Re: harmonics, anharmonics (was: Midterm Question)

["JACK L. URETSKY (C)1998; HEP DIVISION, ARGONNE NATIONAL LAB ARGONNE, IL 60439" <JLU@HEP.ANL.GOV>]

Hi John Denker-
        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.
        If f(t) is not a "pure" frequency function like cos(wt), then it
will have many frequency components.  The case of pushing the swing, for
example, can be modeled by a square wave function with a single period
for the square wave.  The Fourier decomposition of the square wave will
then have a spectrum of pure frequencies, as in your example.  But as long
as f(t) is not some function of y (or proportional to y) the system is, by
definition, "linear".
                        Regards,
                                Jack

***************************************************************
It is the nonlinearity that creates the frequency multiplication effect.
To understand this, take your favorite nonlinear function (e.g.
exponential) and consider exp(sin(x)).  Expand the nonlinear function as a
Taylor series.  There will be terms in sin**2(x), sin**3(x), et cetera.
Then use elementary trigonometry identities to express sin**2(x) in terms
of sin(2x) et cetera.

> It is multiplying the frequency that I relate to -
> the frequency at which I apply impulses

Right.

> however the swing system would
> have to be described by multiple frequencies

Right.  There are many terms in the Fourier series.
***********************************


        "I scored the next great triumph for science myself,
to wit, how the milk gets into the cow.  Both of us
had marveled over that mystery a long time.  We had
followed the cows around for years - that is, in the
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                Mark Twain, Extract from Eve's
                Autobiography