The properties of Cylindrical Internal-Surface Acoustic Waves (CISAW) propagating on the inside surface of a high purity tubular fused quartz fiber are derived from basic principles using a variational method. The change in the wave vector due to gravity, of the CISAW is also derived. I was unable find descriptions in technical or scientific literature of the modes of CISAW propagating on the inside surface of a tubular fiber. There is also no description in the technical or scientific literature of the effect of gravity, on CISAWs. Therefore, these derivations are completely new. The group velocity of CISAW is less than the velocity of light, and the effect of gravity on the wave vector of the CISAW is much larger than the effect of gravity on the light wave vector. Therefore, gravity detectors based CISAW would be much smaller than the present km-sized gravity detecting LIGO. One can build an array of CISAW gravity wave detectors which would be able to detect gravity wave images. It would be interesting what new discoveries could be made with such an array.