In Water Quality Modeling, there is a basic differential equation describing concentration $s(x,t)$ in an advective nondispersive stream. Its formula and solution can be seen in Fig.1.
I tried to examine this solution by caculating reversely, that is: $$\frac{\partial s}{\partial x}=\frac{W(t-t^*)}{Q}\cdot (-\frac{K}{U})\cdot exp(-\frac{Kx}{U})$$
Then the right part of equation became:$$-U\frac{\partial s}{\partial x}-Ks=...=0$$
This is in conflict with the left part ($\frac{\partial s}{\partial t}$), which is not zero.
Could someone tell me what's wrong?
Terrific catch. Note that the text (Fischer?) states "only the waste load is assumed to be varying in time." Presumably, slowly enough that the downstream concentration can be regarded as being in a steady state and that, as you surmised, $\frac{\partial s}{\partial t} =0$.