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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

Abstract

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.

Keywords

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

CWT

Continuous wavelet transform

N

Sample size

NCCC

Non-stationary cross-correlation coefficient

NCSDP

Non-stationary cross-spectral density phase

NCSDPE

Non-stationary cross-spectral density phase envelope

SNR

Signal-to-noise ratio

T

Integration window

TOF

Time-of-flight

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

Wavelet cross-correlation function, Eq. 2

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

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

WECC

Wavelet envelope cross-correlation coefficient

x(t), y(t)

Continuous signals

ΔL

Distance between two transducers

ΔT

Time-of-flight of a disturbance between two sensors

\(\bar{\tau}\)

Time delay in wave propagation between two transducers

σ

Standard deviation

\(\psi\)

Wavelet function (mother wavelet)

\(\psi_{a,b}\)

Translated and dilated (daughter) wavelet

*

Complex conjugate

(^)

Estimate

Notes

Acknowledgments

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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