertz [1, 4, 9]; it has units of pressure 2 Hertz: S p (f )= p2 rms (C.5) Δf If discrete frequencies in the time history are spaced at 1 Hz frequency intervals (1 Hz bandwidth), then the spectral density is the mean square pressure at each frequency. The integral relationship on the right-hand side of Equation C.3 is called Parseval s equation. Overall mean square is the integral of the spectral density over its frequency range (Equation C.3). (Other definitions of spectra used in the literature include the two-sided spectrum with frequencies from minus infinity to plus infinity with one half the values of single-sided spectrum, spectrum with frequency in radian per second instead of Hertz, and rms and peak spectra rather than mean square spectra. Anyone of these spectra can be converted to another at constant bandwidth.) C.4 Logarithmic Frequency Scales and Musical Tunings Based on historical developments in the tuning of stringed instruments [1, 9 12], the audible frequency range is divided into proportional frequency bands called octaves. The upper frequency limit, f a, of an octave is twice the lower frequency limit, f b, f b f a = 2. Octaves are not linear scales; higher-frequency bands are wider than lower-frequency bands. The logarithmic center frequency, f c =(f a f b ) 1 2, is always less than the arithmetic mean frequency, 1 2 (f a + f b ). The third-octave, tenth-octave, and twelfth-octave bands are subintervals of one octave. An octave is spanned by three 1/3-octave bands and 12 1/12-octave bands. Center frequencies of successive 1/3-octave bands are approximately in the ratio 5:4 [1]. Frequency band 1 octave 1/3 octave 1/10 octave 1/n octave 2 f 2 Upper frequency/ n lower frequency, f b f a Center frequency, f a f a 1 2n f f c =(f a f b ) 1 2 a a Bandwidth, (f b f a ) f c (2 1 n 1)2 1 2n

4 Appendix C: Standard Octaves and Sound Pressure 425 To convert a measurement from a wider to a narrower frequency band, one usually assumes the spectral densities or the rms pressures are equal in the smaller bands. A one-octave band SPL is converted to three 1/3-octave bands by subtracting 10 log 10 (3) =4.77 db from the one-octave band SPL. Equation C.3 is applied to convert several smaller bands to a single larger band, if the pressures in each of the smaller bands are independent. Standard 1-octave and 1/3-octave bands in Table C.2 are endorsed by the Acoustical Society of America [9]. The one-third-band limit frequencies are nice integers, approximately equal to 10 3+n 10,wheren is a positive or negative integer. One thousand hertz is a band center, whereas classical musical scales are based on the note A 4 = 440 Hz and do not include a 1000 Hz band frequency [1, 9]. Table C.2 Standard one and one-third octave bands The frequency bands can be extended by multiplication or division by powers of 10 [9].

5 426 Appendix C: Standard Octaves and Sound Pressure Classical music uses 12 proportional frequencies (notes) per octave in the ratio of small integers, f m/n,wherem and n are integers. Seven of these notes are the familiar do, re, mi, fa, so, la, andti (do) that are seven successive white keys per octave on a piano [2, 4, 9]. C D E F G A B (C) do re mi fa so la ti do 1:1 9:8 5:4 4:3 3:2 5:3 15:8 2:1 The frequency ratio 1:1 is called unison, 3:2 is a perfect fifth, 5:4 is a major third, 6:5 is a minor third, 4:3 is a fourth, 8:5 is a minor sixth, and 2:1 is an octave [2, 7, 8]. The ratio between notes is not quite consistent. For example, D:C is 9:8 but E:D is 10:9. However, it is found by a slight tampering with the exact ratios that the tuning requirements of stringed musical instruments, which sound at discrete frequencies, can be fairly well met by twelve notes per octave approximately 1 12th octave apart in the ratio of = , which is called a half step.twohalfsteps , five , and so on. Temperament is adjustment in the exact frequencies (above) to produce a 12-note musical scale for stringed instruments that is pleasing to the human ear [1, 12]. C.5 Human Perception of Sound (Psychological Acoustics) Sound pressures impinging on the eardrum membrane are transmitted by vibration of a mechanical linkage of small bones to the fluid-filled canals of the inner ear. The brain processes electrical impulses from vibrating hair cells that line these canals to register sound. Humans are most sensitive to sound at 1000 Hz. The acoustic reference pressure inairof20mparms( psi rms, 0 db) is the onset of hearing of a young healthy human adult at 1000 Hz (Eq. 6.7; [1, 5, 10]). The A-scale decibel weighting factors shown in Table C.2 mimic the frequency sensitivity of the human ear to moderate and loud sounds. A-scale weightings are numerically added to the SPL levels in the frequency bands to produce sound levels in db (A) [5, 6]. Community and job noise standards are based on sound in A-scale decibels. The US Environmental Protection Agency (EPA) gives maximum permissible daily sound exposure levels and their durations [14]. Exposure, h /2 1/4 SPL, db (A) For multiple levels in 1 day, the sum SPL i exposure i must be less than unity. These levels accept the possibility of some hearing loss above 4000 Hz ([2], p. 300). Acceptable noise annoyance levels are much lower, typically less than 55 db(a) outdoors [15].

6 Appendix C: Standard Octaves and Sound Pressure 427 Example C.1 An unweighted noise spectrum is equal to 100 db in each of six 1/3-octave bands from 100 to 630 Hz. Convert this spectrum to 1-octave bands with A-scale weighting and calculate the overall sound pressure in db(a). Solution: One-third-octave bands and their A-weighting are provided in Table C.2. Equation 6.8b is used to convert one-octave band levels from decibels to pascals. Mean square pressures are summed over three adjacent 1/3-octave bands (Equation C.3). OASPL is the sum of the bands mean square pressures (Equation C.3). 1/3-octave ctr frequency (Hz) Overall SPL decibels in 1/3 octaves A-scale weight (db) SPL (A) (db) p 2 rms Pa in /3 octaves p 2 rms Pa per octave SPL, db(a) per octave References [1] Pierce, A. D., Acoustics, McGraw-Hill, N.Y., [2] Kinsler, L. E., Frey, A., Coppens, A. B., and Sanders, J. V., Fundamentals of Acoustics, 3 rd ed., John Wiley, N.Y., [3] Beranek, L. Noise and Vibration Control, Institute of Noise Control Engineering, Washington, DC, Revised edition, [4] American National Standard ANSI S , Acoustic Terminology, N.Y., Reaffirmed, Also American National Standard C634-02, Standard Terminology Relating to Environmental Acoustics, [5] American National Standard ANSI S , Reference Quantities for Acoustical Levels, N.Y., Reaffirmed Also, Internationals Standards Organization, ISO 1683:1983, Acoustics Preferred Reference Quantities for Acoustic Levels. [6] American National Standard ANSI S , Sound Level Meters, N.Y. Also European Standard EN 60651, Sound Level Meters and IEC 651, January 1979 and British Standard BS EN 60651, Specification for Sound Level Meters, [7] Carey, W. M., Standard Definitions for Sound Levels in the Ocean, IEEE Journal of Ocean Engineering, vol. 20, pp , [8] Blevins, R. D., Probability Density of Finite Fourier Series with Random Phases, Journal of Sound and Vibration, vol. 208, pp , 1997.

7 428 Appendix C: Standard Octaves and Sound Pressure [9] American National Standard ANSI S , Preferred Frequencies, Frequency Levels, and Band Numbers for Acoustical Measurements, N.Y., Reaffirmed [10] Helmholtz, H., On the Sensations of Tone, Dover, N.Y., Revised Edition of [11] Lamb, H., The Dynamical Theory of Sound, 2 nd ed., Dover, N.Y., Reprint of 1925 edition. [12] Isacoff, S., Temperament, The Idea that Solved Music s Greatest Riddle, Alfred Knopf, N.Y., [13] American National Standard C634-02, Standard Terminology Relating to Environmental Acoustics, [14] USA Code of Federal Regulations, Occupational Safety and Health Standards, 1970, CFR (b). [15] Crocker, M. J., Noise Control, in Handbook of Acoustics, M. Crocker (ed.), John Wiley, N.Y., 1998.