I Is "scalar wave" a legitimate term?

[sophiecentaur]

It depends on the initial velocity. You can have motion in any direction with force in any direction, depending on the initial velocity.

In sound you can and do have molecules that are moving in the opposite direction of the pressure gradient. They are accelerating in the direction of the pressure gradient but not moving that direction.

I know that you know the difference between acceleration and velocity.

I also know that the CM of a small mass of gas will accelerate in the direction of the pressure gradient. The pressure gradient in the plane of the wave is zero and also the acceleration (that's referred to as a wave front). Which is why I jumped a step. The pressure and displacement are in time quadrature - just like any other longitudinal wave. Molecular motion is a red herring here - just as with electron motion in a metal; just consider bulk movement of an elemental volume.

I don't see why you don't just try the line of masses and springs model as a starter for any longitudinal wave. Google "is sound a longitudinal wave?". How is a longitudinal wave described? You say pressure is in all directions so something must be longitudinal; how about displacement?

 

[PeterDonis]

You say pressure is in all directions so something must be longitudinal; how about displacement?

You're mixing up different levels of description, and making things a lot harder than they need to be in consequence.

As @Dale has said, pressure is a scalar field. Describing a fluid using pressure means we are ignoring the fact that the fluid is made of individual molecules with individual motions. We are averaging all that out and that leaves us with a fluid-level description in terms of scalars like pressure and temperature, and, if the fluid is in bulk motion, a velocity field, which is a vector field.

At that level of modeling, sound waves, as waves of variation in pressure, a scalar, are scalar waves. Whether they are longitudinal or transverse is a separate question and doesn't affect the fact that they are scalar waves. But if you are interested in longitudinal vs. transverse, at this level of modeling, what is longitudinal in sound waves at this level of modeling is the variation in pressure.

If you want to talk about "displacement" in the sense of individual molecules (as opposed to bulk motion of the fluid), then you are at a different level of modeling, in which concepts like "pressure" and "scalar wave" don't even exist. You just have a huge number of molecules and you are (at least claiming to) keep track of the individual motion of each one.

I also know that the CM of a small mass of gas will accelerate in the direction of the pressure gradient.

Sound waves are variations in pressure about an average value. There is no net pressure gradient associated with them; the variations in pressure average out to zero. If you are thinking about individual fluid molecules moving about, again, as above, you are at a different level of modeling for which "pressure" and "pressure gradient" are meaningless.

 

[Dale]

Molecular motion is a red herring here - just as with electron motion in a metal; just consider bulk movement of an elemental volume.

I agree. All of your motion and gradient comments are a red herring, not just the molecular motion stuff.

The only relevant fact is that a scalar wave is defined as a wave of a scalar quantity, pressure is a scalar quantity, therefore a pressure wave is a scalar wave.

Gradients are a red herring. Air motion is a red herring. Scalar wave is a clearly defined term and a pressure wave clearly meets the definition.

 

[sophiecentaur]

So a scalar wave is just a solution $u$ to the equation $$\frac{\partial^2 u}{\partial t^2}=c^2 \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right)$$ There is no implication that $$\frac{\partial u}{\partial x} = 0$$ nor that $$\frac{\partial u}{\partial t}=0$$ The only implication of calling a wave a scalar wave is that $u$ is a scalar function.

I am trying to deal with some Physics here and that (I think) should involve dealing with cause and effect. That wave equation is 'just maths' and ignores what the quantity 'u' is. An animated wave pattern on a TV screen (u = brightness) would be a true scalar wave and the relevant difference here is that a picture of a moving wave transfers no energy. When energy is carried by a wave, there are always two physical quantities involved and how can you ignore displacement?

I take your point about the possibility of describing a sound wave in terms of pressure; it makes practical sense but using displacement is also valid. Gases have a bulk modulus too. The term Longitudinal Wave is accepted for describing a sound wave; you seem not to acknowledge this.

 

[Motore]

The point of this thread was to reaffirm that the term scalar wave is a legitimate term, which it is and a pressure wave was given as an example. So the thread can be closed right? I suggest the mentors split the thread if others want to disccuss other type of waves and what they apply to...

 

[Dale]

I am trying to deal with some Physics here

A big part of physics is the definition of the terms used. Pressure is a scalar, so a pressure wave is a scalar wave by the definitions of the terms used.

That wave equation is 'just maths' and ignores what the quantity 'u' is.

Yes, that is the point of mathematics: to abstract away the unnecessary details. Any quantity that can be represented as a scalar function on space and time is a valid $u$ for this purpose. Pressure is an example.

When energy is carried by a wave, there are always two physical quantities involved and how can you ignore displacement?

It is a red herring, that obviously I was not very good at ignoring yesterday. I will try to ignore the red herring today.

Neither the energy nor the displacement is relevant to the classification as a scalar wave. Pressure is a scalar, therefore by definition a pressure wave is a scalar wave. There is no requirement in the definition of the term scalar wave that depends on either energy or displacement.

The term Longitudinal Wave is accepted for describing a sound wave; you seem not to acknowledge this.

I have nothing against that. Multiple words often describe the same thing. The fact that sound waves can also be described as longitudinal waves is another red herring. Scalar waves and longitudinal waves are not mutually exclusive categories. It has no bearing on the fact that pressure is a scalar and therefore pressure waves are scalar waves.

 

[Vanadium 50]

Usually, I don't like formal "mathy" explanations, but I am having a hard time reconstructing the argument. It seems to be:

1. Given a scalar U that follows the wave equation,
2. \nabla U is always a vector V
3. Therefore, there are no "true" scalar waves - there is always a vector V "behind the scenes".

Is that correct? I

 

[renormalize]

Therefore, there are no "true" scalar waves - there is always a vector V "behind the scenes".

And behind the vector there's a symmetric tensor $\nabla_{i}\nabla_{j}U$, followed in turn by a rank-3 tensor, and so on ad infinitum. Where do we stop? What's the point?

 

[PeterDonis]

I am trying to deal with some Physics here and that (I think) should involve dealing with cause and effect.

To do that you need to know the source of the waves, which appears nowhere in the wave equation we have been discussing. That wave equation describes how waves propagate after they have already left the source. A different equation describes how waves are generated at the source.

Again, you are making things a lot harder than they need to be by mixing up different aspects of the problem.

When energy is carried by a wave, there are always two physical quantities involved

What are the two physical quantities involved for a sound wave?

how can you ignore displacement?

I already addressed that in my previous post: you are mixing up different levels of description, which just makes a muddle.

 

[PeterDonis]

it makes practical sense but using displacement is also valid

Please give a reference for how sound waves are described in terms of displacement. (Not water waves, sound waves.)

 

[Vanadium 50]

I am 90% sure there is something in Rossing about sound waves as displacement waves as a way to deal with boundary conditions. But I think we are better off dealing with the abstraction here. We don't want to say "Well, you proved/disproved it for X. But what about Y?"

 

[Dale]

There is nothing to prove. Pressure is a scalar so a pressure wave is a scalar wave.

That remains true even if red herrings are vectors and so red herring waves are vector waves. Red herrings and pressure are different.

 

[PeterDonis]

I am 90% sure there is something in Rossing about sound waves as displacement waves as a way to deal with boundary conditions.

Is the displacement a scalar or a vector? @sophiecentaur appears to be assuming it's a vector, but I don't think that's the case for a displacement treatment of sound waves (as opposed to, say, water waves).

For analyzing the behavior of, say, an old-fashioned microphone or an old-fashioned loudspeaker, a displacement treatment can be useful since the devices in question have an actual membrane whose amplitude of displacement from their equilibrium positions are the receiver or source of sound waves. But it's still a scalar wave model.

 

[Vanadium 50]

@sophiecentaur has been here for a while, so I am trying to get at the root of the misunderstanding rather than just shutting him down. "It;s a red herring" is isomorphic to "you just don't get it", which may well be true, but just because a statement is true doesn't mean it's helpful. And it's not like he's deliberately setting out to confuse.

 

[Dale]

"It;s a red herring" is isomorphic to "you just don't get it",

Here “red herring” is just isomorphic to ”distraction”. I know that @sophiecentaur gets the idea that pressure is a scalar and therefore a pressure wave is a scalar wave.

He correctly pointed out that the molecular displacement is a distraction (red herring was his term and I understood it as meaning distraction). That made me realize that so were the macroscopic displacements and the pressure gradients.

 

[PeterDonis]

Since the simplest acoustic approximation has P-P0=rho0 c0 u,I am going out on the limb and say every introductory textbook.

In that equation, $u$ is a scalar. So this model would still be a scalar wave model.

 

[PeterDonis]

Thread closed for moderation.