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Re: definition of "wave"



At 12:35 AM 1/27/00 -0500, David Bowman wrote:

My (maybe idiosyncratic) definition of a wave is an oscillatory behavior
of a *field* such that the oscillations occur in the value of the field
(or its components) about a mean value in *both* space *and* time. Thus
at each fixed point in space the field oscillates in time about its mean
value, and at each fixed instant of time the field oscillates in space
about its mean value.

That definition has many strengths and a few weaknesses.

1) A favorable example is a big block of metal subjected to alternate
heating and cooling on one face. The internal temperature will be a
function of space and time, and it will oscillate as a function of time,
but it will never oscillate in space. So the proposed definition
successfully excludes this non-wavelike situation.

2a) On the other hand, a legitimate wave need not oscillate. Consider a
long string under tension, with absorbers at each end. Suppose I pluck
it. The result is a little wave packet with an everywhere-upward
displacement that runs along. Most people would consider it a wave.

2b) You can consider example (2a) as a superposition of oscillatory waves,
but there are solitons (solitary waves) which *cannot* be considered
superpositions. A soliton in a trough of water is stable if it has a
rising edge; the corresponding falling-edge waveform is unstable.

... it
would consider purely evanescent "waves" (i.e. oscillations characterized
by purely imaginary wave numbers but real frequencies) not as waves since
they do not oscillate in space (they, rather, only decay/grow in space)
at a given instant of time.

They don't "only" decay... A typical evanescent wave (such as in a
waveguide beyond cutoff, or when light is totally internally reflected from
the side of an aquarium) oscillates in space and time for many cycles while
it is decaying.

So the proposed definition _includes_ evanescent waves, which suits me fine.

In order to count as a wave in my book the
oscillation would need to have a nonzero real part of the frequency *and*
a nonzero real part of the wave number/vector, or be some sort of
superposition of multiple components made of such things.

Are you assuming linearity? There are lots of nonlinear waves. Without
linearity it's tricky to talk about superposition.

If the oscillatory function is only a function of time (such as the
position of the mass center for a harmonic oscillator or the AC voltage
on the terminals of an electrical outlet) but is not a field which is a
function of space as well, then it is not a wave to my way of thinking.

Agreed.

This definition can make the status of a propagating shock front
somewhat problematic as such a situation need not involve oscillations
in time or space at all, but rather might only include a propagating
localized change in the mean value of the field. My definition would,
I guess, not count such a shock as a wave.

I'd want to see a stronger argument before declaring shockwaves to be not
waves.

OTOH, a propagating localized pulse can be thought of as a legitimate
superposition of spatially oscillatory waves of a field about a fixed
mean value. Such a pulse *does* oscillate in space and time, albeit
with possibly few total excursions in its value. But in such a case the
field's value has at least one oscillation of both a rise and a fall in
space. So I would count a pulse as a wave.

Actually it is easy to construct a pulse that has a rise but no fall.

--------------

I think we don't yet have a simple bug-free definition of "wave" -- but
we're getting close. I'm surprised how hard it is to find one.

It provides a lesson on the unimportance of definitions. Students often
demand a definition of this or that, and they get really ticked off if they
don't get one. But I suspect most real-world physicists get along just
fine without having a precise definition of "wave". Biologists can't even
agree on the definition of "plant" and "animal".