A one-dimensional exact solution for the advective, transient thermal response of river flow is presented for the case of linear, position-dependent volume flow rate. For reasonable restrictions appropriate to river flow it is shown that for the special quasi-linear case of nonlinear heat source functions, the same equations are produced as those obtained from the method of characteristics for linear source functions. This approach avoids heat source linearization with the associated degradation in accuracy and enables the use of an accurate numerical solution procedure for a wider range of advective flows. The proposed solution method was used to investigate the thermal response due to longitudinal variation of volume flow rate, river speed, width, and depth. ¿ 2000 American Geophysical Union