  The oscillatory dynamics of nanoelectromechanical systems (NEMS) is at the
heart of many emerging applications in nanotechnology. For common NEMS, such as
beams and strings, the oscillatory dynamics is formulated using a
dissipationless wave equation derived from elasticity. Under a harmonic ansatz,
the wave equation gives an undamped free vibration equation; solving this
equation with the proper boundary conditions provides the undamped
eigenfunctions with the familiar standing wave patterns. Any harmonically
driven solution is expressible in terms of these undamped eigenfunctions. Here,
we show that this formalism becomes inconvenient as dissipation increases. To
this end, we experimentally map out the position- and frequency-dependent
oscillatory motion of a NEMS string resonator driven linearly by a
non-symmetric force on one end at different dissipation limits. At low
dissipation (high Q factor), we observe sharp resonances with standing wave
patterns that closely match the eigenfunctions of an undamped string. With a
slight increase in dissipation, the standing wave patterns become lost and
waves begin to propagate along the nanostructure. At large dissipation (low Q
factor), these propagating waves become strongly attenuated and display little,
if any, resemblance to the undamped string eigenfunctions. A more efficient and
intuitive description of the oscillatory dynamics of a NEMS resonator can be
obtained by superposition of waves propagating along the nanostructure.
