# Longitudinal waves and vector potentials.

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1. Nov 26, 2015

### muscaria

Hello,
I was wondering if anybody knew of any material (books, papers etc..) which considers a possible connection between longitudinal waves and vector potentials, at least mathematically. I have been scouting about, but failed to find anything substantial. I understand that there seems to be no need for such a treatment in the common physical systems which display longitudinal modes - these being associated with oscillating pressure/density variations of a medium from equilibrium. In these setups, one is concerned with bare "canonical type" momenta as constituting the wave motion, vector potentials being irrelevant. Also in standard electromagnetic theory, one is concerned with transverse waves, and the fact there are no longitudinal modes is reflected (as I "understand" it anyway) in the validity of the Coulomb gauge. Although there may be no present physical setup which gives rise to longitudinal mode solutions resulting from certain vector potential field properties (I'm thinking of vector potentials generally, not strictly of the common dynamic E-M field type), I was wondering if someone had come across any form of mathematics which describes or hints at such a connection. I imagine that if such a connection were to exist, the longitudinal modes would require a non vanishing divergence of the vector potential field, which oscillates in magnitude over space and time. Just as the longitudinal modes associated with sound are scalar waves due to density variations, an oscillating $\nabla\cdot\textbf{A}$ scalar function may give rise to longitudinal modes? Again, mathematically speaking. I am investigating features of a physical system which I think may support such modes, but am unaware of the mathematics I need to look into this.
Anyway, if anybody has any information regarding a mathematical connection between longitudinal waves and vector potentials or has a feeling they may have come across something which hints at these topics - even slightly - I would really appreciate it! Any comments of any kind would also be very welcome.
Thank you very much and hope this message has reached you in good spirits.

2. Nov 26, 2015

### Staff: Mentor

I think that the Proca action describes a "massive EM" field. It has a longitudinal polarization and a four-vector potential. I am only barely aware of it, so won't be able to do much more than point you in that direction.

3. Nov 26, 2015

### muscaria

Thanks very much Dale, I'll look into it! Much appreciated.

4. Nov 27, 2015

### Chandra Prayaga

The electric field waves in an anisotropic medium have a longitudinal component. In general, the D vector D = ε0E + P is a transverse wave, E is not.