### 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 . 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

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).

#### Reflection and Refraction

The first equality in Eq.(C.56) implies that the
*angle of incidence equals angle of reflection:*

We now wish to find the wavenumber in medium 2. Let denote the phase velocity in wave impedance :

*refraction*of the plane wave as it passes through the impedance-change boundary. The refraction angle depends on ratio of phase velocities . This ratio is often called the

*index of refraction*:

*Snell's Law*(of refraction).

#### Evanescent Wave due to Total Internal Reflection

Note that if
, the horizontal component
of the wavenumber in medium 2 becomes *imaginary*. In this case,
the wave in medium 2 is said to be *evanescent*, and the wave in
medium 1 undergoes *total internal reflection* (no power travels
from medium 1 to medium 2). The evanescent-wave amplitude decays
exponentially to the right and oscillates ``in place'' (like a
standing wave). ``Tunneling'' is possible given a
medium 3 beyond medium 2 in which wave propagation resumes.

To show explicitly the exponential decay and in-place oscillation in an evanescent wave, express the imaginary wavenumber as . Then we have

Thus, an imaginary wavenumber corresponds to an exponentially decaying
evanescent wave. Note that the time dependence (cosine term) applies
to *all points* to the right of the boundary. Since evanescent
waves do not really ``propagate,'' it is perhaps better to speak of an
``evanescent acoustic field'' or ``evanescent standing wave''
instead of ``evanescent waves''.

For more on the physics of evanescent waves and tunneling, see [295].

**Next Section:**

Longitudinal Waves in Rods

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Plane-Wave Scattering