Why do trains have to slow down when crossing in a tunnel?

Question

When two trains cross inside a tunnel, the velocity of the air around the train increases so the pressure decreases (Bernoulli's Principle). Using that logic, the windows are pushed outwards because the pressure outside is inferior then the pressure inside the train (which continues to be $P_{\rm atm}$, right?).

The problem is when two trains cross on a tunnel the relative velocity between them will be $v = v_{\rm train\,1} - v_{\rm train\,2}$. So the velocity will be inferior compared to the velocity of the train going by himself.

My logic must be wrong because I'm pretty sure trains only slow down when crossing the tunnel together, right?

Answer 1

I think you're confused. When two trains cross each other, the velocity one sees the other is (classically): $$ \vec{v}_{1,2} = \vec{v}_{2,t} - \vec{v}_{1,t}. $$ But they go in opposite directions, so the absolute value (i.e. speed) is simply the sum $$ v_{1,2}=v_{1,t}+v_{2,t}, $$ which is larger than both speeds in the tunnel reference frame (which is what you mean by "going by himself").

This speed $v_{1,2}$ is what matters for the Bernoulli principle, so if the trains don't slow down, the difference in pressure may derail them.