Re: definition of "wave"

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding:

> At 05:49 PM 1/26/00 -0500, Ludwik Kowalski wrote:
> > Can somebody bring an acceptable general definition of a wave
> > for classical physics?
>
> The usual pat answer is:
>         A wave is a disturbance that propagates leaving the
>         medium (if any) behind.

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.

Such a definition would include propagating and standing waves, damped
and positive gain waves, and solitons, etc. as actual waves.  But 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.  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.

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.
If we don't have a field defined over (at least one dimension of) space
we don't have a wave--just an AC time dependence of some function.

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.

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.

Sometimes one hears that a wave needs a medium to propagate in.  The way
I see it the 'medium' is really the field itself (defined over space and
time), and the wiggles/oscillations in the value of the field are the
wave.  The wave's "wave function" is just the field considered as a
function (or a multiplet set of functions if the field has multiple
components) of space and time.

David Bowman
David_Bowman@georgetowncollege.edu