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Plane-Wave Scattering

Figure C.15: Plane wave propagation in a medium changing from wave impedance $ R_1$ to $ R_2$ .
\includegraphics{eps/planewavescat}

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.C.15. (Assume $ R_1$ and $ R_2$ are real and positive.) The physical constraints on the wave are that

Since power is pressure times velocity, these constraints imply that signal power is conserved at the junction.C.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 junction)}\\ v^{+}_1+v^{-}_1 &=& v^{+}_2\quad\mbox{(velocity in = velocity out)} \end{eqnarray*}

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

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

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 right of the junction is $ p_2=p^+_2+p^-_2= p^+_2$ , since $ p^-_2=0$ according to our set-up.


Subsections
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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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