A typical equation to describe transport phenomena is the telegrapher's equation: $$ \big[\partial_t^2 + \frac{1}{\tau}\partial_t - A^2 \partial_x^2\big]P(x,t)=0.$$ In the physics literature (https://arxiv.org/abs/cond-mat/0606116) it is often stated that at short times, $t\ll \tau$, the equation is dominated by its wave-like part: $$ \partial_t^2 P(x,t) \approx A^2 \partial_x^2 P(x,t),$$ whereas at long times, $t \gg \tau$, the equation is dominated by its diffusive part: $$ \partial_t P(x,t) \approx \tau A^2 \partial_x^2 P(x,t) .$$ Is there a means by which I nondimensionalize time to better understand these limits? Or is there some other method? I don't understand why we can neglect a derivative with respect to a variable just because the variable itself is small, although it is clear this procedure gives the "right" answer.
Thanks!