  The resonant modes of two-dimensional elastic elliptical objects are studied
from a modal formalism by emphasizing the role of the symmetries of the
objects. More precisely, as the symmetry is broken in the transition from the
circular disc to the elliptical one, the splitting up of resonances and level
crossings are observed. From the mathematical point of view, this observation
can be explained by the broken invariance of the continuous symmetry group O(2)
associated with the circular disc. The elliptical disc is however invariant
under the finite group C2v and the resonances are classified and associated
with a given irreducible representation of this group. The main difficulty
arises in the application of the group theory in elastodynamics where the
vectorial formalism is used to express the physical quantities (elastic
displacement and stress) involved in the boundary conditions. However, this
method significantly simplifies the numerical treatment of the problem which is
uncoupled over the four irreducible representations of C2v. This provides a
full classification of the resonances. They are tagged and tracked as the
eccentricity of the elliptical disc increases. Then, the splitting up of
resonances, which occurs in the transition from the circular disc to the
elliptic one, is emphasized. The computation of displacement normal modes also
highlights the mode splittings. A physical interpretation of resonances in
terms of geometrical paths is provided.
