  Scalar induced gravitational waves contribute to the cosmological
gravitational wave background. They can be related to the primordial density
power spectrum produced towards the end of inflation and therefore are a
convenient new tool to constrain models of inflation. These waves are sourced
by terms quadratic in perturbations and hence appear at second order in
cosmological perturbation theory. While the focus of research so far was on
purely scalar source terms we also study the effect of including first order
tensor perturbations as an additional source. This gives rise to two additional
source terms: a term quadratic in the tensor perturbations and a cross term
involving mixed scalar and tensor perturbations. We present full analytical
expressions for the spectral density of these new source terms and discuss
their general behaviour. To illustrate the generation mechanism we study two
toy models containing a peak on small scales. For these models we show that the
scalar-tensor contribution becomes non-negligible compared to the scalar-scalar
contribution on smaller scales. We also consider implications for future
gravitational wave surveys.
