Recently, the ray tracing method has been used to derive the non-singular cylindrical invisibility cloaks for out-of-plane shear waves, which is impossible via the transformation method directly owing to the singular push-forward mapping. In this paper, the method is adopted to design a kind of non-singular acoustic cloak. Based on Hamilton's equations of motion, eikonal equation and pre-designed ray equations, we derive several constraint equations for bulk modulus and density tensor. On the premise that the perfect matching conditions are satisfied, a series of non-singular physical profiles can be obtained by arranging the singular terms reasonably. The physical profiles derived by the ray tracing method will degenerate to the transformation-based solutions when taking the transport equation into consideration. This illuminates the essence of the newly designed cloaks that they are actually the so-called eikonal cloaks that can accurately control the paths of energy flux but with small disturbance in energy distribution along the paths. The nearperfect invisible performance has been demonstrated by the numerical ray tracing results and the pressure distribution snapshots. Finally, a kind of reduced cloak is conceived, and the good invisible performance has been measured quantitatively by the normalized scattering width.