  We report on broadband-to-narrowband elastic wave filtering resulting from
time-periodic modulation of the stiffness of a one-dimensional elastic
waveguide. Time modulation produces flat dispersion bands at frequencies that
are multiple integers of half the modulation frequency. These flat bands lead
to the selective reflection of a broadband incident wave at the interface
between a non-modulated medium, and one with time-modulated stiffness
properties. This results from the vanishing group velocity at the flat band
frequencies, which prevents their propagation into the modulated domain. Thus,
the considered modulated waveguide is understood as a single port system, in
which a broadband incident wave (input) results in a narrowband reflected wave
(output) at a frequency defined by modulation. The appearance of the flat bands
for a time-modulated waveguide is here illustrated analytically and through
numerical simulations. The filtering characteristics of a
non-modulate/modulated interface are observed experimentally by implementing a
square-wave modulation scheme that employs an array of piezoelectric patches
bonded to an elastic waveguide subject to transverse motion. The patches are
shunted through a negative electrical capaticance that, when connected,
implements a stiffness reduction for the resulting electromechanical waveguide.
Switching the capacitance on and off effectively modulates the stiffness of the
waveguide, and illustrates the filtering characteristics associated with
time-modulation of the equivalent elastic properties. We envision that a
similar approach could be extended to investigate other properties of
time-modulated elastic metamaterials, such as non-reciprocity and one-way
filtering of elastic waves.
