Rewrite the second equation as
In the neighborhood of the weak discontinuity, the quantities to the first order (such as ) reduces the above equation to
At this point, it is worth mentioning that in general, disturbances in gases are propagated with respect to the gas at the local sound speed. In other words, in the fixed frame, the disturbances are propagated at the speed  (the other possibility is  although it is of no interest here). If the gas is at rest , then the disturbance speed is . This is just a normal sound wave propagation. If however  is non-zero but a small quantity, then one find the correction for the disturbance propagation speed as  obtained using a Taylor series expansion, where  is a necessarily a positive constant (for ideal gas, , where  is the specific heat ratio). This means that the above equation can be written as
whose solution is
where  is a constant. This determines  implicitly in the neighborhood of the week discontinuity where  is small. This equation shows that at , , , but all higher-order derivatives are discontinuous. In the above equation, subtract  from the left-hand side and  from the right-hand side to obtain
which implies that  if  is a small quantity. It can be shown that the relation  not only holds for small , but throughout the rarefaction wave.