Re: harmonics, anharmonics (was: Midterm Question)

[John Denker <jsd@MONMOUTH.COM>]

At 10:12 PM 7/6/99 -0500, Doug Craigen wrote:
> You have a child on a swing of some length
> and therefore work out its period.  We can pump the swing by pushing on
> the child with that period, but we can also push every second time they
> come back (twice the period half the frequency), every third time etc.
> So I've asked them to identify those frequencies.

OK, as you say, that's a trivial mathematical excercise.

We have
   drive_frequency(N) = drive_frequency(1) / N                  (1)
other things being equal.

> The thing I've been pondering is that as compared to other "sympathetic
> vibration" / "resonance" / ... type examples I can think of, in this
> case (or the comparable spring question) the system responds at its own
> frequency which is a harmonic of the frequency at which you pump it....
> So it *appears* to be multiplication of a frequency.  (i.e. if an
> optical analog existed it would amount to something such as putting 800
> nm in and getting 400 nm out.)

Optical frequency doubling definitely exists.  Have you seen those green
diode lasers?  They are really infrared lasers, doubled.

The analogy to the swingset is complicated.

We have
   output_frequency(N) = output_frequency(1) * N                        (2)

> However, is it really accurate to think of this as multiplying the
> pumping frequency?

Pretty much yes.  (Piano strings are a different story; see below.)

There's a theorem (Floquet's theorem, Bloch's theorem) that says any
periodic signal can be composed using a Fourier series. In each case
(swingset and optical doubling crystal) you are driving the system with a
periodic signal.

Also in each case there is a nonlinearity:
 *) The force on the swingset is a highly nonlinear function of whether
your hand is touching or not.
 *) In the optical doubling crystal, some of the atoms are held in the
crystal by highly anharmonic potentials.

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.

> (since it is increasing and
> decreasing in the amplitude of its oscillations),

I wouldn't have said that.  You have to consider one cycle to be the period
between one *push* and the next.  In this sense, the amplitude is
unchanging from cycle to cycle.

Within each cycle, there are peaks of N different heights, because of the
way the different harmonics combine.

> and the pumping itself
> is periodic impulses which should be fourier decomposed into a set of
> frequencies, one of which is the resonant frequency of the swing.

I wouldn't have said that either.  The swing will respond at the frequency
of the push, no matter whether that is on-resonance or off-resonance.

It is true that that *magnitude* of the response will be bigger if the
swing is resonant at the frequency of the push (or a harmonic thereof), but
it *never* happens that you apply a periodic push at an off-resonance
frequency and get a response at the resonant frequency.  Floquet's theorem.

=============================

To answer a question that you didn't ask:

A rather different analysis applies to piano strings.

For one thing, the excitation is a blow from a hammer.  It is totally
nonperiodic, so Floquet's theorem does not apply.

If you actually look at the waveform of a piano string, you see that it is
*not* periodic.  Even if we have an idealized piano string with negligible
damping, the motion is not periodic.  If you analyze the signal with a full
Fourier transform (as opposed to series) you find that the partials occur
at distinctly *anharmonic* frequencies.  The following formula is a
simplified model, but it gives you the idea:
    output_frequency(N) ~ output_frequency(1) * N^1.005         (3)

(Half a percent in the exponent may not seem like a lot of anharmonicity,
until you realize that a 6 percent change in frequency is an entire
half-step on the musical scale.)

The usual superficial analysis of waves on a string derives the wave
equation from two bits of physics:  the tension and the mass.  This gives a
nondispersive wave equation;  the speed of waves is independent of
frequency, so all standing waves occur at the harmonic frequencies give by
expression (2) above.

The real physics invovles not just tension and mass, but also the
*stiffness* of the steel piano-wire.  This results in a speed of waves that
increases with frequency.  Therefore the higher partials occur at
anharmonic frequencies, as modelled by expression (3).

Cheers --- jsd