Going back to the poles of the mass-spring system in Eq.(F.39), we see that, as the imaginary part of the two poles, , approach zero, they come together at to create a repeated pole. The same thing happens at since both poles go to ``the point at infinity''.
It is a well known fact from linear systems theory that two poles at the same point in the plane can correspond to an impulse-response component of the form , in addition to the component produced by a single pole at . In the discrete-time case, a double pole at can give rise to an impulse-response component of the form . This is the fundamental source of the linearly growing internal states of the wave digital sine oscillator at dc and . It is interesting to note, however, that such modes are always unobservable at any physical output such as the mass force or spring force that is not actually linearly growing.
Physical Perspective on Repeated Poles in the Mass-Spring System
Linearly Growing State Variables in WD Mass-Spring Oscillator