  The splitting of invariant manifolds of whiskered (hyperbolic) tori with two
frequencies in a nearly-integrable Hamiltonian system, whose hyperbolic part is
given by a pendulum, is studied. We consider a torus with a fast frequency
vector $\omega/\sqrt\varepsilon$, with $\omega=(1,\Omega)$ where the frequency
ratio $\Omega$ is a quadratic irrational number. Applying the
Poincar\'e-Melnikov method, we carry out a careful study of the dominant
harmonics of the Melnikov potential. This allows us to provide an asymptotic
estimate for the maximal splitting distance, and show the existence of
transverse homoclinic orbits to the whiskered tori with an asymptotic estimate
for the transversality of the splitting. Both estimates are exponentially small
in $\varepsilon$, with the functions in the exponents being periodic with
respect to $\ln\varepsilon$, and can be explicitly constructed from the
continued fraction of $\Omega$. In this way, we emphasize the strong dependence
of our results on the arithmetic properties of $\Omega$. In particular, for
quadratic ratios $\Omega$ with a 1-periodic or 2-periodic continued fraction
(called metallic and metallic-colored ratios respectively), we provide accurate
upper and lower bounds for the splitting. The estimate for the maximal
splitting distance is valid for all sufficiently small values of $\varepsilon$,
and the transversality can be established for a majority of values of
$\varepsilon$, excluding small intervals around some transition values where
changes in the dominance of the harmonics take place, and bifurcations could
occur.
