PHYS212 : Lectures

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Subsections

# 4.3 Reflection, Refraction, and Transmission

## 4.3.1 Plane Waves in Three Dimensions

• 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
 (4.41)

• 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

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 © 2003-2009, Lewis A. Riley Updated Wed Jan 18 09:51:28 2006