Sums over Histories

The second way is to use the ``sum over histories'' idea developed in the lecture notes. The reflection and transmission amplitudes at the interface of two half-infinite strings can be obtained by analogy with sound waves. For example, for waves going from the light string to the heavy string we have

$$R_{01} = \frac{z_0-z_1}{z_0+z_1};$$

$$T_{01} = \frac{2z_0}{z_0+z_1},$$ (6)

where the impedance for strings is $z=\mu c$.

(a)

What are the transmission and reflection coefficients for waves going from the heavy string to the light string, $R_{10}$ and $T_{10}$?

(b)

What is the complex amplitude (at $x=a$) of the wave that was reflected from the interface between the strings without entering the light string? What about the wave that passed into the light string, was reflected from the boundary, and then passed back into the heavy string?

(c)

Challenge problem! Generalizing your result in part (b), obtain the full complex amplitude of the right-moving wave in the heavy string at $x=a$ by summing over all possible back-and-forth reflections within the light part of the string. Use this result to find $r$.
HINT: To simplify your expression, use the formula for the sum of a geometric series (which also works for complex numbers): $\underline{a}+\underline{a}\, \underline{x}+\underline{a}\, \underline{x}^2+... = \underline{a}/(1-\underline{x})$.