Spatial Attenuation

As described above (Section 3.2.1), division of sequential signal pulses in the frequency domain, also known as the pulse-overlap method [Kinra & Dayal 1988], is a straight forward method of determining the frequency dependent attenuation of harmonic waves propagating in space. This method of attenuation measurement is very well suited to pulse-echo configurations because the transfer functions of the two pulses contain identical information save for the material transfer function of the difference in path length between them. The key assumption to this method is that both pulses contain the same spectral content where the spectral amplitude of the second pulse is lower than the amplitude of the first pulse. As shown above and in Figure 5-11b, this is generally not the case with coda wave spectra as the transducer separation distance is increased.

Indeed, frequently the spectral amplitude increases with separation distance as can be seen. Thus the pulse-overlap method is not applicable for coda wave attenuation measurement and is included in here only for completeness.

The resulting frequency dependent amplitude attenuation of coda waveforms collected from the unidirectional CFRP plate with the transducer pair aligned to 0˚ fiber orientation is shown in Figure 5-11a while the corresponding spectra are shown in Figure 5-11b. Negative amplitude attenuation values are readily apparent. It is a tendency to erroneously assume that negative attenuation values mean energy is being added to the system and that the signals are growing in strength as they propagate, which would violate conservation laws. A negative attenuation does not indicate the addition of energy, but in actuality it merely means that the amplitude of the second point is greater than the amplitude of the first point. Keep in mind that coda wave signals are not propagating waves but are spatially sampled, time-dependent superposition field. Thus coda wave signals need not follow the same restrictions as propagating harmonic waves. Also note that

amplitude measurements taken from various locations on a plate that is resonating will also yield negative attenuation as illustrated in Figure 5-12. This is a simple illustration that traditional definitions for ultrasonic wave parameters do not necessarily apply to coda waves.

The most basic definition of attenuation is the decay of waveform (or wavelet) peak amplitude with increasing travel distance. This definition is logical for a single traveling wave mode as it measures the loss of information with travel distance due to scattering and absorption, but may not be very applicable to a superposition of multiple wave modes. One of the best methods

(a) (b)

Figure 5-11. Attenuation calculated by frequency domain division with a) typical attenuation curves and b) typical spectra of waveforms collected at different transducer separations.

Measurements were made on a 0.9 cm unidirectional CFRP plate with transducer separations as listed. The signal was transmitted and received with a 1.25 cm, 1MHz longitudinal transducer.

Figure 5-12. Illustration of positive and negative attenuation due to measuring the amplitudes of a plate resonance mode at various locations.

of characterizing the loss of information with travel distance for coda waves would be to use a parameter that characterizes the entirety of the waveform, such as spectral energy. In this case the decay of energy correlates to the decay of information. Both of these attenuation measures were applied to the coda wave components collected from the isotropic, quasi-isotropic, and transversely isotropic plates. The attenuations are determined by fitting a line to the natural logarithm of the peak amplitude (or spectral energy) versus transducer separation plots using least squares. The slope of the line is the attenuation while the standard error of the fit is the measurement error. Typical attenuation results and corresponding coefficients of determination (R²) are plotted in Figure 5-13. R² > 0.6 is considered a good fit, and the majority of the data yield R² values greater than 0.9, which indicates that the reductions in amplitude and spectral energy are well described by an exponential decay. The lowest R² value observed is 0.4.

The measured attenuation of the peak amplitudes and the spectral energies are plotted in Figure 5-14 & Figure 5-15. The spatial attenuation of a waveform (or wavelet) in an isotropic material, regardless of whether it is the attenuation of the peak amplitude or the energy, is expected to be insensitive to orientation angle.

Figure 5-13. Typical results (Figure 5-14a1) from fitting an exponential to the data sets using least squares where  is the value of the exponential decay and R² is the coefficient of

determination between the fit line and the data set.

(a1) (b1)

(a2) (b2)

(a3) (b3)

(a4) (b4)

Figure 5-14. Attenuation of peak amplitude and spectral energy due to increasing transducer separation with fiber angle for a 30x30x0.9 cm a) unidirectional CFRP plate and b) Plexiglas plate. Signal was transmitted with a 1.25 cm, 1 MHz longitudinal transducer and received with 1)

1 MHz L, 2) 1MHz S0, 3) 1MHz S90, and 4) pinducer transducers.

(a) (b)

(c)

Figure 5-15. Attenuation of peak amplitude and spectral energy due to increasing transducer separation with fiber angle for a) a 0.9 cm unidirectional CFRP plate, b) a 0.9 cm quasi-isotropic CFRP plate and c) a 0.9 cm Plexiglas plate. Signal was transmitted and received with a 1.25 cm,

1 MHz longitudinal transducer.

The spatial attenuations of coda wave peak amplitudes and spectral energies may provide a measure of the decay of the information carried in waveform as the waveform spreads, but gives no indication of the spatial decay of specific frequency components. Typically, the frequency dependent attenuation is determined by the pulse-overlap method, but this method is not applicable as demonstrated previously. To overcome this, the coda wave spectra are equipartitioned using Tukey windows with 12.5% tapers on both sides then inverse transformed back into the time domain. The natural logarithm of the peak amplitudes of the time domain waveforms corresponding to a specific frequency bin are then plotted versus transducer separation and a line is fit to the data using least squares linear regression. The attenuation for that particular center

frequency is then the slope of the line and the error in the attenuation measurement is the standard error of the fit.

Attenuation results and corresponding coefficients of determination (R²) are plotted in Figure 5-16. R² > 0.6 is considered a good fit, and the majority of the data yield R² values greater than 0.8, which indicates that the reduction in amplitude is well described by an exponential decay.

Several R² values are quite low (mostly along the 0˚ orientation for the unidirectional CFRP). This is typically caused by a single datum that is significantly out of alignment with the other data during the regression, which is usually caused by significant differences in the spectra with increasing separation. The spectra for the 0˚ orientation for the unidirectional CFRP exhibits the most variation with respect to transducer separation. This is similar to one of the issues that invalidated the pulse-overlap method for coda waves. Poor regression fits usually result in negative attenuations.

Figure 5-16. Example results from fitting an exponential to the frequency windowed data sets using least squares where  is the value of the exponential decay and R² is the coefficient of

determination between the fit line and the data set.

(a1) (b1)

(a2) (b2)

(a3) (b3)

(a4) (b4)

Figure 5-17. Frequency-windowed attenuation of peak amplitude due to increasing transducer separation with fiber angle for a) a 0.9 cm unidirectional CFRP plate and b) a 0.9 cm Plexiglas plate. Signal was transmitted with a 1.25 cm, 1 MHz longitudinal transducer and received with 1)

1 MHz L, 2) 1MHz S0, 3) 1MHz S90, and 4) pinducer transducers.

(a) (b)

(c)

Figure 5-18. Frequency-windowed attenuation of peak amplitude due to increasing transducer separation with fiber angle for a) a 0.9 cm unidirectional CFRP plate, b) a 0.9 cm quasi-isotropic CFRP plate and c) a 0.9 cm Plexiglas plate. Signal was transmitted and received with a 1.25 cm,

1 MHz longitudinal transducer.

Frequency dependent attenuation results are plotted in Figure 5-17 & Figure 5-18 for the three materials and the wave components. The difficulty in measuring the coda wave attenuation for the unidirectional CFRP sample at 0˚ orientation is clear, and most of the data must be discarded. Attenuation of the longitudinal component increases with frequency until 0.5 MHz then plateaus. It also increases with fiber angle until 45˚ then decreases slightly until 90˚. This agrees with the peak amplitude observations. In a similar manner, there is also disagreement between the longitudinal attenuation measured with the 12.7 mm and 2 mm transducers. The differences in peak amplitude was due to the phase cancellation, but the differences in attenuation are due to the increased noisiness of the pinducer spectra. The attenuation of the shear components shows a general upward trend without plateauing.

For an ideal, symmetrically expanding wave field the isotropic and quasi-isotropic attenuation curves should be insensitive to orientation angle. This is not the case, however, with the measured coda wave attenuation as the data exhibits significant variation with orientation. The superposition field is not composed of ideal, symmetrically expanding wave fields and as such there is no reason to suppose that the coda wave field itself is circularly symmetric or that the attenuation should be insensitive to orientation.