Before revision:  We use the problems of two ideal fluid waveguides as examples, i.e., one with a constant sound speed in shallow water and one with a Munk sound speed profile in the deep ocean. The correctness of the  discrete PE model using the CSM to solve a simple underwater acoustic propagation problem is verified.
After revision:  We use the problems of two ideal fluid waveguides as examples, the discrete PE model using the CSM to solve a simple underwater acoustic propagation problem is verified.