Question

Is there a clear interpretation for energy and momentum transport in the simple wave equation?

Answer 1

I'm looking at some classical field theory concepts. I was working through the 1D simple wave equation Lagrangian density and stress-energy tensor. The equations of motion are μϕtt=Tϕxxμϕtt=Tϕxx, TT being the tension, μμ the mass density, and a subscript denoting differentiation with respect to that coordinate. This has Lagrangian density L=12μϕ2t−12Tϕ2xL=12μϕt2−12Tϕx2, and gives a stress-energy tensor from Noether's theorem

Tμ   ν=∂L∂(∂μϕ)∂νϕ−δμνL=[12μϕ2t+12Tϕ2x−Tϕtϕxμϕtϕx−12μϕ2t−12Tϕ2x]T   νμ=∂L∂(∂μϕ)∂νϕ−δνμL=[12μϕt2+12Tϕx2μϕtϕx−Tϕtϕx−12μϕt2−12Tϕx2]

So you can see TttTtt is just the energy density which I'll denote ee. The statement that ∂μTμ   ν=0∂μT   νμ=0 is the statement: ∂xe=∂t(μϕtϕx)∂xe=∂t(μϕtϕx) and ∂te=∂x(Tϕtϕx)∂te=∂x(Tϕtϕx). I'm pretty sure the first one is a statement about momentum transport (it has units of momentum density) and the second is about energy transport (it has units of power density), but I'm drawing a blank on how to clearly physically interpret it. Is there any clear way to say "energy is moving to the right" or "momentum is moving to the left" given these equations?

Answer 2

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density and stress-energy tensor. The equations of motion are μϕtt=Tϕxxμϕtt=Tϕxx, TT being the tension, μμ the mass density, and a subscript denoting differentiation with respect to that coordinate.

This has Lagrangian density

L=12μϕ2t−12Tϕ2xL=12μϕt2−12Tϕx2,

and gives a stress-energy tensor from Noether's theorem

Tμ ν=∂L∂(∂μϕ)∂νϕ−δμνL=[12μϕ2t+12Tϕ2x−Tϕtϕxμϕtϕx−12μϕ2t−12Tϕ2x]T
νμ=∂L∂(∂μϕ)∂νϕ−δνμL=[12μϕt2+12Tϕx2μϕtϕx−Tϕtϕx−12μϕt2−12Tϕx2]

So you can see TttTtt is just the energy density which I'll denote ee.

The statement that ∂μTμ ν=0∂μT νμ=0 is the statement: ∂xe=∂t(μϕtϕx)∂xe=∂t(μϕtϕx) and ∂te=∂x(Tϕtϕx)∂te=∂x(Tϕtϕx).

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