Characterization of animal bones by acoustic microscopy is a relatively new field of research. Non-uniformity of bones requires a complete analysis of the interaction between the focused acoustic beam and the bone specimens for its characterization. The complete analysis produces the V(z) curve taking into account the lens angle and the elastic properties of the bone. V(z) oscillations may be generated by surface skimming Rayleigh waves and/or P-waves in different parts of the bone depending on the bone properties. In this paper after a short review of different applications of the acoustic microscopy technique a complete theory of the V(z) curve synthesis is briefly presented. A number of V(z) curves are analytically generated following this theory. These curves sometimes show a regular oscillatory shape as predicted by the simple ray theory. However, often the synthesized V(z) curves show irregularities. The synthesized V(z) curve for silicon has been compared with the experimental results to have a confidence on the analytical computation. Then additional numerical exercises are carried out to understand why sometimes the V(z) curve shapes are irregular, and whether the P-wave speed and/or the Rayleigh wave speed of the material can be extracted from such irregular shaped V(z) curves. After understanding the V(z) phenomenon the wave speeds at different points of a bone are experimentally measured from its V(z) curves.