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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.
Figure C.18:
Sinusoidal plane wave scattering at an
impedance discontinuityoblique angle of incidence
.

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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