The salinity profile in a coastal sea-river system is the result of balancing of salt transport due to mixing of the water by tides and the freshwater discharge from the river. When for example during times of drought freshwater discharge suddenly declines, the salinity in the river may increase significantly and threaten drinking water supplies and agriculture. The time it takes for the salinity to reach the new equilibrium may be considerable. Thorough understanding of the adaptation process is essential for taking appropriate prevention measures. However, much about this adjustment process is still unknown, in particular regarding the interaction between the river and the adjacent sea. In this study, a simplified linear model is developed to describe a well-mixed river and the adjacent sea where the water flows radially away from the mouth of the river. The unique assumptions, most notably a time-independent intrusion length, allow for a unique analytical approach using the method of eigenfunction expansion. The smallest eigenvalues are found to define a time scale of the adjustment process. The eigenfunctions provide insight about how the salinity adjustment time varies within the system. The adjustment time does not only depend on the parameters that describe the new equilibrium, but also on the initial salinity. The eigenvalue time scales corresponding to the sea and to the river can then be used to determine the time scale of the coupled system. It is found that when the sea adjusts much faster than the river, the time scale of the river is leading the adjustment. When the eigenvalue time scale of the sea is similar to or larger than the eigenvalue time scale of the river, the adjustment time is significantly increased by the sea. When salinity is increasing, the sea restricts the inflow of salt in the river, while for decreasing salinity, the sea keeps transporting salt to the river by mixing which delays the adjustment. The model is applied to analyse a sudden decrease in freshwater discharge in the Rotterdam Waterway, which is part of the Dutch Rhine-Meuse delta. Results show that the salinity response is delayed compared to freshwater discharge, indicating that the adjustment time of the river indeed plays an important role. Further research should consider a more general dispersion relation or an extra vertical dimension to better describe estuaries that are not well-mixed. Using numerical methods there are many other possibilities to extend the model. The effect of tides on the presence and dynamics of the fresh water bulge in the sea is of specific interest.