^{51}If you would like to prove this to yourself, you may do so by substituting path length (L), fluid velocity (v), and sound
velocity (c) for the times in the flow formula. Use t_{up} = L _
c−v and t_{down} = L _
c+v as your substitutions, then
algebraically reduce the flow formula until you find that all the c terms cancel. Your final result should be
Q = 2kv
L .