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In this paper, we propose a new model of the Kuramoto type to study nonlinear wave generation and propagation on a circle network with excitable node dynamics, and reveal certain common features of traveling wave solutions, which are independent of local dynamics but strongly related to the diffusive coupling of neighboring nodes. In view of the stability and size of the basin of attraction, regular nonlinear waves and, in the case of a large number of nodes, circulating pulses are the most important ones. The period T and the functional expression for regular nonlinear waves are computed analytically, which matches well with the numerical result. A new type of solution, the special nonlinear wave existing only for the discrete node dynamics, is studied and compared with the regular solution. [ABSTRACT FROM AUTHOR]