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) |

(8) |

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

Download vir_tube.pdf

REALSIMPLE Project — work supported by the Wallenberg Global Learning Network .

Released

Center for Computer Research in Music and Acoustics (CCRMA), Stanford University