The axisymmetric longitudinal waves propagating in the long infinite cylindrical rod composed of material and structural constants, combinedly called as general incompressible materials, are derived using the perturbation reduction method as the far-field equation in the form of KdV equation in Dai and Huo (2002). In this work, the F expansion method is applied to the far-field equation and the properties of longitudinal strain waves travelling in the cylindrical rod are studied. The behaviour of the strain waves is analysed with the material and structural constants. To support the study, the graphs are drawn in the two and three dimensional surfaces. The obtained longitudinal strain waves are in the form of Jacobi elliptic function and necessary condition for the existence of each wave is provided. © 2020 International Association for Mathematics and Computers in Simulation (IMACS)