Reflection coefficient: Acoustics vs Mechanics

Question

I recently tried to derive the reflection coefficient $R$. This is not a complicated task, however after making some literature research I found two derivations which arrive at seemingly different results:

1. The first derivation is from Griffiths ‚Introduction to Electrodynamics‘. In chapter 9.1.4 he derives the reflection coefficient of a 1D string to be $$R=\frac{c_2–c_1}{c_1+c_2}$$ where $c_{i}=\frac{\omega_i}{k_i}$ is the wave velocity, $k$ the wave number and $\omega$ the angular frequency.
2. The second derivation is from this webpage and refers to the transmission and reflection of sound waves. There, they arrive at the result $$R=\frac{r_2–r_1}{r_2+r_1}= \frac{\rho_2c_2–\rho_1c_1}{\rho_2c_2+\rho_1c_1} $$ with $r_i=\rho_i c_i$ the so-called acoustic impedance and $\rho$ and $c$ are the density and wave velocity of the medium.

Both derivations look very similar and I am not sure why the solution differ. Maybe someone of you has an idea where the difference comes from?

My attempt

I first thought that there may be some differences in the boundary conditions, because we also treat different systems (strings vs sound waves). However, it seems that the boundary conditions in both derivations seem to be the same. For example: The continuity condition used by Griffiths ($f(0^–,t)=f(0^+,t)$) seems to be the same as for acoustic waves ($\dot{f}(0^–,t)=\dot{f}(0^+,t) $), because they only differ by a factor $\omega$ which is a constant in space an cancles on both sides.

Answer 1

The first derivation of the 1D string makes the assumption that that the tension in both strings is equal and the only difference is mass ( density per unit length).

The second derivation assumes mass is different and the force (pressure) between the two media is different as well.

If you derive the 1D problem assuming the string tensions are different your result will be closer to the second derivation.

Edit to address Lockharts comment

The easiest way to compare these two derivations is by using the concept of impedance. This is used in the second derivation. This pdf explains impedance for mechanical waves in different media including strings, air. https://homepage.physics.uiowa.edu/~fskiff/Physics_044/Impedance.pdf

The reflection coefficient of any wave can be expressed as a function of impedance:

$$ R = \frac{Z_1 - Z_2}{Z_1+Z_2}\text{ (1)}$$ where Z is the impedance. This is the formula used in the second derivation.

For a string, the impedance is given by:

$$Z = T/v$$ where T is the tension of the string and v is the phase velocity.

The Griffith's derivation states that the tension is the same in both strings so : $$ Z_1 = T/v_1 \text{ and } Z_2 = T/v_2$$ Plugging these values into equation (1) gives the result given by Griffith's.

If the tension is not the same in both strings then:

$$ Z = \frac{T_1v_2 - T_2v_1}{T_1v_2+T_2v_1}$$

Which closely resembles the results of the second derivation. Hope this helps.