The propagations of one-dimensional nonlinear acoustical waves are studied numerically and experimentally in this paper. The finite element method (FEM) is used to simulate the propagations of nonlinear acoustic waves. The FEM equation of one-dimensional nonlinear acoustic wave is derived according to the theory of nonlinear acoustics. A three-dimensional matrix appearing in the nonlinear FEM equation represents the nonlinear part of the nonlinear acoustic equation and indicates the complex propagation characteristics of nonlinear acoustic waves. However, there is no corresponding matrix in the linear FEM equation. The matrix correlates with the nonlinear properties of propagating waves such as wave distortion, high order harmonic wave generation and transformation of energy from basic frequency to high order harmonic frequency, etc. Then, an FEM program is coded to compute the propagations of the one-dimensional nonlinear acoustic waves. The results show that the nonlinear acoustic waves are distorted obviously during propagation. After fast Fourier transform processing the original wave signal, the basic frequency signals and high order harmonic signals both appear in the frequency-region signals. To prove the correctness of the FEM results, nonlinear acoustic experiments in water are carried out under different conditions. In the first experiment, the distance between the transmitting and receiving transducers is kept unchanged, but the transmitting transducer is excited with different energies. So with propagation distance fixed, the influences of different exciting energies on the nonlinear properties of acoustic waves are obtained from this experiment. In the second experiment, with the exciting energy fixed, the influences of different propagation distances on the nonlinear properties of acoustic waves are obtained by changing the distance between the transmitting and receiving transducers. Then the numerical results and the experimental results are compared and analyzed carefully. The result shows that the waveforms and the spectra of simulated nonlinear waves are in good agreement with those of experimental signals. These results prove the correctness of the proposed numerical method. It is also noticed that the propagation properties of basic frequency wave and the second order harmonic waves are different. The amplitude of basic frequency wave decreases gradually, but the amplitude of second order harmonic wave first increases and then decreases after propagating some distance. The amplitude of the second harmonic wave changes with propagation distance and energy of the input source amplitude. The relationship between the amplitude of second harmonic wave and the propagation distance is numerically fitted. We find a fitting equation of the relation between high order harmonic acoustic wave and propagation distance, which also brings clear physical meaning for the high order harmonic waves. Finally, the properties of nonlinear acoustic wave propagation in solid are preliminarily discussed. This study provides theoretical and experimental evidence for the nonlinear acoustic wave traveling in liquid.