  Thermal resonance, in which the temperature amplitude attains a maximum value
(peak) in response to an external exciting frequency source, is a phenomenon
pertinent to the presence of underdamped thermal oscillations and explicit
finite-speed for the thermal wave propagation. The present work investigates
the occurrence condition for thermal resonance phenomenon during the
electron-phonon interaction process in metals based on the hyperbolic
two-temperature model. First, a sufficient condition for underdamped electron
and lattice temperature oscillations is discussed by deriving a critical
frequency (a material characteristic). It is shown that the critical frequency
of thermal waves near room temperature, during electron-phonon interactions,
may be on the order of terahertz ($10-20$ THz for Cu and Au, i.e., lying within
the terahertz gap). It is found that whenever the natural frequency of metal
temperature exceeds this frequency threshold, the temperature oscillations are
of underdamped type. However, this condition is not necessary, since there is a
small frequency domain, below this threshold, in which the underdamped thermal
wave solution is available but not effective. Otherwise, the critical damping
and the overdamping conditions of the temperature waves are determined
numerically for a sample of pure metals. The thermal resonance conditions in
both electron and lattice temperatures are investigated. The occurrence of
resonance in both electron and lattice temperature is conditional on violating
two distinct critical values of frequencies. When the natural frequency of the
system becomes larger than these two critical values, an applied frequency
equal to such a natural frequency can drive both electron and lattice
temperatures to resonate together with different amplitudes and behaviors.
However, the electron temperature resonates earlier than the lattice
temperature.
