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Let us assume that a wave propagates in the direction perpendicular to the flat surface of discontinuity. When the characteristic impedance of the medium of medium 0 (where the incident and reflected wave exists) is mush larger than medium 1 (where the transmitted wave exists). How do the pressure and intensity reflection and transmission coefficient behave?

I believe that in this case, the power will not be transmitted. And all the sound waves will turn into reflection.

I don't know how to represent it math terms for R and $\tau$.

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    $\begingroup$ Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. $\endgroup$
    – Markoul11
    Dec 23, 2021 at 11:02

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Assuming that your media have characteristic acoustic impedances $r_0$ and $r_1$ you end up with the following reflection and transmission coefficients

\begin{align} &R = \frac{r_1 - r_0}{r_1 + r_0}\, , \\ &T = \frac{2r_1}{r_1 + r_0}\, . \end{align}

The intensity reflection and transmission coefficients are

\begin{align} &R_I = \left(\frac{r_1 - r_0}{r_1 + r_0}\right)^2\, , \\ &T_I = \frac{4 r_1 r_0}{(r_1 + r_0)^2}\, . \end{align}

You can see that the intensity reflection coefficient does not depend on the sign of $r_1 - r_0$, that is, it does not matter which one is larger. The same power is reflected. The reflection coefficient does depend on the sign of $r_1 - r_0$ and it tells us something about the (relative) phase of the reflected wave.

For more detail I suggest checking the following reference:

  • Kinsler, L. E., Frey, A. R., Coppens, A. B., & Sanders, J. V. (2000). Section 6.2 in Fundamentals of acoustics. John wiley & sons.
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