Transmission and Reflection in Strings (II)

A thin string (mass per unit length $\mu_0$, wave speed $c_0$) of length $a$ is attached to a wall at $x=0$. The other end of the string is attached to a thicker string (mass per unit length $\mu_1$, wave speed $c_1$), see Fig. 2. A sinusoidal wave of amplitude $A$ and wavevector $k_1$ travels in from the right along the thick string and is then reflected. Using complex representation, the motion of the string can be described by

$$y_0(x<a,t) = \Re [e^{-i\omega t} (\underline{B} e^{-i k_0 x} + \underline{C} e^{i k_0 x})],$$

$$y_1(x>a,t) = \Re [\underline{A} e^{-i\omega t} (e^{-i k_1 (x-a)} + \underline{r} e^{i k_1 (x-a)})].$$ (4)

where $\underline{r}$ is the reflection coefficient. In this problem, you will find $\underline{r}$ in two different ways.

Figure: Transmission and reflection in strings.