### 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.- 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:*

*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

*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].

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Longitudinal Waves in Rods

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Plane-Wave Scattering