Recently, the development of a novel ultrasonic inspection technique that detects radial fatigue cracks on the far side of so-called “weep” holes in thin airframe stiffeners was reported [1]. These cracks tend to be located on the upper part of the weep hole (at 12 o’clock position) therefore are not readily detectable by conventional ultrasonic inspection techniques from the lower skin of the wing. The new technique utilizes circumferential creeping waves propagating around the inner surface of the hole to perform the inspection. However, the wet wing has to be emptied and dried out before inspection because even a small amount of fluid fuel trapped in these rather small (approximately 6–7 mm in diameter) holes would strongly affect the propagation of circumferential creeping waves. We have searched the literature for published results on circumferential creeping wave propagation around fluid-filled cylindrical cavities in elastic media. Surprisingly, although the analytical solution of this canonical problem can be readily constructed from existing building blocks, very little was found in terms of numerical results that could be used to gain better understanding of the phenomenon. This motivated us to attack the problem by numerically solving the dispersion equation and constructing the corresponding dispersion and attenuation curves for a specific case of interest, namely, for that of a water-filled cylindrical hole in aluminum.