  Inspired by Barnard 68, a Bok globule, that undergoes stable oscillations, we
perform multi-phase hydrodynamic simulations to analyze the stability of Bok
globules. We show that a high-density soft molecular core, with an adiabatic
index $\gamma$ = 0.7 embedded in a warm isothermal diffuse gas, must have a
small density gradient to retain the stability. Despite being stable, the
molecular core can still collapse spontaneously as it will relax to develop a
sufficiently large density gradient after tens of oscillations, or a few $10^7$
years. However, during its relaxation, the core may abruptly collapse triggered
by the impingement of small-amplitude, long-wavelength ($\sim$ 6 $-$ 36 pc)
sound waves in the warm gas. This triggered collapse mechanism is similar to a
sonoluminescence phenomenon, where underwater ultrasounds can drive air bubble
coalescence. The collapse configuration is found to be different from both
inside-out and outside-in models of low-mass star formation; nonetheless the
mass flux is close to the prediction of the inside-out model. The condition and
the efficiency for this core collapse mechanism are identified. Generally
speaking, a broad-band resonance condition must be met, where the core
oscillation frequency and the wave frequency should match each other within a
factor of several. A consequence of our findings predicts the possibility of
propagating low-mass star formation, for which collapse of cores, within a mass
range short of one order of magnitude, takes place sequentially tracing the
wave front across a region of few tens of pc over $10^7$ years.
