Plane-Wave Scattering

Figure G.16: Plane wave propagation in a medium changing from wave impedance $R_1$ to $R_2$.

Consider a plane wave with peak pressure amplitude $p^+_1$ propagating from wave impedance $R_1$ into a new wave impedance $R_2$, as shown in Fig.G.16. (Assume $R_1$ and $R_2$ are real and positive.) The physical constraints on the wave are that

• pressure must be continuous everywhere, and
• velocity in must equal velocity out (the junction has no state).

Since power is pressure times velocity, these constraints imply that signal power is conserved at the junction.^G.5Expressed mathematically, the physical constraints at the junction can be written as follows:

$$\begin{eqnarray*} p^+_1+p^-_1 &=& p^+_2\quad\mbox{(pressure continuous across ju... ...+}_1+v^{-}_1 &=& v^{+}_2\quad\mbox{(velocity in = velocity out)} \end{eqnarray*}$$

As derived in §G.7.3, we also have the Ohm's law relations:

$$\begin{array}{rcrl} p^+_i&=&&R_iv^{+}_i\\ p^-_i&=&-&R_iv^{-}_i \end{array}$$

These equations determine what happens at the junction.

To obey the physical constraints at the impedance discontinuity, the incident plane-wave must split into a reflected plane wave $p^-_1$ and a transmitted plane-wave $p^+_2$ such that pressure is continuous and signal power is conserved. The physical pressure on the left of the junction is $p_1=p^+_1+p^-_1$, and the physical pressure on the left of the junction is $p_2=p^+_2+p^-_2= p^+_2$, since $p^-_2=0$ according to our set-up.