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Re: [Phys-L] definition of "wave"



Regarding the 'definition of wave' discussion:

I find it quite tricky to define "wave". The usual careless
definitions get into trouble because:
-- standing waves don't propagate at all
-- typical circular and spherical waves (unlike plane waves) do
not propagate without some change in shape.
-- some media are dispersive
-- some media are nonlinear
-- some media are dissipative
-- a wave is not necessarily repetitive

The best dodge I've found is this:
We define a wave to be something that exists as the solution
to a wave equation. It is best to focus attention on the wave
equation rather than the wave itself.

The wave equation must support some running-wave solutions,
even if not all solutions are of this form.

For examples and further discussion, see
https://www.av8n.com/physics/wave-intro.htm

If anybody has a better way of looking at this, i.e. simpler
and/or more correct, please let me know.

One problem of defining waves as solutions of a wave equation is that the definition may be somewhat circular, particularly if the concept of a wave equation is defined as such because of the wave-like characteristics of its solutions.

FWIW the way I think of waves and their definition is this way:

Fundamentally, a wave is a temporally and spatially oscillatory behavior in the value of a field. This abbreviated statement admittedly requires some unpacking.

First, what I mean by a field is that it is a function of some sort, whose argument (or domain) is/are events or points in spacetime or a connected sub region of it containing an extended open set along at least the timelike direction *and* at least one spacelike direction. The value (range) of the function for the field can be any geometrically significant object e.g., scalar, spinor, vector, 2-form, higher rank tensor, with various possible symmetry or antisymmetry properties. For instance sound waves in a fluid would have a scalar valued field (i.e. the local pressure or density of the fluid), and EM waves would have a field whose values are the antisymmetric 2nd rank 4-tensor whose component values are made of the E 3-vector & the B pseudo-3-vector components. The Schrödinger wave equation would have a field which is the scalar complex value of the De Broglie amplitude in position space. Waves on a vibrating string/wire would have a field whose domain contains time and the 1-d position along the string, and have a range of the 2-d transverse displacement vector of the position of the string from its local equilibrium position. A wave on a drum head or a water/air interface would have a domain containing time and the 2-d location of the equilibrium position on the surface of the drum head or the interfacial surface. The value or range would be the transverse displacement of the 2-d boundary into the 3rd spatial dimension, and as such could be considered as a single scalar at each position on the 2-d surface. (In a quantum context of field quantization the value of the field may well be an operator-valued distribution, and then the oscillations would be in magnitudes of the projections of the operator along the members of a convenient basis of the Hilbert space in which the operators operate).

Second, what I mean by temporally oscillatory behavior is that the value of the field oscillates in and with time at each fixed spatial point of the field's domain. And what I mean by 'spatially oscillatory behavior' is that the value of the field oscillates in and with displacement in space (along at least one spatial dimension), at each fixed moment of time for the field's domain. IOW the oscillations are 2-fold. The wave oscillates in space (along at least one spatial dimension) at each fixed moment of time, *and* it oscillates in time at each fixed point in space. No requirement is made one way or the other as to if this 2-fold oscillatory behavior constitutes running/propagating or standing wave-like motions.

Note that since my definition requires the field's domain to include *both* time and at least one spatial dimension this means that a function of merely time which oscillates in time does not count as a wave. For instance the AC voltage value across the two terminals of an electrical outlet has a nice wavy temporal behavior over time, but it has no spatial dependence, and as such does not count as a wave. To be a wave, in my mind, it must oscillate in space at each fixed moment of time, as well as oscillate in time at each fixed (relevant) spatial point. (Mathematically the field's domain needs to be at least two dimensional with the internal geometry of that domain having a locally hyperbolic character.)

Typically the dynamics of the field (i.e. the wave equation(s) describing the dynamical behavior of the waves) in question has a few common properties of their own:

First, the field has a possible static 'undisturbed' 'equilibrium' value behavior/solution everywhere spatially if it is not being driven or never was or will be driven inhomogeneously. If the initial conditions on the value of the field are such that the field initially takes on the equilibrium value everywhere in the domain, then the field's value does not change anywhere in time, and that static local value at each place is the equilibrium value of the field at that spatial place. Also the oscillations mentioned earlier are understood to be oscillations *from* the background value of this equilibrium state. IOW we can conceptually take the value(s) of the field and subtract off the static background equilibrium value at each place and consider the waves to be oscillations in the difference between the static background value and the actual instantaneous value of the field over its spacetime domain.

Second, the field locally has some sort of 'restoring force'. If the field value is not the background equilibrium value at some spacetime point it experiences a 'force' or acceleration tending to push its value back to the equilibrium value and act locally to reduce the magnitude of the disturbance away from equilibrium. Also the strength of this restoring tendency increases with the magnitude of the local disturbance away from the equilibrium value.

Third the field has locally an 'inertia' property so that if its value is initially changing with time it tends to 'coast' over time with that same rate, at least until the net disturbance gets so large in magnitude that the 'restoring' effect previously mentioned take over and pushes it back towards the equilibrium value.

David Bowman