Matching at the Boundary

The first way involves using boundary conditions at $x=0$ and $x=a$:

(a)

Using the boundary condition at $x=0$, show that the motion of the light string is described by

$$y_0(x<a,t)=\Re [e^{-i\omega t} \underline{D} \sin(k_0 x)].$$ (5)

How is the coefficient $\underline{D}$ related to $\underline{B}$ and $\underline{C}$?

(b)

Write down the equations that $y_0(x,t)$ and $y_1(x,t)$ should satisfy at $x=a$. What is the physical meaning of each equation?

(c)

Solve these equations to obtain $r$. Express your answer in terms of $c_0$, $c_1$, $k_1$, and $a$.