Inst. of Acoust., Academia Sinica, Beijing 100080, People's Republic of China
In acoustic well logging, the borehole may be considered as a vertical waveguide of infinite cross section with horizontal discontinuities, and the conventional mode expansion cannot be applied. Some authors introduced an artificial large coaxial cylindrical boundary, transforming the problem into one with finite cross section. However, this approach is limited to low frequency. Based on the theory of generalized function, the mixed spectrum of the infinite cross-section waveguide is analyzed and a new hybrid method is proposed in which the acoustic field is expressed as a sum of several discrete modes and an integral of continuous modes with unknown weight functions which are obtained by numerically solving a group of integral equations deduced from the boundary conditions at discontinuities. It is shown that this method makes it possible to calculate up to the frequency for a typical logging environment. Laboratory experiments have been conducted in which a specific method was adopted to form a liquid cylinder surrounded by fluid layers. Both studies reveal that the reflection of the guided wave at the discontinuity is strongly dependent on the wave frequency. When the frequency is low the reflection is prominent, but at a high frequency the reflection is much weaker.