In this study the steady-state dynamic of a linear, homogeneous, un-damped string, coupled with a locally connected spring-dashpot system is analytically investigated. Both ends of the string are assumed to be excited with identical and synchronous harmonic motion. It is shown that the damper introduces mode complexity and leads to frequency shift between the peak amplitudes in different locations of the string. Also it causes phase variations which indicates mode complexity domain. In this study, it is shown that there are different combinations of spring and damper constants in which the mode complexity attains its maximum level. Surprisingly, the combination is unique in each given excitation frequency ratio. In this situation, the damping constant is bounded in a specified range but, the spring constant is increased as the excitation frequency ratio is increased. In such case, all vibration normal modes of the string are completely destroyed and, in turn, traveling waves are formed. Also it is shown that the damping constant which leads to the maximum frequency shift, is not necessarily equal to the one that introduces the maximum mode complexity.