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So in QED a photon has a probable 4D position?Then it's a probability.
So in QED a photon has a probable 4D position?Then it's a probability.
Sounds interesting and quite surprising to me. Can you give one example please? And if we have "found" an equation that explain the wave, better.As Whitham (1974) pointed out, many (if not most) waves are not governed by the "wave equation"!
(He gives a much broader definition, that a wave is "a recognizable signal that is transferred from one part of a medium to another with a recognizable velocity of propagation.")
Perhaps the most ubiquitous example would be surface water waves, which are governed by Laplace's equation,Sounds interesting and quite surprising to me. Can you give one example please? And if we have "found" an equation that explain the wave, better.
Thanks.
The motions of ripples in water are determined by the motions of the water, just as the motions of waves moving along a string are determined by the motions of the string itself.The resulting surface elevation is a solution to the wave equation. The fluid's velocity is not described by the wave equation, but the height of the ripples is. When people talk of water waves they are speaking about the ripples, not the velocity of the water molecules.
Certainly. But "determined by" is not "the same as".The motions of ripples in water are determined by the motions of the water
This is true. Dispersive waves are only approximately modeled by the classical wave equation. The approximation is bad for long times, but for short times it is reasonable. So I would still keep the definition and just indicate that dispersive waves are not ideal waves.The moving ripples in (deep) water are dispersive; solutions to the classical wave equation are not.
The behavior of dispersive water waves is fundamentally different from the hyperbolic waves that arise from the wave equation. You can see this easily and within a short time (relative to the wave period) from the way ripples crawl across the top of a wave 'bump' due to the difference between phase and group speeds.Dispersive waves are only approximately modeled by the classical wave equation. The approximation is bad for long times, but for short times it is reasonable. So I would still keep the definition and just indicate that dispersive waves are not ideal waves.
Hope this doesn't make this thread too concrete, but I had to ask: the homogenous wave equation reduces to Laplace's equation if the perturbation on the surface of water doesn't depend explicitly on time yes? Also, your second statement (u being conservative), is this only valid for irrotational flows? Could surface water waves still be governed by Laplace's equation even if the velocity field was non-conservative? (I'm thinking of the sea sloshing around).Perhaps the most ubiquitous example would be surface water waves, which are governed by Laplace's equation,
[tex]\nabla^2\varphi = 0[/tex]
where velocity is the gradient of the potential
[tex]\mathbf{u} = \nabla\varphi.[/tex]
Mostly because I don't like any of the alternative definitions that I have seen. Like this one:I admit I do not really understand why one would want to define an entire class of interesting (and very well-studied) waves out of the discussion just because they don't fit an a "wave" equation which happens to arise from the physics of certain other kinds of ideal waves.
IMO that is a poor scientific definition. What makes a signal and a propagation velocity "recognizable"? Does empty space count as a medium? Etc. If you pursue those rigorously you probably wind up with a wave equation anyway."a recognizable signal that is transferred from one part of a medium to another with a recognizable velocity of propagation."
The problem is that many phenomena of interest escape a more precise definition. It's similar to asking "what is turbulence." The "poor scientific definition" that you so dislike is simply one of the working definitions offered by one of the more prominent mathematical physicists to study "waves" in general.Mostly because I don't like any of the alternative definitions that I have seen. Like this one:
"a recognizable signal that is transferred from one part of a medium to another with a recognizable velocity of propagation."
IMO that is a poor scientific definition. What makes a signal and a propagation velocity "recognizable"?
Except that you don't end up with a wave equation -- even approximately -- in the example of deep water waves or in many other examples which nevertheless seem to be full-blown "waves."Does empty space count as a medium? Etc. If you pursue those rigorously you probably wind up with a wave equation anyway.
Just because something is approximately a wave doesn't mean that you cannot study or talk about it.
Hope this doesn't make this thread too concrete, but I had to ask: the homogenous wave equation reduces to Laplace's equation if the perturbation on the surface of water doesn't depend explicitly on time yes?
Water waves theory allows a surface perturbation which is explicitly time-dependent, so it is a somewhat different case.Laplace's equation only works for non-moving waves (time independent).
The idealized water wave theory is built around inviscid flow around rest, no rotation, and so forth. This works fairly well since the Reynolds number is typically pretty big away from boundaries, the kind of waves we're talking about tend to be well above inertial frequency, etc.Also, your second statement (u being conservative), is this only valid for irrotational flows? Could surface water waves still be governed by Laplace's equation even if the velocity field was non-conservative? (I'm thinking of the sea sloshing around).
You have said that, but the examples so far are not very convincing. And in any case, they wouldn't qualify as waves according to the fuzzy definition either, except approximately.Except that you don't end up with a wave equation -- even approximately -- in the example of deep water waves or in many other examples which nevertheless seem to be full-blown "waves."
I see now. It also assumes incompressible flow since that's how the continuity equation reduces to Laplace's right?Water waves theory allows a surface perturbation which is explicitly time-dependent, so it is a somewhat different case.
Here you get Laplace's equation from the continuity equation rather than by assuming no time dependence.
The idealized water wave theory is built around inviscid flow around rest, no rotation, and so forth. This works fairly well since the Reynolds number is typically pretty big away from boundaries, the kind of waves we're talking about tend to be well above inertial frequency, etc.
Before I go too far off on a tangent, what kind of sloshing did you have in mind? Tidal or possibly seiches in the basins?
Regarding dispersive effects, a nonlinear wave equation can be written for amplitude effectsI am unconvinced mathematically. As far as I know a monochromatic sinusoidal plane wave is a solution, so the problem can be formulated as a classical wave equation even if there exists an alternative formulation as Laplace's equation. Furthermore dispersive and nonlinear waves are still solutions to the classical wave equation where c is a function of wavelength or amplitude.
I hope it's getting clearer rather than sloppier!I see no reason to trade a clear definition for a sloppy one.
That is good enough for me.In the limit of shallow water, the dispersive effects go away, and you do in fact recover the 2-d wave equation. So if you are to argue from this standpoint that all water waves are just shallow water waves (and hence obey the wave equation) with some corrections, then I suppose you could make a case for that. But they would not be small corrections in many cases.