# Acoustics: Reflection and Transmission Coefficients

While studying about reflections and transmission coefficient in a discontinuous Media I came across two definition of coefficients one which seems to be correct according to me as it is in accordance with the physics but there is another definition which I have found in many books and used in many papers. Consider a case, when wave is incident from low impedance medium 1 to high impedance medium 2. Then as per my understanding, the amplitude of incident wave should decrease as it goes from low to high impedance ( given by relation 2 below). But if I use relation 1, the amplitude of transmitted wave increases as we go from low to high impedance and that is not possible.

$$Z_1$$ and $$Z_2$$ are impedance of two media

• Relation 1:

$$R= \frac{Z_2 - Z_1}{Z_1 + Z_2},\quad T = \frac{2 Z_2}{Z_1 + Z_2}$$

• Relation 2: $$R = \frac{Z_1 - Z_2}{Z_1 + Z_2},\quad T = \frac{2 Z_1}{Z1 + Z_2}$$

Could someone please clarify my doubts and discrepancy in the relations that I found?

## Reference

1. Kinsler. Fundamentals of Acoustics.

• You didn't give any references to where either formula was used, so we don't know what any of the variables mean, and therefore we can't answer the question properly. The obvious guess why the formulas are different is that the labels 1 and 2 for the two materials are interchanged. Note the sign of R is different, as well as your question about T. Jun 11, 2019 at 10:00
• These formulas are used tor calculating reflection and transmission coefficient when wave hits an interface. Reference for relation 1 : "Fundamnetals of acoustics" by Kinsler et.al (equation 6.2.8) reference for relation 2 is : assets.press.princeton.edu/chapters/s9912.pdf Jun 11, 2019 at 12:40
• Link for the book( reference chapter-6, eqn 6.2.8): google.com/url?sa=t&source=web&rct=j&url=http://… Jun 11, 2019 at 12:44
• You haven't really asked a question yet. You have not pointed out the discrepancy since there is no correlation of index to medium that is consistent throughout. I figure would help. Also, why do you assert that the amplitude increases for T? can you give an example with real medium, air to water, etc? This might help illuminate the issue.
– user196418
Jun 11, 2019 at 13:03
• But index does depend on medium. If a plane wave is incident on the interface of two mediums ( high impedance to low) then amplitude should increase as per the definition of impedance. For example, if a plane wave at normal incidence goes from air to water then amplitude of wave will decrease as water has high impedance then air. Jun 11, 2019 at 13:40

The two expressions look the same to me, with the difference being on the label placed on each medium.

You should focus on the meaning of transmission an reflection coefficient. That is, for a plane wave with unitary amplitude how much of it transfers to the other medium and how much gets reflected. Once you get that, you write down the equations and apply the relations between particle velocity and pressure (given by the characteristic impedance) and boundary conditions.

This is pretty much what is done in this section of Kinsler's book.

## Edit: 2019-06-12

But if I use relation 1, the amplitude of transmitted wave increases as we go from low to high impedance and that is not possible.

Yes, it is possible. The amplitude can go up or down, depending on the impedance contrast. Although, I think I understand what the problem is. You might be thinking that this implies that the power is increasing, but that's not the case. For that, you should look at the intensity (or power) transmission coefficients given by (6.2.10 and 6.2.11):

$$R_I =\left(\frac{Z_2 - Z_1}{Z_2 + Z_1}\right)^2\, ,$$

and

$$T_I = \frac{4 Z_2 Z_1}{(Z_2 + Z_1)^2}\, .$$

Now, $$T_I$$ cannot increase from one medium to the other but $$T$$ can. Let us consider $$Z_1 = 1$$ and $$Z_2 = 4$$, in that case, we have

$$R = \frac{3}{5}\, ,\quad T = \frac{8}{5}\, ,$$

but,

$$R_I = \frac{9}{25}\, ,\quad T_I = \frac{16}{25}\, .$$

If we change the roles, $$Z_1=4$$ and $$Z_2=1$$, we get

$$R = \frac{-3}{5}\, ,\quad T = \frac{2}{5}\, ,$$

but,

$$R_I = \frac{9}{25}\, ,\quad T_I = \frac{16}{25}\, .$$

In the second case, we should interpret the negative sign in the reflection coefficient as a phase change of $$\pi$$ radians.

Notice that in both cases, the transmitted "energy" is less than 1. But this information is not encoded in the transmission coefficient itself, but in $$T_I$$. This makes total sense since the impedance is telling us how "easy" is to move the fluid for a given pressure.

• Yes, the thing is I understand the meaning of impedance. But as per the expression give(6.2.8 and 6.2.9), if the medium 1 is low impedance and medium 2 is high impedance then amplitude of transmitted wave is going to increase. How's that possible? Let us say Z1=1 and Z2=4 then as per the understanding of impedances transmitting amplitude should decrease but this is not what we get from expression 6.2.8/9 above in the book. Jun 11, 2019 at 19:56
• @SumanKumari, there is no reason for thinking that the amplitude should decrease. Actually, when a seismic wave comes to a valley it increases in amplitude for the change in impedance. The amplitude is not a conserved quantity. If you looked at power or momentum density it would be different though. Coming back to Kindler's book, check equations 6.1.5 – 6.1.7, power transmission and reflection coefficients should add up to 1, hence they should not increase when changing from one medium to the other. Jun 11, 2019 at 21:10
• Then what determines whether the amplitude should increase or decrease as the medium changes if it is not impedance?( specifically in terms if acoustic wave) Jun 12, 2019 at 2:38
• @SumanKumari, the impedance contrast is what determines the change in amplitude. The amplitude can go up or down, accordingly. Jun 12, 2019 at 3:21
• If you consider intensity, you will see that the amplitude of the pressure can increase, but the particle velocity may decrease (or the other way around), as long as the intensity is conserved. This is what nicoguaro is trying to point out. Feb 1, 2020 at 14:32