Figure 717 Interaction of Sound Waves with a Surface

Reflected

Absorbed

Inci

Absorbed

Inci

Transmitted absorbed energies. If we divide Eq. 7.42 by E^

Er Et,a

We can express each energy ratio as a coefficient of reflection or transmission. The fraction of the incident energy that is absorbed (or transmitted) at the surface boundary is the coefficient

Et+a

and the reflection coefficient is

Substituting these coefficients into Eq. 7.43,

The reflection coefficient can be expressed in terms of the complex reflection amplitude ratio r for pressure that was defined in Eq. 7.7

and the absorption coefficient is ae = 1 - r2 (7.48)

The reflected energy is

Impedance Tube Measurements

When a plane wave is normally incident on the boundary between two materials, 1 and 2, we can calculate the strength of the reflected wave from a knowledge of their impedances. (This solution was published by Rayleigh in 1896.) Following the approach taken in Eq. 7.3, the sound pressure from the incident and reflected waves is written as p(x) = A ej t- kx) + B ej t+kx) (7.50)

If we square and average this equation, we obtain the mean-squared acoustic pressure of a normally incident and reflected wave (Pierce, 1981)

(p2) = 2 A2 [1 + | r2 | + 2 | r | cos (2kx + <5r)] (7.51)

where 5r is the phase of r. Equation 7.51 describes a standing wave and gives a method for measuring the normal-incidence absorption coefficient of a material placed in the end of a tube, called an impedance tube, pictured in Fig. 7.18.

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