Why pressure is a scalar instead of a vector?

Question

I know by definition that the pressure is the ratio of the force, which is perpendicular to the surface, to the area of that surface and it has no direction. But I am confused by the two questions concerning the 'direction' of pressure.

According to the definition, the force that corresponding to the atmosphere pressure is everywhere and to all directions in the air. This means that the force due to atmosphere pressure in the vertical direction is the same as that in the horizontal directions. This can not be understood intuitively because the gravity only acts vertically.
For a compressional acoustic wave, such as that in the air or in the water, the direction of wave propagation is the same as that of particle vibrations. For a plane wave, the acoustic pressure at one point in space can be sensed from any directions since it is a scalar. Here comes the question: in the plane that is normal to the propagating direction, the particle velocity is zero wheras the acoustic pressure is not. But a nonzero pressure implies a nonzero force and a nonzero force will induce a movement. Consequently, the acoustic wave becomes transverse instead of compressional. Why?

Hope someone can help me. Thank you very much.

Because if it's a vector then it's force. And air is a fluid. — Hot Licks, Jul 13 '18 at 02:00
Pressure is technically a tensor. Well, it's part of the stress tensor, anyway. — probably_someone, Jul 13 '18 at 02:08
Possible duplicate of Mathematical Reasoning for Fluid Pressure as a Scalar — John Rennie, Jul 13 '18 at 03:27
@John Rennie The question here is different from those two. Actually, the second part is concerned with physics of acoustic wave. — ecook, Jul 13 '18 at 07:57
Actually, pressure is a scalar "magnitude" associated with a 2nd order tensor known as the stress tensor (kind of like the magnitude of a vector). A 2nd order tensor has multi-directional character. The pressure term in the expression for the stress tensor represents the isotropic part of the stress tensor, and so has equal influence in all directions. In addition to the pressure contribution to the stress tensor, there are also viscous contributions that come into play when the fluid is experiencing a deformation; these contributions are typically not isotropic. — Chet Miller, Jul 14 '18 at 14:56

score 1 · Accepted Answer · answered Jul 13 '18 at 02:25

1

Air pressure is not the direct effect of gravity; gravity only makes it so that pressure increases downwards. Pressure is the force exerted by molecules on neighboring molecules just because of the mutual repulsion when they get close. In a gas, where molecules are far apart, it's caused by collisions; in a liquid or solid the molecules are all bunched up, so pressure is just their resistance to being compressed.

This explains the fact that pressure acts in all directions. Gravity doesn't have anything to do with it per se; the force at a given point is the same in all directions because the molecules are constantly moving around and bumping into each other, and this moving and bumping is (at a given height) independent of direction. It does get stronger as you go down, because this bumping has to support the weight of the air above.

Regarding your second question, the answer depends on what velocity you're speaking of. Air molecules are constantly moving around randomly in all directions (remember that this is the source of pressure!). But on a relatively large scale (say, of the order of the wavelength of a sound wave), this random movement averages out, and we consider the air to be a rest. The back-and-forth movement in a wave happens on top of this random vibration, but it is not the back-and-forth itself that produces pressure.

answered Jul 13 '18 at 02:25

Javier

28,106

Thank you for the explanation from the microscopic perspective. The velocity of the back-and-forth movement caused by external plane, acoustic wave has the same direction of propagation. But how is the pressure of this external wave generated and why is this external_source generated pressure a scalar? Besides, how is the pressure increased when the acoustic wave is present? – ecook Jul 13 '18 at 07:39
@ecook the pressure wave is generated because somewhere air is being pushed and hence compressed. Compressed air has a higher pressure, because the molecules are closer and bump more into each other. This increased bumping makes the air want to move to a region of lower pressure, and this is how the wave propagates. – Javier Jul 13 '18 at 13:57
I am still not clear why the acoustic wave propagates to ONLY one direction wheras the acoustic pressure can be sensed from any directions in space. For example, If a balloon is pressed and released continuously from x direction, it would deform and compress all molecules around it. In this way, the 'wave' will propagate to all directions instead of one. Please tell me why this is wrong. Much thanks – ecook Jul 13 '18 at 15:03
@ecook A wave does propagate in all directions, in general. The sound out of a speaker comes out in a half-spherical wave (towards the front). You need very careful conditions to produce a plane wave: essentially, you need a big plane moving all together. – Javier Jul 13 '18 at 15:05
But even when the conditions for a plane wave are satisfied, the direction of particle velocity may also be perpendicular to the propagation in the case of a compressional wave according to the microscopic explanation. Why? – ecook Jul 13 '18 at 15:13
@ecook There are two velocities here, microscopic and macroscopic. The microscopic velocities are random and in all directions, and they are the source of pressure. This doesn't move particles very far, because they bump into each other; it's much like a vibration. The macroscopic velocity is that due to the wave, and it is collective: many particles moving together in some direction, on top of their random microscopic movements. The microscopic movement changes direction all the time and averages to zero: if a particle moves perpendicular to the wave at some time, it will soon bounce back. – Javier Jul 13 '18 at 16:23
But the back-and-forth movement generated by a sound projector will only compress the molecules in the direction of wave progation, say x, the molecules are not compressed in the y direction. How is the pressure 'in the y direction' inceased since it is not subjected to the movement of external wave? – ecook Jul 17 '18 at 07:16

Why pressure is a scalar instead of a vector?

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