Abstract: By utilizing the eigen function expansion method and Laplace transformation, under gravitational action, the wave equation of vertical vibration of one-dimensional string which one end is fixed and the other end oscillates harmonically with a small amplitude is solved strictly. The explicit solution is acquired, and the resonant eigen frequencies that born the standing-waves of string (or called elastic rope) is got. It is also represented that the eigen frequencies and the standing-waves antinodes are all independent to gravity action. The linear changing relationship between the standing-waves loop numbers and the resonant frequency is observed and computed experimentally. Using the wave velocity expression in the wave equation derived theoretically together with the tension force and the length of elastic rope which measured while vibrating, the line mass density of rope is figured out. The result agrees well to the experimental value. From another angle, these works are well verified the correctness of wave equation of vertical vibration of one-dimensional string which elucidated by theory. 

Key words:  gravitational action, one-dimensional horizontal string, eigen function expansion method, Laplace transformation, standing-wave