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Subsections
 Acoustic plane waves in three dimensions are described by

(4.39) 
with a wavenumber vector

(4.40) 
 In three dimensions, the displacement of the medium by a plane
wave is a vector function with components given by
 The wavenumber vector and the displacement vector
are collinear. is always parallel to the
velocity of the wave, while oscillates along the
velocity vector.
4.3.2 Planar Boundaries
Figure 4.2:
Wave fronts of plane waves incident on, reflected from, and
refracted by a planar interface between media.

Snell's Law
 Consider a planar boundary at between two materials with
, and , .
 The frequency is the same on both sides of the boundary, since
the transmitted wave is generated by the oscillations of the incident
and reflected waves at the boundary.
 With identical frequencies and different phase velocities
and
, we expect different wavenumbers in
the two media.
 The reflected and incident waves have wavenumbers of the same
magnitude,

(4.42) 
because they have the same frequency and are in the same
medium. Reflection means

(4.43) 
and since nothing about the medium changes along

(4.44) 
 Eq.'s 4.43 and 4.44 give the law
of reflection  that the angle of incidence equals the angle of
reflection, as shown in Fig. 4.2.
 The pressure and displacement wave functions must be continuous
in order to satisfy the wave equation. Along the boundary, at ,
and choosing , this constraint is expressed
which with Eq. 4.44 simplifies to

(4.45) 
 Eq. 4.45 can only hold for all if the
exponents are equal. That is,
or
which can be written in terms of the phase velocities in the two media
as

(4.46) 
which is known as Snell's law.
Reflection and Transmission Coefficients
 The equality of the exponents in Eq. 4.45
relates the pressure amplitudes across the boundary

(4.47) 
 An identical argument yields

(4.48) 
for the component of the displacement normal to the boundary.
 Generalizing Eq. 4.4 to three dimensions, we
can make the following connection between the displacement component
normal to the boundary and the pressure,
and therefore
which with Eq.'s 4.48 and Eq. 4.43
yields

(4.49) 
 Using the expression of Eq. 4.14 of the
phase velocity, and defining the acoustic impedance

(4.50) 
Eq. 4.49 can be written in the form

(4.51) 
 The sum of Eq.'s 4.47 and 4.51
gives the transmission coefficient

(4.52) 
 Dividing Eq. 4.47 by gives an
expression relating the transmission coefficient to the
reflection coefficient

(4.53) 
 Although and depend on both
and , Snell's law (Eq. 4.46)
can be used to relate them to a single angle.
 There are two characteristic angles associated with a planar
interface, Brewster's angle, the angle at which no reflection
occurs, and the critical angle, the angle above
which total internal reflection occurs. You will consider these for
homework.
 For normal incidence,
, and the incident, reflected, and transmitted waves lie along a
single axis. Eq.'s 4.52 and 4.53 become

(4.54) 
and

(4.55) 
 Caution! The reflection and transmission coefficients
and above describe amplitudes, not intensities. You will
use these coefficients to derive similar coefficients for intensity
for homework.
Next: 4.4 The Doppler Effect
Up: 4. Acoustic Waves
Previous: 4.2 Energy, Power, and
Copyright © 20032009, Lewis A. Riley

Updated Wed Jan 18 09:51:28 2006
