Physical realization

As described above, transmission and reflection from a change in medium is quite general and applies to all types of waves. For concreteness, we focus on the propagation of sound as an example. As we move along in the analysis we will also describe briefly the analogous results for other types of waves in a series of footnotes.

Figure 1 illustrates our realization. The system consists of two regions, Region 0 (all points $x<0$) and Region 1 (all points $x>0$). The material in Region 0 may be a gas, fluid, or solid, and is characterized by an equilibrium mass density, bulk modulus and pressure of $\rho_0$, $B_0$, and $P_0$, respectively. Region 1 consists of some other medium with corresponding quantities $\rho_1$, $B_1$ and $P_1$. Finally, if necessary to keep the media from mixing (if they are gas or fluid), we may include a thin, massless membrane at $x=0$. We let this membrane be completely free to move without friction.

Figure 1: Physical realization for wave behavior at a change in medium: a massless membrane of cross-sectional area $A$ separates regions filled with two different media.

Before any waves are introduced into this combined system, we allow the two media to reach equilibrium with each other. In equilibrium, the membrane will not move, and so the net force on it in the x-direction, $F=A\cdot(P_0-P_1)$, where $A$ is the total area of the membrane, must be zero. Hence, $P_0=P_1$, and the pressures on either side are in balance. Because these two quantities are equal, we shall henceforth refer to them both as $P \equiv P_0=P_1$¹.