We have developed linearized complex eikonal equations for orthorhombic media and the analytic solution for homogeneous acoustic media. The linearized formulations are developed from the highly nonlinear complex eikonal equation using a perturbation analysis. The analytic solution is obtained by expanding the complex traveltime in a Taylor series and constructing its coefficients. A critical factor for deriving this analytic expression is to apply the complex point-source method to the analytic expression for the background anisotropic medium. The analytic expressions show that the complex traveltime is a function of the anisotropic parameters, wave velocity, and spatial coordinates. We give the numerical results of the complex traveltime for different anisotropic parameters and initial beam widths. Furthermore, numerical examples are investigated for various coefficients as a function of the coordinates.