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