  The partial or complete confinement of waves in an open system is omnipresent
in nature and in wave-based materials and technology. Here, we theoretically
analyze and experimentally observe the formation of a trapped mode with perfect
mode conversion (TMPC) between flexural waves and longitudinal waves, by
achieving a quasi-bound state in the continuum (BIC) in an open elastic wave
system. The latter allows a quasi-BIC in a semi-infinite background plate when
Fano resonance hybridizes flexural and longitudinal waves and balances their
radiative decay rates. We demonstrate that when the Fabry-P\'erot resonance of
the longitudinal wave is realized simultaneously, the TMPC formed by the
elastic BIC approaches infinite quality factor. Furthermore, we show that
quasi-BIC can be tuned continuously to BIC through the critical frequency of
mode conversion, which offers the possibility of TMPC with an arbitrarily high
quality factor. Our reported concept and physical mechanism open new routes to
achieve perfect mode conversion with tunable high quality factor in elastic
systems.
