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Plane-Wave Scattering at an Angle

Figure C.18 shows the more general situation (as compared to Fig.C.15) of a sinusoidal traveling plane wave encountering an impedance discontinuity at some arbitrary angle of incidence, as indicated by the vector wavenumber $ \underline{k}_1^+$ . The mathematical details of general sinusoidal plane waves in air and vector wavenumber are reviewed in §B.8.1.

Figure C.18: Sinusoidal plane wave scattering at an impedance discontinuity--oblique angle of incidence $ \theta _1^+$ .
\includegraphics{eps/planewavescatangle}

At the boundary between impedance $ R_1$ and $ R_2$ , we have, by continuity of pressure,

\begin{eqnarray*}
k_1\sin(\theta_1^+)
&=&k_1\sin(\theta_1^-)
\;=\;k_2\sin(\theta_2^+)
\end{eqnarray*}

as we will now derive.

Let the impedance change be in the $ \underline{x}=(0,y,z)$ plane. Thus, the impedance is $ R_1$ for $ x\le0$ and $ R_2$ for $ x>0$ . There are three plane waves to consider:

By continuity, the waves must agree on boundary plane:

$\displaystyle \left<\underline{k}_1^+,\underline{r}\right> = \left<\underline{k}_1^-,\underline{r}\right> = \left<\underline{k}_2^+,\underline{r}\right>
$

where $ \underline{r}=(0,y,z)$ denotes any vector in the boundary plane. Thus, at $ x=0$ we have

$\displaystyle k_{1y}^+\,y + k_{1z}^+\,z
= k_{1y}^-\,y + k_{1z}^-\,z = k_{2y}^+\,y + k_{2z}^+\,z.
$

If the incident wave is constant along $ z$ , then $ k_{1z}^+=0$ , requiring $ k_{1z}^- = k_{2z}^+ = 0$ , leaving

$\displaystyle k_{1y}^+\,y = k_{1y}^-\,y =k_{2y}^+\,y
$

or

$\displaystyle \zbox {k_1\sin(\theta_1^+) =k_1\sin(\theta_1^-) =k_2\sin(\theta_2^+)} \protect$ (C.56)

where $ \theta$ is defined as zero when traveling in the direction of positive $ x$ for the incident ( $ \underline{k}_1^+$ ) and transmitted ( $ \underline{k}_2^+$ ) wave vector, and along negative $ x$ for the reflected ( $ \underline{k}_1^-$ ) wave vector (see Fig.C.18).



Subsections
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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA