When a long line of initially stationary and unstressed touching particles is hit on one end by another particle, dynamical self-organized impulsive waves will be generated and propagating through the granular chain. In this paper we develop a continuous function to represent the initial impulse wave. Then the partial differential wave equations for chains of cylindrical granules, which can take such function as one of its solutions, are constructed and exactly coincided with the wave equation obtained by using Taylor series expansion. The properties of the impulse wave propagating, such as the attenuation of the wave and the forward momentum transfer are studied in detail based on our theoretical model, and the analytical solutions are supported by numerical simulations.