The boundary condition.

  Physically, it is also clear that we will need to know the entropy that flows into the pipe at the left end, giving a boundary condition

 

s(0,t) = s_i(t),

where s_i is some given function of time. But the entropy that flows out of the pipe at the left end is obviously no longer of any concern, so we do not need a boundary condition at the left end of the bar. Note that having only one boundary condition is consistent with the fact that the convection equation (6) is only first order in the x-direction