To differ from the usually applied Eulerian method for describing the motion of fluid, the governing equations complete in the Lagrangian form for describing three-dimensional progressive and short-crested waves system are derived in this paper. A systematical ordering expansion by an appropriate perturbation approximation is developed, and the exactly satisfactory solutions in a form of functional, up to third-order progressive waves and up to second-order short-crested waves, are obtained. The kinematic properties of the waves, including the surface profile, pressure, the paths of fluid particles, and the mass transport velocity, are then described directly. The obtained solution for the short-crested waves system is successfully verified by reducing to two special cases, one is the two-dimensional simple progressive waves, and the other is the two-dimensional standing waves. Also, the analytical results are compared with experimental data including the surface profiles, the pressures and the paths of fluid particles for validation.

Three-dimensional short-crested waves in Lagrangian form was already solved by Wang(2007). By employing the technique of perturbation analysis, the solution for the entire wave filed was obtained and the results are verified to be correct to second-order. The period of the trajectory of fluid particle in short-crested wave field was manifested in Lagrangian form. Consequently, all the characteristics of the flow field can be vividly described including the moving trajectory of fluid particle. To distinguish two different ways that short-crested waves might take place, Wang(2007)¡¦s results were extended to perturbation¡¦s third-order. The mechanism of resonance phenomenon is then clearly explained. In this study, the analytical results for the three-dimensional short-crested wave field correct to third-order were explicitly derived. The fluid particle with different initial positions or different phases has different moving trajectories. Besides, the period of the trajectory of fluid particle varies with different water depths. These are obviously revealed in our perturbation solutions. The three-dimensional short-crested wave system is successfully verified by reducing to two special cases, two-dimensional progressive waves and standing waves. Also, the analytical results were compared with experimental data including the surface profiles, the pressures, and the paths of fluid particles for validation. Furthermore, the mechanism of resonance phenomenon and the property of angular frequency were explained. Thus, the exactness and generality of the results are firm certified.