- The fluvial response time dictates the duration of fluvial channel adjustment in response to changing climatic and tectonic conditions. However, when these conditions vary continuously, the channel cannot equilibrate and the response time is not well defined. Here I develop an analytical solution to the linear stream power model of fluvial incision that predicts the channel topography as a function of time-dependent climatic and tectonic conditions. From this solution, a general definition of the fluvial response time emerges: the duration over which the tectonic history needs to be known to evaluate channel topography. This new definition is used in linear inversion schemes for inferring climatic or tectonic histories from river long profiles. The analytic solution further reveals that high-frequency climatic oscillations, such as Milankovitch cycles, are not expected to leave significant fingerprints on the long profiles of fluvially incised detachment-limited rivers.