Abstract
The spatial autocorrelation (SPAC) method requires a special circular array where several observation points are equally spaced on the circumference. In order to look for the possibility of developping a new method of array observation with fewer restriction of arrangement than the SPAC method, we proposed a formula of the complex coherence function (CCF) of the Rayleigh wave measured on a couple of observation points located at any place. This formula was derived on the basis of an analytical solution of Lamb's problem, aiming to study the relation between wave source and observation point. The formula was given as simple discrete representation consisting of the Bessel function of the first kind of zero order J0 (kr) (k: wavenumber, r: radius of array) and an infinite series with higher-order Bessel functions. In the SPAC method, by regarding the SPAC coefficient from directional average of CCFs (real part) as J0 (kr), phase velocities of Rayleigh waves (wave number k) are calculated.We first studied the relationship between the values of CCF and wave sources located far from a couple of observation points, and found that the values of CCF strongly varies depending on the direction with increase in kr, and also found that such directional properties were mainly caused by the variation of the infinite series in the formula of CCF. Furthermore, we applied the formula to the SPAC method for revealing the mechanism of the directional average and the reason why the SPAC method requires the special circular array with sensors equally spaced on a circle. The results are summarized as follows: 1) The values of the infinite series gets lower enough to be negligible after the directional average of CCFs, so that the SPAC coefficient can be approximated to J0 (kr). 2) From the inverse analysis on the condition that the values of the infinite series is equal to zero, it was found that the condition was satisfied not only in the usual SPAC arrays but also in some extra circular arrays consisting of observation points not equally spaced on the circumference.This result suggests the possibility of array design with fewer restriction of arrangement of observation points, using a new algorithm for obtaining J0 (kr) without the operation of directional average used in the SPAC method.
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