ICSV 14 Cairns • Australia 9-12 July , 2007 PROPAGATION PHASE REPRESENTATION IN 3 D SPACE USING POLES AND ZEROS IN COMPLEX FREQUENCY PLANE
This paper describes the transfer function, including the coherent sound field, as the increased distance between source and observation points, from the point of view of the number of zeros in the complex frequency plane. The authors estimated the propagation phase obtained by linear regression analysis of the narrow-frequency-band phase characteristics of the minimumphase component of the transfer function. The direct sound field estimated from the propagation phase of the minimum-phase characteristics consequently seems to be the coherent field within the critical distance. We determined that the zeros drift based on complex-integration analysis. When the sound source distance (SSD) increases, the zeros move toward a higher frequency; thus, an accumulated phase is produced following the propagation phase. When the SSD is long, even in the coherent field, we could confirm a pole-line around the theoretical value.
Phase variabilities and zeros in a reverberant transfer function
The phase trend of a transfer function (TF) has a nearly constant group delay in a reverberant sound field under high modal overlap conditions. This phase trend is predictable from the number of nonminimum phase zeros of the TF. This paper investigates the phase variability from such a phase trend in a reverberant TF. These phase fluctuations, which are generally due to random occurrences of both poles and zeros, can be expressed using the group delay, which is the first derivative of the accumulated phase. Theoretical analysis of the group delay implies that its fluctuations have a ‘‘1/f’’ type of spectrum and that the variance of the group delay is independent of the frequency band and decreases as the damping of the system increases. The results are confirmed by the power spectrum and correlation analysis using measured TF records. Accumulated phase variances are also investigated on the basis of the Δ statistic [F. Dyson and M. Mehta, ‘‘Statistical Theory of the Energy Levels of Complex Systems. IV,’’...
Propagant phase in reverberant environments
The phase of the transfer function between two points in an extended system can be easily measured if it is taken to be the accumulated phase obtained by smoothly raising the measurement frequency from zero to the reference frequency. Lyon et al., in an extended series of papers [most recently J. Acoust. Soc. Am. 95, 286–296 (1994)], have examined the behavior of this accumulated phase in systems of two and three dimensions and have elucidated the concept of a reverberant phase which is independent of the separation between the two measurement points, provided they are far enough apart, but which rises sharply with increasing frequency. In some applications, for example, in nondestructive testing of extended structures, it is important to be able to observe simple wave‐propagation behavior and in particular to measure the propagant phase as a function of frequency and position. The conditions under which this is possible are investigated, and are shown to impose constraints on the ratio between the propagation distance and the size of the structure under test, and either the material damping coefficient or the reflection coefficient at the domain boundaries. These results, which represent an extension of those of Lyon et al., are discussed in terms of the distribution of zeros of the transfer function in the complex frequency plane. Many platelike structures of practical interest are found to satisfy these conditions, so that measurement of propagant phase behavior can provide the basis for a useful technique of nondestructive examination.