Understanding Waves: Is There Hope?

[olivermsun]

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.

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 moving ripples in (deep) water are dispersive; solutions to the classical wave equation are not.

 

[Dale]

The motions of ripples in water are determined by the motions of the water

Certainly. But "determined by" is not "the same as".

The moving ripples in (deep) water are dispersive; solutions to the classical wave equation are not.

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.

 

[olivermsun]

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 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.

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.

 

[Lavabug]

Perhaps the most ubiquitous example would be surface water waves, which are governed by Laplace's equation,
\nabla^2\varphi = 0
where velocity is the gradient of the potential
\mathbf{u} = \nabla\varphi.

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).

As for the OP: first start by looking at individual pulses. Think of tying a rope to a stake in the ground and giving it a jolt. You can think of a progressive wave as a sequence of jolts that move from point A to B. Ie: shock wave from a loud event, ripples on a pond etc. In these cases, the "jolt" is a rapid compression-decompression of air, or an elevation in water height, respectively, instead of a mountain formed on a rope by yanking it.

Consider fixing tying both ends of the rope or string to two stakes, then strumming the rope/string. This is essentially a musical instrument string. The waves are formed by two identical progressive waves that propagate in opposite directions along the string. Their sum produces what is called a standing or harmonic wave, which as its name implies doesn't go anywhere, as you've bounded both ends of the string.

You can extend this idea of standing waves to to 2-D by looking at how a drumskin vibrates, since its also "bounded" or " its ends are also fixed", so to speak.

I hope my wording is helpful.

 

[Dale]

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.

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"? 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.

 

[EnergyHobo]

@Lavabug I believe you are correct with both statements. Laplace's equation only works for non-moving waves (time independent).

I believe the wave equation is something like:
\nabla^{2} \varphi = k\frac{\partial^{2}\varphi}{\partial^{2}t}

Then the derivative with respect to time would equal 0, which is Laplace's equation.

Also, rotational fields are non-conservative. I can't answer the second part of that question. Hope that helps.

 

[olivermsun]

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"?

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.

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.

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."

 

[olivermsun]

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?

Laplace's equation only works for non-moving waves (time independent).

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.

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).

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?

 

[Dale]

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."

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.

 

[olivermsun]

Irrotational flow, Laplace's equation, surface boundary condition of balance between the potential and displacement*gravity ... do you find it unconvincing in a mathematical sense or a phenomenological one?

Do you agree that water waves are waves?

 

[Dale]

To a good approximation, yes. Although, I am not an expert on water waves.

I 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 see no reason to trade a clear definition for a sloppy one.

 

[Lavabug]

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?

I see now. It also assumes incompressible flow since that's how the continuity equation reduces to Laplace's right?

Didn't have in mind the driving force behind the sloshing, just in general.

 

[olivermsun]

I 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.

Regarding dispersive effects, a nonlinear wave equation can be written for amplitude effects
c(u)^2 \nabla^2 u = \frac{\partial^2 u}{\partial t^2},
but how would you formulate this to include frequency-dependent dispersion? You could prescribe a phase speed for every Fourier component
c = \frac{\omega}{\kappa}
but how would you determine c as a function of x, t?

By contrast, the dispersion of deep water waves already captured by the linearized system which arises from the Laplace equation, and as a result you get the dispersion relationship right out of it. You don't need to introduce nonlinearity to describe this behavior.

It might be also be helpful to realize the "wave" propagates not just along the surface, but also through the body of water. That is, the stuff between the surface and the bottom matters (which is why the dispersion is depth-dependent).

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..

I see no reason to trade a clear definition for a sloppy one.

I hope it's getting clearer rather than sloppier!

 

[Dale]

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.

That is good enough for me.