Region 1

Eq. (5) also gives the solution in Region 1,

$$s_1(x_1 \ge 0,t) = T_{0 \rightarrow 1} t_0\left(\frac{c_0}{c_1}(x_1-c_1 t)\right)$$

$$= T_{0 \rightarrow 1} {\mbox{Re\,}}\left(\underline{A} e^{ik_0\left(\frac{c_0}{c_1}(x_1-c_1) \right)}\right)$$

$$= {\mbox{Re\,}}\left(\underline{A} T_{0 \rightarrow 1} e^{i\frac{c_0 k_0}{c_1} x_1} e^{-i c_0 k_0 t} \right)$$

$$= {\mbox{Re\,}}\left(\underline{A} T_{0 \rightarrow 1} e^{i k_1 x_1} e^{-i \omega t} \right)$$

$$= {\mbox{Re\,}}\left(e^{-i\omega t} \underline{A} T_{0 \rightarrow 1} e^{i k_1 x_1} \right)$$

$$= {\mbox{Re\,}}\left( e^{-i\omega t} \underline{Q}_1(x_1) \right),$$ (10)

where, in the second line from the bottom, we have used the dispersion relation of Region 1 to identify $(c_0/c_1) k_0=\omega/c_1$ as just the wave vector $k_1$ from Region 1. Also, in the last line we define a new complex amplitude for the wave motion in Region 1,

$$\underline{Q}_1(x_1\ge 0) \equiv \underline{A} T_{0 \rightarrow 1} e^{i k_1 x_1}.$$ (11)

Thus, we once again find that all points in the system respond with simple harmonic motion at the same driving frequency $\omega$, so that knowledge of $\underline{Q}_1(x_1)$ completely determines the motion in Region 1. For instance, the amplitude of the motion for each point in Region 1 is $\left\vert\underline{Q}_1(x_1)\right\vert = \left\vert\underline{A} T_{0 \right... ...t\vert\underline{A}\right\vert \cdot \left\vert T_{0 \rightarrow 1} \right\vert$, just the amplitude of the incoming wave times the magnitude of the transmission amplitude.

Finally, we see that we can again write this result directly in terms of relative complex amplitudes. Relative to the motion at point $x=a$, the motion of the transmitted wave at $x_1$ should contain a factor of $e^{ik_0(-a)}$ to give the motion of the incoming wave at $x=0$, a factor of $T_{0 \rightarrow 1}$ to give the motion of the transmitted wave at $x=0$ relative to the motion of the incoming wave at the same point, and a factor of $e^{ik_1x_1}$ to give the motion of the traveling wave after propagating the distance $\vert L\vert=x_1-0$ from $x=0$ to the point $x_1$. We thus can immediately write a result equivalent to (10),

$$s_1(x_1,t)= {\mbox{Re\,}}\left( e^{-i\omega t} \underline{A}... ...} \cdot T_{0\rightarrow 1} \cdot e^{ik_1 x_1} \right] \right).$$

Note that in the case we do not have a sum of terms because only a single wave contributes to the motion in Region 1. (See Figure 1.)