Does a standing wave have zero momentum?

[jcap]

I understand that a standing wave in one dimension is a superposition of a traveling wave going one way with a traveling wave going the other way.

Does that mean that the momentum at every point along the combined standing wave is zero?

For example if one has elecromagnetic standing waves confined in a metal box does that mean they have zero momentum everywhere inside the box?

 

[BvU]

Depends on what you are looking at. e.g. here you see a lot of motion, so no zero momentum. But the macroscopic longitudinal momentum transport for identical waves traveling in opposite directions is zero.

 

[lightarrow]

Depends on what you are looking at. e.g. here you see a lot of motion, so no zero momentum. But the macroscopic longitudinal momentum transport for identical waves traveling in opposite directions is zero.

What is "momentum transport"?

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lightarrow

 

[BvU]

With a traveling wave you can transport energy (and thus also momentum). Think of the picture in the link -- but for the wave moving to the right only.
(Equivalent link here, longitudinal waves picture in the lower left -- but I can only see a small part of it with my browser...).

 

[muscaria]

With a traveling wave you can transport energy (and thus also momentum).

For physical fields, isn't the energy transport you are talking about precisely the momentum the OP is asking about? i.e. the linear momentum of an elementary volume of the E-M field (Poynting vector) representing a flow of energy density, with a corresponding continuity equation which relates both quantities and describes local energy conservation. Transporting momentum would require stress in the E-M field. Basically I'm just pointing out that energy flow is linear momentum and momentum flow is stress.

 

[muscaria]

I think the OP's question brings up the difference between linear momentum and canonical momentum - i.e. the momentum in field space. Although a transverse standing wave in the E-M field has non-zero canonical momentum ("field momentum") everywhere apart from the nodes, there is no linear momentum of the field given that there is no spatial translation of the field itself over time - there's no Noether current.
So the linear momentum of the E-M field, namely the quantity which scatters charges and corresponds to energy flow in the field and the momentum we usually think of which knocks things around, is precisely the Noether current associated with space translational symmetry. This quantity would be zero for the setup devised by the OP given that nothing is being translated in physical space. However there is a translation in field space encapsulated by the canonical momentum which describes the change in the field value at any point over time and is evidently not zero for E-M standing waves, otherwise the standing waves wouldn't oscillate in time.

 

[BvU]

Well, and here I am thinking OP is in the early stages of learning about waves ?! jcap, tell us some more !

 

[muscaria]

May well be, it's a good question to ask because there seems to still be some kind of momentum given that the standing waves oscillate in time, yet at first sight one might think that the oppositely traveling waves have momenta that cancel. The point is the linear momenta of the waves (which generate wave translations) cancel, but we still have momentum in field space.. there are two different momenta which one has to consider and the OP's question points to that.