In this work,Green-Naghdi (GN) equations with general weight functions were derived in a simple way. A wave-absorbing beach was also considered in the general GN equations. A numerical solution for a level higher than 4 was not feasible in the past with the original GN equations. The GN equations for shallow water waves were simplified here, which make the application of high level (higher than 4) equations feasible. The linear dispersion relationships of the first seven levels were presented. The accuracy of dispersion relationships increased as the level increased. Level 7 GN equations are capable of simulating waves out to wave number times depth . Numerical simulation of nonlinear water waves was performed by use of Level 5 and 7 GN equations, which will be presented in the next paper. In this work, Green-Naghdi (GN) equations with derived weight in a simple way. A wave-absorbing beach was also considered in the general GN equations. A numerical solution for a level higher than 4 was not feasible in the past with the original GN equations. The GN dispersion for shallow water waves were illustrated here, which makes the application of high level (higher than 4) equations feasible. The linear dispersion relationships of the first seven levels were presented. The accuracy of dispersion relationships Increased as the level increased. Level 7 GN equations are capable of simulating waves out to wave number times depth. Numerical simulation of nonlinear water waves was performed by using Level 5 and 7 GN equations, which will be presented in the next paper.