  Recent experiments have pointed to the formation of a new state of matter,
the electron quadrupling condensate in Ba$_{1-x}$K$_x$Fe$_2$As$_2$ . The state
spontaneously breaks time-reversal symmetry and is sandwiched between two
critical points, separating it from the superconducting and normal metal
states. The adjacent two critical points make acoustic effects a promising tool
to study such states because of their sensitivity to symmetry-breaking
transitions. We report a theory of the acoustic effects of systems with an
electron quadrupling phase and new ultrasound velocity measurements of
Ba$_{1-x}$K$_x$Fe$_2$As$_2$ single crystals. The presented theory for the
electron quadrupling state gives the same type of singularities that are
observed in experiment.
