It may seem disturbing that such a simple, passive, physically
rigorous simulation of a mass-spring oscillator should have to make
use of state variables which grow without bound for the limiting cases
of simple harmonic motion at frequencies zero and half the sampling
rate. This is obviously a valid concern in practice as well.
However, it is easy to show that this only happens at dc and ,
and that there is a true degeneracy at these frequencies, even in the
physics. For all frequencies in the audio range (e.g., for typical
sampling rates), such state variable growth cannot occur. Let's take
closer look at this phenomenon, first from a signal processing point
of view, and second from a physical point of view.
A signal processing perspective.
Going back to the poles of the mass-spring system in Eq. (N.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.
A physics perspective.
In the physical system, dc and infinite frequency are in fact strange
cases. In the case of dc, for example, a nonzero constant force
implies that the mass is under constant acceleration. It is
therefore the case that its velocity is linearly growing. Our
simulation predicts this, since, using
Eq. (N.43) and Eq. (N.42),
The dc term is therefore accompanied by a linearly growing
term
in the physical mass velocity. It is therefore
unavoidable that we have some means of producing an unbounded,
linearly growing output variable.