Solution within each region

We already have the general solutions to (2),

$$s_0(x<0,t) = t_0(x-c_0 t) + a_0(x+c_0 t)$$ (4)

$$s_1(x>0,t) = t_1(x+c_1 t) + a_1(x-c_1 t).$$

Here, to properly solve the respective wave equation, we use the wave speed appropriate to each region, $c_0$ and $c_1$, respectively. As a matter of notation, rather than labeling the forward and backward traveling pulse shape functions in each region as $f(x-ct)$ and $g(x+ct)$, respectively, we now name them so that $t_0(x-c_0 t)$ describes a pulse in Region 0 traveling toward the interface at speed $c_0$, $a_0(x+c_0 t)$ describes a pulse in Region 0 traveling away from the interface, and $t_1(x+c_1 t)$ and $a_1(x-c_1 t)$ describe pulses moving in Region 1 either toward or away from the interface, respectively. We introduce this labeling here because, ultimately, we are interested in knowing what reflections and transmissions comes out from the interface, given what we send into it.

There are two reasons why (2) is not a general solution to the overall set of equations of motion. First, it has too many arbitrary parameters. Each point on either side of the interface (or molecule, if you prefer to count discrete objects) obeys a second order (in time) equation of motion. The general solution thus must have two arbitrary parameters for each point in space. However, there are four arbitrary functions in (4), each of which can take any value for each value of its argument. Thus, there are four adjustable parameters per point in space, rather than just two. We must therefore be able to eliminate two of the four functions in (4) using some other constraints in order to leave ourselves with just two adjustable functions.

The second difficulty with (4) gives us the additional constraints to resolve the first difficulty. Although (4) solves the equations of motion for the interior points, it does not necessarily solve the equations of motion for the membrane (1,3). These conditions translate into relations among the $t()$'s and $a()$'s which will allow us to eliminate the extra freedom while satisfying the equation of motion for all points in the system.