To calculate the transmitted and reflected velocity waveform of elastic longitudinal wave with arbitrary function form propagating through nonlinear joint by Fourier series theory. Based on the discontinuous displacement model and the momentum balance equation at the wave front, the governing equation for the propagation of an elastic longitudinal wave through the nonlinear joint is developed, in which the joint deformation is described by the Barton-Bandis model. With assuming that the minimal positive period of stress waveform function remains unchanged as stress wave propagates across the nonlinear joint, the arbitrary Fourier series solutions of transmitted and reflected waves are obtained using the Fourier series method and the periodic extension method, and the Fourier series solutions are validated. Based on Fourier series solutions, the dependence of the amplitude and phase on the order number of the harmonic waves generated by the transmitted wave at the joint is analyzed. It is shown that the relationship between the amplitude and the order number of the harmonic waves follows a negative exponential law. The attenuation index of the amplitude is a quadratic function of the order number when the order number is less than 7, whereas the attenuation index of the amplitude is a linear function of the order number when the order number is less than 7. A linear relationship exists between the phase and the order number for harmonic waves of different orders.