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I am working with a numerical modelling program allowing to simulate wave propagation in a drill string.

The reflected waves are modelled using the reflection coefficient between the drill bit and the underlying rock, defined as the difference between the impedances of both media, $Z_1$ and $Z_2$, divided by the sum of these impedances: $$\frac{Z_1-Z_2}{Z_1+Z_2}.$$

So far so good. However, the primary waves rely on another "reflection coefficient" defined as $$\frac{Z_1Z_2}{Z_1+Z_2}.$$

What bugs me here is that this coefficient is not dimensionless: it is actually homogeneous with an impedance!

Assuming that this is not a typo from the author of the code, what would be the physical meaning of a non-dimensionless reflection coefficient?

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  • $\begingroup$ I don't think it is a reflection coefficient. It is, however, the harmonic average of the two impedances, or the effective impedance for the two materials in parallel instead of series. Maybe it is saying that you should replace $Z_2$ with the $Z_1Z_2/Z_1+Z_2$ impedance? $\endgroup$
    – Michael M
    Apr 26, 2021 at 11:40
  • $\begingroup$ Thanks for your reply, Michael. Your suggestion is an interesting one. This "coefficient" (for the lack of a better word) is used to define the amplitude of the primary wave originating at the interface between media 1 and 2... So maybe the author was trying to compute the impedance of an effective medium by computing the harmonic average of the two impedances. $\endgroup$
    – Sheldon
    May 2, 2021 at 19:56

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Yes, a dimensionless reflection coefficient makes sense (see https://en.wikipedia.org/wiki/Reflection_coefficient).

From a physical point of view you can think about it as the relative amplitude with respect to the input amplitude.

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  • $\begingroup$ Thanks for your reply nicoguaro, but I was asking whether a non-dimensionless RC was making any sense. $\endgroup$
    – Sheldon
    May 5, 2021 at 1:25
  • $\begingroup$ @Sheldon, I would say no. Do you have the reference? $\endgroup$
    – nicoguaro
    May 6, 2021 at 2:12

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