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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 H.19 shows the more general situation (as compared to Fig.H.16) 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 §F.5.4.

Figure H.19: Sinusoidal plane wave scattering at an impedance discontinuity--oblique angle of incidence $ \theta _1^+$.
\begin{figure}\input fig/planewavescatangle.pstex_t \end{figure}

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

$\displaystyle \zbox {k_1\sin(\theta_1^+) =k_1\sin(\theta_1^-) =k_2\sin(\theta_2^+)} \protect$ (H.58)

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 (