We consider the numerical approximations of the complex amplitude in a coupled bay-river system in this work. One half-circumference is introduced as the artificial boundary in the open sea, and one segment is introduced as the artificial boundary in the river if the river is semi-infinite. On the artificial boundary a sequence of high-order artificial boundary conditions are proposed. Then the original problem is solved in a finite computational domain, which is equivalent to a variational problem. The numerical approximations for the original problem are obtained by solving the variational problem with the finite element method. The numerical examples show that the artificial boundary conditions given in this work are very effective.