Experiments in Fluids

, Volume 52, Issue 1, pp 167–178 | Cite as

Wavelets for uncertainty estimates of propagating shock and detonation waves

  • F. K. Lu
  • A. A. Ortiz
Research Article


The propagation speed of a shock or detonation wave in a shock or detonation tube is usually determined by a time-of-flight method by dividing the distance between two transducers with the propagation time of the disturbance signal. Some arbitrariness is inherent in determining the propagation time by this method. A new method based on Haar and Morlet wavelet transforms is reported. The method was applied to shock and detonation waves representing a step and a decaying spike discontinuity. The wavelet methods can be applied to the step discontinuity provided that the SNR ratio is good. The wavelet methods worked well for a decaying spike in the presence of noise.


Detonation Wave Shock Tube Continuous Wavelet Transform Morlet Wavelet Shock Mach Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of symbols


Continuous wavelet transform


Sample size


Non-stationary cross-correlation coefficient


Non-stationary cross-spectral density phase


Non-stationary cross-spectral density phase envelope


Signal-to-noise ratio


Integration window



\(\widehat{\rm WC}_{xy}(a, \tau)\)

Wavelet cross-correlation function, Eq. 2

\(\widehat{\rm WR}_{xy}(a, \tau)\)

Wavelet cross-correlation coefficient (WCCC), see Eq. 9


Wavelet envelope cross-correlation coefficient

x(t), y(t)

Continuous signals


Distance between two transducers


Time-of-flight of a disturbance between two sensors


Time delay in wave propagation between two transducers


Standard deviation


Wavelet function (mother wavelet)


Translated and dilated (daughter) wavelet


Complex conjugate





This work is partly funded by the National University of Singapore via Research Collaboration Agreement No. TL/AE/2008/01. We also gratefully acknowledge the detailed comments by the reviewers that have improved this paper.


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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of Mechanical and Aerospace Engineering, Aerodynamics Research CenterUniversity of Texas at ArlingtonArlingtonUSA

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