The skewness of the first time derivative of a pressure waveform, or derivative skewness, has been used previously to describe the presence of shock-like content in jet and rocket noise. Despite its use, a quantitative understanding of derivative skewness values has been lacking. In this paper, the derivative skewness for nonlinearly propagating waves is investigated using analytical, numerical, and experimental methods. Analytical expressions for the derivative skewness of an initially sinusoidal plane wave are developed and, along with numerical data, are used to describe its behavior in the preshock, sawtooth, and old-age regions. Analyses of common measurement issues show that the derivative skewness is relatively sensitive to the effects of a smaller sampling rate, but less sensitive to the presence of additive noise. In addition, the derivative skewness of nonlinearly propagating noise is found to reach greater values over a shorter length scale relative to sinusoidal signals. A minimum sampling rate is recommended for sinusoidal signals to accurately estimate derivative skewness values up to five, which serves as an approximate threshold indicating significant shock formation.
References
1.
D. T.
Blackstock
, “Nonlinear propagation of jet noise
,” in Proceedings of the Third Interagency Symposium on University Research in Transportation Noise
, University of Utah, Salt Lake City, UT (1975
), pp. 389
–397
.2.
C. L.
Morfey
and G. P.
Howell
, “Nonlinear propagation of aircraft noise in the atmosphere
,” AIAA J.
19
, 986
–992
(1981
).3.
K. L.
Gee
, V. W.
Sparrow
, M. M.
James
, J. M.
Downing
, C. M.
Hobbs
, T. B.
Gabrielson
, and A. A.
Atchley
, “The role of nonlinear effects in the propagation of noise from high-power jet aircraft
,” J. Acoust. Soc. Am.
123
, 4082
–4093
(2008
).4.
K. L.
Gee
, J. M.
Downing
, M. M.
James
, R. C.
McKinley
, R. L.
McKinley
, T. B.
Neilsen
, and A. T.
Wall
, “Nonlinear evolution of noise from a military jet aircraft during ground run-up
,” AIAA paper No. 2012–2258 (2012
).5.
B. P.
Petitjean
and D. K.
McLaughlin
, “Experiments on the nonlinear propagation of noise from supersonic jets
,” AIAA paper No. 2003–3127 (2003
).6.
B. P.
Petitjean
, K.
Viswanathan
, and D. K.
McLaughlin
, “Acoustic pressure waveforms measured in high speed jet noise experiencing nonlinear propagation
,” Int. J. Aeroacoust.
5
, 193
–215
(2006
).7.
B.
Greska
and A.
Krothapalli
, “On the far-field propagation of high-speed jet noise
,” Proceedings of NCAD2008
, paper No. NCAD2008–73071 (2008
).8.
W. J.
Baars
, C. E.
Tinney
, M. S.
Wochner
, and M. F.
Hamilton
, “On cumulative acoustic waveform distortions from high-speed jets
,” J. Fluid Mech.
749
, 331
–366
(2014
).9.
J. E.
Ffowcs-Williams
, J.
Simson
, and V. J.
Virchis
, “ ‘Crackle’: An annoying component of jet noise
,” J. Fluid Mech.
71
, 251
–271
(1975
).10.
M. R.
Shepherd
, K. L.
Gee
, and A. D.
Hanford
, “Evolution of statistical properties for a nonlinearly propagating sinusoid
,” J. Acoust. Soc. Am.
130
, EL8
–EL13
(2011
).11.
S. N.
Gurbatov
and A. N.
Malakhov
, “Statistical characteristics of random quasi-monochromatic waves in non-linear media
,” Sov. Phys. Acoust.
23
, 325
–329
(1977
).12.
K. L.
Gee
, T. B.
Neilsen
, and A. A.
Atchley
, “Skewness and shock formation in laboratory-scale supersonic jet data
,” J. Acoust. Soc. Am.
133
, EL491
–EL497
(2013
).13.
P.
Mora
, N.
Heeb
, J.
Kastner
, E. J.
Gutmark
, and K.
Kailasanath
, “Near and far field pressure skewness and kurtosis in heated supersonic jets from round and chevron nozzles
,” in Proceedings of ASME Turbo Expo 2013
, paper No. GT2013-95774 (2013
).14.
S. A.
McInerny
, K. L.
Gee
, J. M.
Downing
, and M. M.
James
, “Acoustical nonlinearities in aircraft flyover data
,” AIAA paper No. 2007–3654 (2007
).15.
S. A.
McInerny
, M.
Downing
, C.
Hobbs
, and M.
Hannon
, “Metrics that characterize nonlinearity in jet noise
,” AIP Conf. Proc.
838
, 560
–563
(2006
).16.
O. V.
Rudenko
and S. I.
Soluyan
, Theoretical Foundations of Nonlinear Acoustics
(Plenum
, New York
, 1977
).17.
S. N.
Gurbatov
, A. N.
Malakhov
, and N. V.
Pronchatov-Rubtsov
, “Evolution of higher-order spectra of nonlinear random waves
,” Radiophys. Quantum Electron.
29
, 523
–528
(1986
).18.
D. A.
Hennessy
, “Crop yield skewness under law of the minimum technology
,” Am. J. Agr. Econ.
91
, 197
–208
(2009
).19.
J.
Chen
, H.
Hong
, and J. C.
Stein
, “Forcasting crashes: Trading volume, past returns, and conditional skewness in stock prices
,” J. Financ. Econ.
61
, 345
–381
(2001
).20.
K. R.
Sreenivasan
and S.
Tavoularis
, “On the skewness of the temperature derivative in turbulent flows
,” J. Fluid Mech.
101
, 783
–795
(1980
).21.
S.
Tavoularis
, J. C.
Bennett
, and S.
Corrsin
, “Velocity-derivative skewness in small Reynolds number, nearly isotropic turbulence
,” J. Fluid Mech.
88
, 63
–69
(1978
).22.
C. W.
Van Atta
and R. A.
Antonia
, “Reynolds number dependence of skewness and flatness factors of turbulent velocity derivatives
,” Phys. Fluids
23
, 252
–257
(1980
).23.
K. L.
Gee
, V. W.
Sparrow
, A. A.
Atchley
, and T. B.
Gabrielson
, “On the perception of crackle in high-amplitude jet noise
,” AIAA J.
45
, 593
–598
(2007
).24.
S. A.
McInerny.
“Launch vehicle acoustics. II—Statistics of the time domain data
,” J. Aircraft
33
, 518
–523
(1996
).25.
K. L.
Gee
, T. B.
Neilsen
, M. B.
Muhlestein
, A. T.
Wall
, J. M.
Downing
, M. M.
James
, and R. L.
McKinley
, “Propagation of crackle-containing jet noise from high-performance engines
,” Noise Control Eng. J.
64
, 1
–12
(2016
).26.
P.
Mora
, N.
Heeb
, J.
Kastner
, E.
Gutmark
, and K.
Kailasanath
, “Impact of heat on the pressure skewness and kurtosis in supersonic jets
,” AIAA J.
52
, 777
–787
(2014
).27.
M. B.
Muhlestein
and K. L.
Gee
, “Experimental investigation of a characteristic shock formation distance in finite-amplitude noise propagation
,” POMA
12
, 045002
(2014
).28.
M. B.
Muhlestein
, K. L.
Gee
, T. B.
Neilsen
, and D. C.
Thomas
, “Evolution of the average steepening factor for nonlinearly propagating waves
,” J. Acoust. Soc. Am.
137
, 640
–650
(2015
).29.
S.
Earnshaw
, “On the mathematical theory of sound
,” Trans. R. Soc. London
150
, 133
–148
(1860
).30.
E.
Fubini
, “Anomalie nella propagazione di ande acustiche de grande apiezza
” (“Anomalies in acoustic wave propagation of large amplitude”), Alta Frequenza
4
, 530
–581
(1935
) [English translation: R. T. Beyer, Nonlinear Underwater Sound, 118–177 (1984)].31.
R. D.
Fay
, “Plane sound waves of finite amplitude
,” J. Acoust. Soc. Am.
3
, 222
–241
(1931
).32.
W. J.
Baars
and C. E.
Tinney
, “Shock-structures in the acoustic field of a Mach 3 jet with crackle
,” J. Sound Vib.
333
, 2539
–2553
(2014
).33.
D. T.
Blackstock
, M. F.
Hamilton
, and A. D.
Pierce
, “Progressive waves in lossless and lossy fluids
,” in Nonlinear Acoustics
(1998), Chap. 4, pp. 65–150.34.
R. O.
Cleveland
, “Propagation of sonic booms through a real, stratified atmosphere
,” Ph.D. dissertation, Department of Mechanical Engineering, The University of Texas at Austin
, Austin, 1995
.35.
Loubeau
, A.
Sparrow
, V. W.
Pater
, L. L.
, and Wright
, W. M.
, “High-frequency measurements of blast wave propagation
,” J. Acoust. Soc. Am.
120
, EL29
–EL35
(2006
).36.
S. A.
McInerny
and S. M.
Olcmen
, “High intensity rocket noise: Nonlinear propagation, atmospheric absorption, and characterization
,” J. Acoust. Soc. Am.
117
, 578
–591
(2005
).37.
D. T.
Blackstock
, “Connection between the Fay and Fubini solutions for plane sound waves of finite amplitude
,” J. Acoust. Soc. Am.
39
, 1019
–1026
(1966
).38.
J. A.
Gallagher
and D. K.
McLaughlin
, “Experiments on the nonlinear characteristics of noise propagation from low and moderate Reynolds number supersonic jets
,” in 7th Aeroacoustics Conference
(1981), pp. 1981
–2041
.39.
S. N.
Gurbatov
and O. V.
Rudenko
, “Statistical phenomena
,” in Nonlinear Acoustics
, edited by M. F.
Hamilton
and D. T.
Blackstock
(Academic Press
, San Diego
, 1998
), Chap. 13
, pp. 377
–398
.40.
F. M.
Pestorius
and D. T.
Blackstock
, “Propagation of finite-amplitude noise
,” in proceedings of Finite-Amplitude Wave Effects in Fluids
, Copenhagen (1973
), pp. 24
–29
.41.
M. B.
Muhlestein
and K. L.
Gee
, “Evolution of the temporal slope density function for waves propagating according to the inviscid Burgers equation
,” J. Acoust. Soc. Am.
139
(2), 958
–967
(2016
).© 2016 Acoustical Society of America.
2016
Acoustical Society of America
You do not currently have access to this content.
