At 08:34 AM 7/2/00 -0400, Ludwik Kowalski asked about the definition of "wave".
1) In mathematics, definitions are crucially important. OTOH physics is a
natural science, not an exact science, and definitions are somewhat less
crucial. In biology, another natural science, there is not even a
universally-accepted definition that delineates "plant" versus
"animal". But we all know the difference between a carrot and a parrot.
2) Since you asked, here's what I look for in a wave:
a) The canonical prototypical wavefunction is something of the form
q = f(x - ct) (equation 1)
where q is the displacement, x is the position, t is the time, c is the
wave-speed, and f() is a shape function.
There are a goodly number of wave equations that have such wavefunctions as
their solution.
There is something special about this functional form: The wavefunction
f() is in one sense a function of two variables (x and t), but in another
sense we can eliminate x and t in favor of a single variable, the phase
(phi = x - ct).
This means there is a guaranteed relationship between the x-derivative and
the t-derivative of q. To me, this is the hallmark of wave propagation:
(d/dx) q = (-c d/dt) q
no matter whether the shape function f() describes a sine wave, or a shock
wave, or whatever.
There is no assumption of linearity or superposition; if f(x - ct) is a
solution it is quite possible that 2f(x - ct) is _not_ a solution of the
same wave equation; for example sound waves in air become quite nonlinear
when the sound-field pressure excursions become comparable to one Atm.
Also note that there is no requirement that f() be periodic. Example: a
flash of light. Example: A soliton in a trough of water.
OTOH because of possible dispersion there is no requirement that
nonperiodic solutions exist; that is, it may be that the wave equation
only allows wavefunctions f() which are monochromatic periodic functions of
the phase. Example: water waves on the ocean are highly
dispersive. Transverse flexional waves on a springy rod are highly
dispersive. Optical waves in flint glass are highly dispersive.
IMHO all other notions of wave are generalizations of this canonical wave,
and the farther we get away from this prototype, the more dubious becomes
the usage of the word "wave".
Classical light waves in a vacuum are special: They are the ideal waves in
that they satisfy equation 1 for _practically any_ shape function
f(). That is, they exhibit no dispersion and no nonlinearity.
b) Equation 1 can be generalized to forms that describe decaying waves, e.g.
q = f(x - ct) / |x|
which might describe a sound wave or light wave spreading in 3
dimensions. We can even have more-strongly decaying waves
q = f(x - ct) exp(-|x|)
which might describe evanescent waves, including massive particles below
the mass shell.
These evanescent waves are a pretty marginal case. You can rationalize
calling them "waves" by various good and bad arguments:
--) They have a definite direction of propagation; see below.
--) A wave might start out as a real wave, then go through a _thin_ piece
of material with a different index of refraction (wherein it is evanescent)
and then become real again on the other side.
--) You can wave your hands and say that the equation that generates the
evanescent waves is "similar" to the equation that generates real
waves. This is a completely bogus argument, of course.
c) I absolutely draw the line at diffusion. The diffusion equation is not
a wave equation, and its solutions are not waves, even though some people
think the equation "looks similar" to the wave equation. I say this even
though diffusive "disturbances" can be periodic in time and
decaying-periodic in space, and you can certainly have a signal that
"propagates" by diffusion. (Example: moths send signals by diffusion of
pheromones.)
Here's the difference: Let's consider a system governed by the wave
equation, such as plain old linear transverse waves on a flexible string
under tension. Suppose I show you a snapshot of a wave packet in the
middle of the string and ask "what comes next" ... You can't answer,
because you can't tell from the snapshot whether the waves are propagating
leftward or rightward.
Contrast this with a system governed by the diffusion equation, such as a
locally periodic "packet" or pattern of red and blue dye diffusing through
a gel. If I show you a snapshot of the dye pattern and ask "what comes
next" you _can_ answer. You don't need to ask whether the diffusion was
"propagating" left or right; diffusion has no momentum, no memory of where
it has been or where it is going.
d) Standing waves are a red herring. They can be considered a fortuitous
superposition of leftward plus rightward propagating waves. In the space
of all possible rightward and leftward wave patterns, standing waves are a
subset of measure zero. Pedagogically speaking, I would not emphasize them
and would not cite them as an example of typical waves.
==================
Summary: If it satisfies the equation
(-c d/dt) q = (d/dx) q
then it's a wave.
If it satisfies the equation
(-c d/dt) q = (some "slowly varying" function of x) (d/dx) q
it's probably a wave, whatever that means.
Otherwise, it's probably not a wave.
Sound reasonable? Or have I missed something important?