  General Relativity predicts that black holes do not possess an internal
structure and consequently cannot be excited. This leads to a specific
prediction about the waveform of gravitational waves, which they emit during a
binary black hole inspiral and to the vanishing of their Love numbers. However,
if astrophysical black holes do possess an internal structure, their Love
numbers would no longer vanish, and they could be excited during an inspiral by
the transfer of orbital energy. This would affect the orbital period and lead
to an observable imprint on the emitted gravitational waves waveform. The
effect is enhanced if one of the binary companions is resonantly excited. We
discuss the conditions for resonant excitation of a hypothetical internal
structure of black holes and calculate the phase change of the gravitational
waves waveform that is induced due to such resonant excitation during
intermediate- and extreme-mass-ratio inspirals. We then relate the phase change
to the electric quadrupolar Love number of the larger companion, which is
resonantly excited by its smaller companion. We discuss the statistical error
on measuring the Love number by LISA and show that, because of this phase
change, the statistical error is small even for small values of the Love
number. Our results provide a strong indication that the Love number could be
detected by LISA with remarkable accuracy, much higher than what can be
achieved via tidal deformation effects. Our results further indicate that
resonant excitation of the central black hole during an extreme- or
intermediate-mass-ratio inspirals is the most promising effect for putting
bounds on, or detecting, non-vanishing tidal Love numbers of black holes.
