Recipe

Find a transmission-line or acoustics system's impedance and speed

Use when: you have $L,C$ for a transmission line, or $\rho_0,c$ for acoustics, and need the characteristic impedance and propagation speed

Goal: the impedance ( $Z_0$ or $\rho_0c$ ) and the wave speed, correctly matched to the physical system given

Taught in Linear transmission-line and acoustics examples

Quick recipe

Identify which physical system you're given
Translate one system's formula with the whole dictionary, if needed
Compute the impedance and speed with matched units

1
Decide whether the data describes a transmission line ( $L,C$ ) or acoustics ( $\rho_0,c$ ). The two systems use different symbols for the same underlying $2\times2$ structure.

Example

$L=0.5\,\mu\text{H/m}$ , $C=200\,\text{pF/m}$ describes a transmission line, while $\rho_0=1.2\,\text{kg/m}^3$ , $c=343\,\text{m/s}$ describes acoustics.
- transmission line — find $Z_0=\sqrt{L/C}$ and $c_0=1/\sqrt{LC}$ .
- acoustics — the impedance $\rho_0c$ is already given directly, and the eigenvalues are $\lambda=\pm c$ .
Why: The impedance formula depends on which system's variables are being used. Picking the wrong system gives the wrong formula from the start.
2
To reuse a formula from one system in the other, swap in the WHOLE dictionary at once: $L\leftrightarrow\rho_0$ , $1/C\leftrightarrow\rho_0c^2$ . Do not swap in just one symbol alone.

Example

Translating $Z_0=\sqrt{L/C}$ to acoustics: $L\to\rho_0$ and $1/C\to\rho_0c^2$ , so $Z_0\to\sqrt{\rho_0\cdot\rho_0c^2}=\sqrt{\rho_0^2c^2}=\rho_0c$ .

Why: Each matrix entry in $A$ mixes several physical constants together. Swapping symbols one at a time ignores which entry each one replaces.
3
Plug the identified system's own numbers into its own formula. Keep every unit consistent. For instance, change $\mu\text{H}$ and $\text{pF}$ to base SI units before dividing.

Example

$Z_0=\sqrt{(0.5\times10^{-6})/(200\times10^{-12})}=\sqrt{2500}=50\,\Omega$ ; $c_0=1/\sqrt{(0.5\times10^{-6})(200\times10^{-12})}=10^8\,\text{m/s}$ .

Why: The impedance and speed formulas depend closely on units. Units left in the wrong scale give a wrong number, even with the right formula.

Common mistakes

Applying the transmission-line formula directly to acoustics data, or vice versa, without translating through the dictionary first.
Translating the dictionary symbol by symbol, instead of swapping into the whole formula at once.
Leaving units in the wrong scale, such as $\mu\text{H}$ or $\text{pF}$ , before dividing. This gives a wrong impedance, even with a correct formula.
Treating $Z_0$ and $\rho_0c$ as two different things, rather than the same role played by two different physical systems.