A novel method is presented to exactly solve the progressive and standing waves for N–periodic stepwise strings. A new two–way state–flow graph model is introduced to describe the relations of waves propagating in a periodic system. Based on the model, general solutions of the wave response in the periodic string are derived by a topology scheme. Thus, the harmonic driving waves in the string with one end fixed and one end free are determined and represented in a concise closed form in terms of the properties of a single period. Analytical standing waves of the string with free–free, free–fixed and fixed–fixed constraints are also presented. Finally, examples of a three–period string with two segments in each period and a special uniform string are examined to demonstrate the present method.