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### Plane-WaveScattering 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 . The mathematical details of general sinusoidal plane waves in air and vector wavenumber are reviewed in §B.8.1.

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

as we will now derive.

Let the impedance change be in the plane. Thus, the impedance is for and for . There are three plane waves to consider:

• The incident plane wave with wave vector
• The reflected plane wave with wave vector
• The transmitted plane wave with wave vector
By continuity, the waves must agree on boundary plane:

where denotes any vector in the boundary plane. Thus, at we have

If the incident wave is constant along , then , requiring , leaving

or

 (C.56)

where is defined as zero when traveling in the direction of positive for the incident ( ) and transmitted ( ) wave vector, and along negative for the reflected ( ) wave vector (see Fig.C.18).

Subsections
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