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Physical Audio Signal Processing
    Digital Waveguide Theory
       Scattering at Impedance Changes
          Plane-Wave Scattering at an Angle

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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:

  • The incident plane wave with wave vector $ \underline{k}_1^+$
  • The reflected plane wave with wave vector $ \underline{k}_1^-$
  • The transmitted plane wave with wave vector $ \underline{k}_2^+$
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
Previous: Reflection Coefficient
Next: Reflection and Refraction

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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