Sonoelasticity is a rapidly evolving medical imaging technique for visualizing hard tumors in tissues. In this novel diagnostic technique, a low-frequency vibration is externally applied to excite internal vibrations within the tissue under inspection. A small stiff inhomogeneity in a surrounding tissue appears as a disturbance in the normal vibration eigenmode pattern. By employing a properly designed Doppler detection algorithm, a real-time vibration image can be made. A theory for vibrations, or shear wave propagation in inhomogeneous tissue has been developed. A tumor is modeled as an elastic inhomogeneity inside a lossy homogeneous elastic medium. A vibration source is applied at a boundary. The solutions for the shear wave equation have been found both for the cases with tumor (inhomogeneous case) and without tumor (homogeneous case). The solutions take into account varying parameters such as tumor size, tumor stiffness, shape of vibration source, lossy factor of the material, and vibration frequency. The problem of the lowest detectable change in stiffness is addressed using the theory, answering one of the most critical questions in this diagnostic technique. Some experiments were conducted to check the validity of the theory, and the results showed a good correspondence to the theoretical predictions. These studies provide basic understanding of the phenomena observed in the growing field of clinical Sonoelasticity imaging for tumor detection.