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.
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).
The first equality in Eq.(C.56) implies that the angle of incidence equals angle of 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''.
Longitudinal Waves in Rods