Final form for general solution

Finally, we may substitute the forms (12,13) for the outward moving pulses into (4) to find a valid general solution, one which satisfies the internal equations of motion for each region and for the membrane and which contains the required number of adjustable parameters (two adjustable functions). Using the definitions (14,15) of the various reflection/transmission amplitudes, the final result is

$$s_0(x \le 0,t) = t_0(x-c_0 t) + R_{0 \rightarrow 1} t_0\left(-(x+c_0 t)\right) + T_{1 \rightarrow 0} t_1\left(\frac{c_1}{c_0} (x+c_0 t) \right)$$ (16)

$$s_1(x \ge 0,t) = t_1(x+c_1 t) + R_{1 \rightarrow 0} t_1\left(-(x-c_1 t)\right) + T_{0 \rightarrow 1} t_0\left(\frac{c_0}{c_1} (x-c_1 t)\right).$$