Particular Solutions

From the final general solution (16), it is relatively easy to determine particular solutions for when a single pulse is incident either from Region 0 or Region 1, thereby confirming the interpretations giving in Section 3.3.3. For instance, if a pulse of shape $f(u)$ comes in from Region 0, then $t_0(u)=f(u)$ and $t_1(v)=0$. Thus,

$$s_0(x \le 0,t) = f(x-c_0 t) + R_{0 \rightarrow 1} f\left(-(x+c_0 t)\right)$$ (17)

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

Thus, we have a reflected pulse, horizontally flipped and vertically scaled by $R_{0 \rightarrow 1}$, traveling to the left in Region 0 with speed $c_0$, and we have a transmitted pulse, stretched horizontally by $c_1/c_0$ and scaled vertically by $T_{0 \rightarrow 1}$, traveling to the right in Region 1 at speed $c_1$. Alternately, if a pulse of shape $f(u)$ comes in from Region 1, then $t_1(u)=f(u)$ and $t_0(v)=0$, and

$$s_0(x \le 0,t) = T_{1 \rightarrow 0} f\left(\frac{c_1}{c_0} (x+c_0 t) \right)$$ (18)

$$s_1(x \ge 0,t) = f(x+c_1 t) + R_{1 \rightarrow 0} f\left(-(x-c_1 t)\right).$$

In this case, we then have a reflected pulse, horizontally flipped and vertically scaled by $R_{1 \rightarrow 0}$, traveling to the right in Region 1 at speed $c_1$, and a transmitted pulse, horizontally stretched out by a factor $c_0/c_1$ and vertically scaled by $T_{1 \rightarrow 0}$, traveling to the left in Region 0 at speed $c_0$.