Degrees of Freedom

The degrees of freedom for this system are the sound displacement for each point in either region and the displacement of the membrane. We denote the displacements in Region 0 as $s_0(x<0,t)$ and in Region 1 as $s_1(x>0,t)$. The membrane follows the motion of the material of the neighboring points on either side of it. Its position, $x_{\mathrm{mem}}(t)$ thus can be determined equally well in two different ways, either by considering Region 0 or Region 1, $x_{\mathrm{mem}}(t)=s_0(x=0,t)=s_1(x=0,t)$.

Thus, the degrees of freedom (minimal set of variables needed to describe the state of the system at any time $t$) for the combined system are the values $s_0(x\le 0,t)$ and $s_1(x\ge 0,t)$, with the extra constraint (because this description is slightly redundant) of consistency,

$$s_0(x=0,t)=s_1(x=0,t),$$ (1)

which, mathematically, is considered our first boundary condition at $x=0$.²