  The discovery of the topological insulators has fueled a surge of interests
in the topological phases in periodic systems. Topological insulators have bulk
energy gap and topologically protected gapless edge states. The edge states in
electronic systems have been detected by observing the transport properties the
Hgte quantum wells. The electromagnetic analogues of such electronic edge
states have been predicted and observed in photonic crystals, coupled
resonators and linear circuits. However, the edge state spectrums of the two
dimensional insulators and their electromagnetic analogues haven't been
directly measured and thus the gaplessness of the edge states hasn't been
experimentally confirmed. Here I show the classical strings are more convenient
choice to study the topological phases than the electromagnetic waves. I found
that classical strings with periodic densities can simulate a variety of
topological phases in condensed matter physics including two dimensional
topological insulators, three dimensional topological semimetals and weak
topological insulators. Because the eigenfrequencies and eigenfunctions of the
strings can be easily measured, not only the gapless edge state spectrum but
also the bulk topological invariant can now be directly observed. Further more
I show the topological phase transitions can be simulated by the strings. My
results show the richness of topological phases of mechanical waves. I
anticipate my work to be a starting point to study the topological properties
of mechanical waves. Even richer topological phases other than those have been
found in electronics systems can be explored when we use more general
mechanical waves e.g. waves in membranes with periodic densities. These phases
may provide guides to hunt novel topological properties in other branches of
science.
