Re: harmonics, anharmonics

[John Denker <jsd@MONMOUTH.COM>]

At 08:06 AM 7/8/99 -0500, JACK L. URETSKY  wrote
>        I think that "nonlinearity" is an incorrect term for what you
> are describing.  What we have is a driven system of amplitude y, described
> by the equation:
>        y'' + (p^2)y = f(t).
>
>        You are discussing the forcing function f(t), as I understand it.
> The system is quite linear in y.  This is different from frequency doubling
> in nonlinear optics, which occurs because the system is non-linear in the
> amplitude y.

I discussed two cases:  the swingset and the optical crystal.  We agree
that the optical crystal has a response that is nonlinear in y.  The
swingset is nonlinear for large amplitudes too, but we agree that in the
case in question, this nonlinearity is not the interesting nonlinearity.

>        If f(t) is not a "pure" frequency function like cos(wt), then it
> will have many frequency components.

Right.  That's what I said.

> The case of pushing the swing, for
> example, can be modeled by a square wave function with a single period
> for the square wave.

There are many ways of pushing the swing.  One could imagine a linear
coupling, such as a very long spring that provides a driving force
essentially independent of the position of the pendulum.

But as I said, there is an important nonlinearity in the coupling in the
case in question.

> The Fourier decomposition of the square wave will
> then have a spectrum of pure frequencies, as in your example.

Right.

> But as long
> as f(t) is not some function of y (or proportional to y) the system is, by
> definition, "linear".

It depends on what you include in the definition of "system".  I chose to
include the physics of the driving mechanism in my system and explicitly
identified the point in my system where the nonlinearity occured.  If you
wish to draw a boundary that excludes this nonlinearity from your "system",
then what's left is linear.  You draw your boundaries, I'll draw mine.

Cheers --- jsd