# Acoustic amplitude transmission through multi layer medium

I have an acoustic wave with amplitude $A$ in ice. It passes a water-filled crevasse, and I want to calculate the impact of this passage on the magnitude of the amplitude (I do not care for the phase).

With $r = \frac{Z_1 - Z_2}{Z_1 + Z_2}$ and $t = \frac{2Z_2}{Z_2 + Z_1}$ I get

$t_{all} = \underbrace{\frac{2Z_w}{Z_w + Z_i} \cdot \frac{2Z_i}{Z_i + Z_w}}_{\text{(1)}} \cdot \underbrace{\sum_{n=0}^\infty \left(\frac{Z_i - Z_w}{Z_i + Z_w}\right)^{2\cdot n}}_{\text{(2)}}$

(1) accounts for the transmission from ice into water and out again. (2) accounts for the back and forth reflection of the sound in the crevasse.

Plugging in $Z_w = 1.48$ and $Z_i = 3.5$, I find $t_{all} = 1$.

Calculation the whole thing for intensities rather than amplitudes, I find $T_{all}=0.72$.

I understand that when dealing with amplitudes, $t > 1$ is possible due to different imepedances and conservation of energy (and happening here for the water -> ice transition). However, I dont understand why after the whole process, it all cancels out and gives me $t_{all} = 1$. I definitly loose some energy due to reflection, and since I beginn in ice and end in ice, this should be mirrored in a loss in amplitude as well.

## 1 Answer

If you consider a longitudinal acoustic wave propagating in ice through a slab of water of thickness $L$, you have the following (complex) pressure amplitude reflection factor $r$ at the water slab $$r=\frac {Z_i-Z_s}{Z_i+Z_s} \tag 1$$ where $Z_i$ is the characteristic wave impedance of ice and $Z_s$ is the input impedance $Z_s$ of the water slab at the ice/water interface at $x=0$. This input impedance $Z_s$ can be considered to be the transformed load impedance of the slab at the transition to the (infinite) ice at $x=L$, which is again the characteristic impedance of ice $Z_i$. The input impedance of the water slab at x=0 is given by $$Z_s=Z_w \frac{Z_i+jZ_w \tan{k_w L}}{Z_w+jZ_i \tan{k_w L}} \tag2$$ where $Z_w$ is the characteristic wave impedance of water, and $k_w=\frac{2\pi}{\lambda_w}$ is the angular wavenumber (wave vector) determined by $\lambda_w$, the acoustic wavelength in water. Thus you do not need to consider any infinite back and forth reflections in the slab. The transmission coefficient $t$ at $x=0$ is $$t=1-r=\frac {2Z_s}{Z_i+Z_s}$$ These formulas are directly equivalent to formulas for electric transmission lines or plane EM waves.(See e.g., S. Ramo et al., Fields and Waves in Modern Communication Electronics, 3rd ed., John Wiley 1994)

As far as I can see, your formulas do not contain the thickness of the crevasse or the wavelength in water. Therefore, they cannot be correct. Check the values for $r$ and $t$ obtained by the formulas given here.

• Already found this useful, thank you. You might save acoustics-oriented folks some trouble and express $\beta$ as the angular wavenumber $k$, instead? Just a thought for ease of reading – D. Betchkal Feb 21 '18 at 5:59
• D. Betchkal - Do you mean that what is here $\beta$, which is the angular wavenumber (or wave vector) , should be better named $k$? This would be no problem. Wave vectors in all fields are mostly designated with $k$. – freecharly Feb 21 '18 at 6:31