The exhaust systems such as used for automotive internal combustion engines can be replaced by the four terminal constants (A, B, C and D) of an electrical equivalent circuit as their whole silencing systems. The four terminal matrices for the cases of straight tubes and branched ones in the typical tube systems are derived by considering the attenuation constant δ of plane waves in a tube. It also includes the four terminal matrices of the plane waves propagating without any losses in a tube. The general equation for insertion loss of silencing systems having expansion chamber type muffler are derived by these four terminal matrices. Calculated results of the insertion loss by these equation are compared with the experimental results of pure sound tests, and they are found to be in good agreement. From these results, it is confirmed that the insertion loss in the resonance frequency ranges of a silencing system is numerically calculated by the four terminal matrices considering the attenuation constant δ of plane waves propagating in a tube with dissipation caused by wall friction.