Re: definition of "wave"

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding John D's comments:

> That definition has many strengths and a few weaknesses.

I guess that's better than the other way around.

> 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.

Yeah, a major motivation for my definition was to keep diffusive phenomena
(with a real diffusion coefficient) from being considered as waves.  Note,
though, that a formal diffusion equation with an *imaginary* diffusion
coefficient *is* a wave equation and has solutions that are waves.  The
canonical case in point is the time-dependent Schrodinger equation.

> 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.

It seems to me that the result is a pair of oppositely moving pulses that
coincide at the moment the pluck-shaped waveform is released from rest.
But, anyway, each such a running pulse *is* an oscillation.  The medium
is distended as the leading edge of pulse passes, and then it returns to
its original configuration as the trailing edge of it passes.  And at
each instant of time the pulse has a rising edge and a falling edge in
space.  I would consider a rise/fall pair as an oscillation.  IOW, I
would consider it an oscillation in the form of a single reciprocation
in space and in time.  If the medium did not return to its old average
location, then I would not want to call this a wave.  In that case it is
just an inhomogeneous displacement (as in the case of the pure shock
front I mentioned before).

> 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.

I assume you mean that the soliton is stable if its leading edge (on the
side of its direction of motion) is rising.  Presumably, there is a
trailing edge that falls so that the water level can go back to its
original level.  Such a running pulse *can* be written as a
superposition of many harmonic waves (since they form a basis), but such
a decomposition is not useful as a mathematical device in solving a
soliton problem since the equation that governs the soliton behavior is
is nonlinear and the component harmonic waves of the soliton are *not*
each separate solutions of that equation which is only solved by the
whole composite soliton itself.

> > ... 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.

That depends on the particular type of evanescent wave considered.  If
the evanescent wave has a wave number that is complex valued with *both*
a nonzero real and imaginary part, then that wave *is* a wave according
to my definition; it's just a decaying wave.  But if the evanescent
wave has a *pure imaginary* wave number, then it does not oscillate in
space in the decaying region.  I meant this latter type of evanescence
in my quote above.  An example of it is the reflection of a DeBroglie
wave off of a square potential step beyond which is a classically
forbidden region whose potential energy is greater that the total energy
associated with the incident particle.  In the classically forbidden
region the wave function does not oscillate in space; it just decays with
a purely imaginary wave number.

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

My definition includes some evanescent waves and excludes others.

> >   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.

Not necessarily.  I agree that superposition is not a very useful
concept for waves obeying nonlinear equations (especially for strongly
nonlinear ones for which perturbation methods in the strength of the
nonlinearity are not available).  But even in such a case, a
superposition might be considered as existing; its just that the simple
waves so superposed are *not* solutions of the nonlinear equation,
and such a concept is nearly void of any practical value.  It seems that
for a damped--yet fully oscillatory--wave that happens to obey a
nonlinear equation one could still crudely describe it as having a
complex wave number, anyway, as long as the solution's decay envelope is
nearly exponential and the oscillations are nearly sinusoidal (once the
envelope function is divided out from the wave).

> > 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.

I'm glad we agree on this.

> > 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.

My thinking here is that a (purely non-oscillatory) propagating shock
front merely represents an inhomogeneous change in the mean value of
the field, which results in a net displacement of the background average
of the value of the field. Such a net shift does not count as an
oscillation since the value does not return to its previous value.  Also,
such a behavior is not a square integrable disturbance from the
previous mean value, and any Fourier representation of it would be
ill-defined.

> > 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 know.  But I would not want to count such a pulse as being a wave.  If
it propagated it would be like the shock front case I mentioned above.

> --------------
> 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.

Maybe we are close.  I'm not yet convinced that my definition is very
buggy.

> 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".

I agree.

David Bowman
David_Bowman@georgetowncollege.edu