  We calculate the flux of internal gravity waves (IGWs) generated by turbulent
convection in stars. We solve for the IGW eigenfunctions analytically near the
radiative-convective interface in a local, Boussinesq, and cartesian domain. We
consider both discontinuous and smooth transitions between the radiative and
convective regions and derive Green's functions to solve for the IGWs in the
radiative region. We find that if the radiative-convective transition is
smooth, the IGW flux depends on the exact form of the buoyancy frequency near
the interface. IGW excitation is most efficient for very smooth interfaces,
which gives an upper bound on the IGW flux of ~ F_conv (d/H), where F_conv is
the flux carried by the convective motions, d is the width of the transition
region, and H is the pressure scale height. This can be much larger than the
standard result in the literature for a discontinuous radiative-convective
transition, which gives a wave flux ~ F_conv M$, where M is the convective Mach
number. However, in the smooth transition case, the most efficiently excited
perturbations will break in the radiative zone. The flux of IGWs which do not
break and are able to propagate in the radiative region is at most F_conv
M^(5/8) (d/H)^(3/8), larger than the discontinuous transition result by
(MH/d)^(-3/8). The transition region in the Sun is smooth for the
energy-bearing waves; as a result, we predict that the IGW flux is a few to
five times larger than previous estimates. We discuss the implications of our
results for several astrophysical applications, including IGW driven mass loss
and the detectability of convectively excited IGWs in main sequence stars.
