We present radio observations of two well-known double–double radio galaxies, J0041+3224 and J1835+6204, at frequencies ranging from 150 to 8460 MHz, using both the Giant Metrewave Radio Telescope and the Very Large Array. These observations, over a large radio frequency range, enable us to determine the spectra of the inner and outer lobes. Our detailed spectral ageing analysis of their inner and outer lobes demonstrates that the outer doubles of double–double radio galaxies are created by the previous cycle of activity, while the inner doubles are due to the present cycle of activity. The (core subtracted) spectra of the inner doubles of both sources are power laws over a large frequency range. We found that the duration of the quiescent phase of J0041+3224 is between 4 and 28 per cent of the active phase of the previous activity. The outer north-western lobe of J1835+6204 has a compact hotspot and the regions of both the outer hotspots have close to power-law (rather than curved) spectra, which indicates that the outer lobes are still fed by jet material ejected in the previous episode just before the central engine stopped powering the jet. We estimate that the duration of the quiescent phase of J1835+6204 is ≲5 per cent of the duration of the active phase of the previous activity. Therefore, we conclude that the duration of the quiescent phase can be as short as a few per cent of the active phase in radio galaxies of this type.