  Local density-of-states (LDOS) is a fundamental spectral property that plays
a central role in various physical phenomena such as wave-matter interactions.
Here, we report on the direct measurement of the LDOS of acoustic systems and
derive from which the fractional topological number in an acoustic
Su-Schrieffer-Heeger system. The acoustic LDOS is quantified here with a
state-of-the-art technique through the measurement of the volume flow rate and
the acoustic pressure with a local excitation-probe configuration. Based on
this method, we study the acoustic Purcell effect and establish experimentally
the important relation between the near-field LDOS and the far-field acoustic
emission power. Moreover, we detect the LDOS in the one-dimensional acoustic
Su-Schrieffer-Heeger model and observe the fractional topological number of the
system. Our work unveils the important role of the LDOS in acoustic phenomena
and paves the way toward characterizing and tailoring the LDOS in topological
systems.
