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I'm studying waves moving through mediums and I come across this type of equation a lot while describing acoustic waves speed and stress on the material,

$$\begin{gather} \frac{\partial}{\partial z}\begin{bmatrix} A(t,z) \\ B(t,z) \end{bmatrix} = -\frac{1}{c(z)} \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \begin{bmatrix} A(t,z) \\ B(t,z) \end{bmatrix} \end{gather}$$

and the trace of the matrix on the right hand side, always equals $0.$ Is there a reason for this which is related to energy conservation or something similar? $A$ and $B$ are left and right going waves and $c(z)$ is the wave speed.

I am working on a problem of my own and trying to follow the same theory for elastic waves in mediums, but when I get a similar equation, the matrix on the right hand side does not have a trace of $0$.

Thanks in advance.

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  • $\begingroup$ I've made this clearer. $\endgroup$ – rodger_kicks Feb 28 at 11:37
  • $\begingroup$ I will link a paper which is essentially the same equation. See equation 3.9. math.uci.edu/~ksolna/research/97_1506aniso.pdf $\endgroup$ – rodger_kicks Mar 1 at 11:41
  • $\begingroup$ Is there a reason that the system has this symmetry? I guess what I'm asking is, do you think it's engineered in this way to simplify further calculations? $\endgroup$ – rodger_kicks Mar 1 at 13:04

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