At asymptotically large source to receiver distances

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At asymptotically large source-to-receiver distances, the time of signal arrival equals the time of source event plus the ratio of distance to V g b ( ω ): (46) where V g b is the group velocity: (47) Group velocity depends on plate material and thickness and on the branch in question and the frequency of interest but not on source type or distance. Time of arrival is different for different branches and frequencies. Thus, at any time t , there are several modes contributing to the waveform, all with different effective amplitudes Q and all with different frequencies. The result is a complex beat pattern. Extracting the modal amplitudes Q from that pattern is difficult because each such mode, having a different frequency, has been affected differently by the transducer sensitivity, the amplifier gain and the source function. Conversely, the waveform in the frequency domain can be understood as an interference pattern between the many branches that have the same frequency but different amplitudes Q and different arrival times. In either domain, the waveform is complex, because there are so many interfering branches at the same point. The best way to analyze these waveforms is in neither the frequency domain nor the time domain but rather by means of a simultaneous time and frequency decomposition. If signal processing can effect this decomposition, the only remaining interferences would be at the occasional places where two branches have the same group velocity and frequency. Much signal processing literature tries to optimize such decompositions. Work has focused on the spectrogram, that is, on the square of the short time fourier transform, and on the wigner transform. The simplest analysis of time versus frequency, the spectrogram, has been found to be adequate. A spectrogram calculated from a theoretical waveform (an evaluation of Eq. 34) is shown in Fig. 4, where amplitudes concentrated along the loci of frequency versus arrival time are evident. Similar spectrograms are V d dk b b g = ω Time of signal arrival Time of source event Distance g = + ( ) V b ω Q k k Q k k b eff stepload ( ) = ( ) ( ) 1 2 ω Q k W h 1 2 2 stepload sign ( ) = ( ) ( ) ρ M v F kdk Q k J kr k t k b b b = × ( ) ( ) [ ] × ( ) [ ] ( ) – cos 0 stepload 0 π ω ω 1 2 1 118 Acoustic Emission Testing F IGURE 4. Spectrogram obtained from a theoretical acoustic emission waveform. Amplitudes are confined to specific loci in the space of time versus frequency. 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 100 200 300 400 500 600 700 800 Frequency (MHz) Time (μs) S0 A0 Legend S0 = asymmetric S wave A0 = symmetric P wave
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shown in this NDT Handbook volume’s chapter on source location. From such spectrograms, the distance to the source can be extracted by comparing the times of arrival of different modes of known group velocity. For example, it is clear that the rayleigh wave arrives around time t R = 325 μs. The S0 mode arrives around t S0 = 60 μs. Thus, distance of the source can be calculated: (48) By analyzing multiple points in such plots, it is possible in principle to obtain highly accurate source distances.
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  • Fall '19
  • Acoustic Emission

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