  We present Kepler photometry and ground based spectroscopy of KIC 4544587, a
short-period eccentric eclipsing binary system with self-excited pressure and
gravity modes, tidally excited modes, tidally influenced p modes, and rapid
apsidal motion of 182 y per cycle. The primary and secondary components of KIC
4544587 reside within the delta Scuti and gamma Dor instability region of the
Hurtzsprung-Russell diagram, respectively. By applying the binary modelling
software PHOEBE to prewhitened Kepler photometric data and radial velocity data
obtained using the William Herschel Telescope and 4-m Mayall telescope at KPNO,
the fundamental parameters of this important system have been determined,
including the stellar masses, 1.98+/-0.07 MSun and 1.60+/-0.06 MSun, and radii,
1.76+/-0.03 RSun and 1.42+/-0.02 RSun, for the primary and secondary
components, respectively. Frequency analysis of the residual data revealed 31
modes, 14 in the gravity mode region and 17 in the pressure mode region. Of the
14 gravity modes 8 are orbital harmonics: a signature of tidal resonance. While
the measured amplitude of these modes may be partially attributed to residual
signal from binary model subtraction, we demonstrate through consideration of
the folded light curve that these frequencies do in fact correspond to tidally
excited pulsations. Furthermore, we present an echelle diagram of the pressure
mode frequency region (modulo the orbital frequency) and demonstrate that the
tides are also influencing the p modes. A first look at asteroseismology hints
that the secondary component is responsible for the p modes, which is contrary
to our expectation that the hotter star should pulsate in higher radial
overtone, higher frequency p modes.
