Ultrasonic Material Characterization in Hostile Environments
Anthony Puckett, Ph.D. Candidate; Michael “Mick” Peterson, Ph.D.
Advanced Inspection and Monitoring Lab, University of Maine
Problem
A number of applications exist in materials research where the ability to obtain real-time properties in a hostile environment would greatly facilitate the development of new materials, material treatments and manufacturing techniques.  These hostile environments include extreme temperatures, both high (2100 ºC) and low, extreme pressures, and corrosive environments.  Among the applications are densification and sintering of ceramics, chemical vapor deposition and infusion processes, float zone refining, and degradation processes such as oxidation.
Approach
In-situ ultrasonic nondestructive monitoring has been previously used in applications for material characterization [Peterson, 1994].  Limitations on the operating temperatures of traditional piezo-ceramics make it is necessary to isolate the transducers from the environment.  Solid cylindrical waveguides have been used by a number of investigators to isolate the ultrasonic transducers from the hostile environment in these applications [Jen et al., 2001].  Large diameter waveguides are required for adequate energy transfer through the couplants and sample.  An ultrasonic pulse is used to acquire length scale information about the material.  The broadband nature of the pulse coupled with the thick waveguide produces multiple dispersive propagating modes. The multiple dispersive modes cause a broadband ultrasonic signal to expand in time, Figure 1.  This work explores generation of a signal with compact support in the time domain with experimental and analytical time reversal techniques. 
Waveguides
Solid cylindrical waveguides made from optical grade fused quartz can be used for temperatures up to 1200 °C.  At higher temperatures (up to 2100 °C) a single crystal sapphire waveguide is required.   At high frequencies geometrical dispersion is present in the waveguides for both cases, although material dispersion is negligible.  Geometrical dispersion in cylindrical waveguides is a result of multiple paths and mode conversion at the boundary.  Spherical waves from a point source can be represented by plane waves.  The plane waves propagate along different length paths from one end of the cylinder to the other resulting in different arrival times.  At each reflection a longitudinal wave can excite a shear and longitudinal wave in order to satisfy the boundary conditions.  These two phenomena cause the dispersion.  Mathematically, the multiple dispersive modes are predicted by the Pochhammer-Chree frequency equation. 
Time Reversal Mirror
A time reversal mirror is used to produce a signal with compact support in the time domain despite the dispersive effects of the cylindrical waveguide.  Time reversal and the time reversal mirror are based on the property of time reversal invariance.  For acoustics the concept is simply described (Ing and Fink, 1998):
“Time reversal invariance of the acoustic wave equation means that, for every burst of sound diverging from a source - and possibly reflected, refracted or scattered by any propagation media - there exists in theory a set of waves that precisely retraces all of these complex paths and converges in synchrony, at the original source, as if time were going backward.”
For the cylindrical waveguide this means that there is a signal that will reconstruct in the waveguide to generate a pulse at the opposite end.  The time reversal mirror (TRM) will determine this signal.  A TRM is the series of steps to recreate a signal at a source regardless of the wave interactions between the receivers(s) and the source.  The source and receiver are the transducers at either end of a cylindrical waveguide, Figure 2. 
Analytic Time Reversal
The time reversal mirror can also be implemented with analytical inputs instead of experimental inputs.  Model based sensors can create a more robust system as well as providing the potential to interrogate modes separately.  An accurate model that describes the dispersion in a waveguide can be used to determine the time reversed signal for any signal.  The analytic model currently being developed predicts the dispersed signal using the propagating modes and the Pochhammer-Chree equations.  The relative amplitudes of each mode are determined from the pressure distribution on the end of the waveguide.  A phase shift is calculated from the phase velocity of each mode at each frequency.  Finally, the modes are summed over the end of the waveguide and averaged.  Currently the models predicts qualitatively the shape of the dispersed signal.  Future work will be required to increase accuracy. Figure 5 shows the analytic model and compares the analytical signal with the actual experimental signal. 
Figure 1. Effect of dispersion in a solid cylindrical waveguide.  The top graph is the excitation signal and the bottom graph is the received signal at the opposite end of the waveguide. 
Trigger
Arbitrary Function Generator
Agilent 33250A
RF Amplifier
ENI A-300
Transducers, 38 mm dia.
Panametrics, model V194
Fused Quartz rod, 10 mm dia.
Oscilloscope
Tektronix TDS 520A
Pre-amplifier
Panametrics,
 model 5660
485 mm length
Experimental TRM
A cylindrical waveguide can be used in both a pulse echo configuration and a through transmission configuration.  The time reversal mirror is effective in both configurations. The pulse echo configuration uses the same transducer to send and receive the ultrasonic signal.  Figure 4 illustrates the experimental set up of the pulse echo configuration.  However, for some applications two waveguides are required since multiple reflections through the sample cannot be received at a single transducer due to the high level of attenuation within the sample.  For these cases a waveguide is placed on either side of the sample in a through transmission configuration. One transducer sends a signal and a second transducer receives the signal with the ultrasonic signal passing through each waveguide once.  The experimental setup would be similar to the one in Figure 2 with a sample between two waveguides replacing the single waveguide. 
Figure 3. Signals in a time reversal mirror.  The signals from top to bottom are: the original excitation signal, the dispersed signal, the time reversed dispersed signal (new excitation signal), the new measured signal, and the new signal reversed in time for comparison to the original excitation signal.
Figure 2. Through transmission TRM experimental setup. 
Future Work
Future work is focused on refining and developing the model.  The analytical model shows promise, but needs to be revised to be more useful.  Experimentally, future work will included applying the presented techniques to actual material characterization processes.   
Figure 6. Current model (top) and comparison of experimental and analytic dispersed signals (bottom). 
References:
•Ing, R.K., and Fink, M. (1998). “Time-reversed lamb waves,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 45,1032-1043 •Jen, C.-K., Franca, D.R., Sun, Z., and Ihara, I. (2001). “Clad polymer buffer rods for polymer process monitoring,” Ultrasonics. 39, 81-89. •Montaldo, G., Roux, P., Derode, A., Negreira, C., and Fink,M. (2001). “Generation of very high pressure pulses with 1-bit time reversal in a solid waveguide,” J. Acoust. Soc. Am. 110, 2849-2857. •Peterson, M.L. (1994). “A High temperature Ultrasonic Process Monitoring System Utilizing Signal Processing to Separate Multiple Wave Guide Modes.” Ph.D. diss., Northwestern University. •Peterson, M.L. (1999). “Prediction of longitudinal disturbances in a multi-mode cylindrical waveguide,” Experimental Mechanics. 39, 36-42.
For the TRM considered in this research the desired signal is a pulse.  The pulse contains sufficiently high frequencies such that the pulse is distorted as it travels down the waveguide due to the different group velocities of the mode in waveguide.  The first step of the TRM is to acquire this distorted or “dispersed” signal.  This signal is reversed in time.  When the reversed signal is now applied as the excitation signal, the received signal has the shape of the original pulse. This signal reversed in time will match the original excitation pulse.  Figure 3 illustrates the signals involved in the time reversal mirror.  An advantage of the waveguide is the symmetry normal to the cylinder axis at the center of the waveguide allowing the receiver and source to be interchangeable. 
Oscilloscope
Tektronix TDS 520A
Trigger
Transducer, 38 mm dia.
Panametrics, model V194
Diplexer,
Ritec, RDX2
Aluminum sample
Pre-amplifier
Panametrics,
model 5660
Arbitrary Function Generator
Agilent 33250A
RF Amplifier
ENI A-300
Fused Quartz rod, 25 mm dia.
228 mm in length
Research has been conducted using TRM with solid cylindrical waveguides by Montaldo et al (2001). The work by Montaldo et al is focused on developing large pressure pulses at point in a fluid for lithotripsy.  Signal accuracy is not required, so multiple transducers are used on the end of a waveguide with a 1-bit method TRM for maximum energy generation. 
The current research is focused on material characterization, and the accuracy of the signals is critical.  Current techniques use a single transducer on the end of the waveguides.  The pulse echo configuration has demonstrated the ability to distinguish a change in acoustic impedance at the end of the waveguide.  Figure 5 compares received signals from the previous experimental setup without and with an aluminum block on the end of the waveguide. 
Figure 4. Pulse echo experimental setup. 
Figure 5. Received signals, without (top) and with aluminum block. 
Acknowledgment
This research is sponsored by the Ballistic Missile Defense Organization through the Office of Naval Research (ONR), Science Officer Dr. Y. D. S. Rajapakse.