The lumped driving-point admittance of the mass-spring system can be defined in terms of wave variables in a manner analogous to distributed systems.
An applied force can be written as
, where is viewed as a force wave traveling into the admittance and
is viewed as a force wave traveling out of the admittance.
In a similar way, we can define
.
In distributed systems, waves have a finite propagation speed and force and velocity wave components can be related by a wave impedance, which is defined by the parameters of the medium of travel.
In lumped systems, however, traveling waves may only be understood to travel instantaneously and there is no particular medium. But an arbitrary reference impedance, , can still be defined such that:
Making this change of variables, the velocity transfer function can be written
Using the bilinear transform to convert from a continuous-time system to a discrete-time system, we obtain
where the continuous-time impulse function mapping has been bandlimited as
and the coefficients are defined as
Because the reference impedance is arbitrary, it is possible to choose a value such that the coefficient , as
In this way, a unit of delay can be factored out of the second-order transfer function, such that
where
with given by
It is then possible to rewrite in terms of , eliminating , as