Plane-Wave Scattering at an Angle

Figure G.19 shows the more general situation (as compared to Fig.G.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 §E.5.4.

Figure G.19: Sinusoidal plane wave scattering at an impedance discontinuity--oblique angle of incidence $\theta _1^+$.

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:

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

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

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

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

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

or

$$\zbox {k_1\sin(\theta_1^+) =k_1\sin(\theta_1^-) =k_2\sin(\theta_2^+)} \protect$$ (G.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 ( $\underline{k}_1^-$) wave vector (see Fig.G.19).