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### Wave Impedance, Reflection from a Radius Mismatch

In the previous section, we identified the right- and left-traveling components of two key quantities describing wave propagation in an acoustic tube: the pressure in the tube ( ), and the volume velocity in the tube ( ). For the right- and left-traveling components, it turns out we can relate them using relatively simple formulas. Using a combination of calculus, Newton's laws of motion, and the law of conservation of matter, it can be shown that the right-traveling pressure and volume velocity components obey the following formula:

 (7)

where is the wave impedance in the tube, given by the following formula:
 (8)

where is the ambient fluid density in the tube, is the velocity of wave propagation (see Equation (5)), and is the cross-sectional area of the tube. Thus, the wave impedance relates pressure to volume-velocity everywhere along a plane wave traveling to the right along the axis of an acoustic tube having cross-sectional area .

Similarly, for the left-traveling wave components, it may be shown that

 (9)

It is next interesting to consider what happens to a traveling pressure waveform in an acoustic tube when it encounters a radius mismatch. In other words, what happens when the waveform is traveling through an initial tube with radius , and all-of-a-sudden is transferred into a tube with a second disparate radius ? It turns out that part of the waveform will be reflected back into the first tube, and the strength of the reflection is given by the following formula:

 (10)

where is the wave impedance in the first tube section, and is the wave impedance in the second tube section. Using the previous formulas, it may be further shown that the reflectance is also given by the following formula for cylindrical tubes:
 (11)

where is the radius of the first tube, and is the radius of the second tube.

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