Acoustic Tubes

In the case of the acoustic tube [Morse, 1981, Markel and Gray, 1976], we have the analogous relations... 

In the acoustic tube literature which involves only a cascade chain of acoustic waveguides, x+ is taken to be traveling to the right along the axis of the tube [Markel and Gray, 1976], In classical network theory [Belevitch, 1968] and in circuit theory, velocity (current) at the terminals of an V-port device is by convention taken to be positive when it flows into the device. 

Smith, 1991] Smith, J. O. (1991). Waveguide simulation of non-cylindrical acoustic tubes. In Proc. 1991 Int. Computer Music Conf, Montreal, pages 304-307. Computer Music Association. 

For example, Ft(s) could be pressure in an acoustic tube and 1 (s) the corresponding volume velocity. In the parallel case, the junction reduces to the unloaded case when the load impedance Rj(s) goes to infinity. 

In nearly all natural wave phenomena, losses increase with frequency. Distributed losses due to air drag and internal bulk losses in the string tend to increase with frequency. Similarly, air absorption increases with frequency, adding loss for sound waves in acoustic tubes or open air [Morse and Ingard, 1968],... 

Valimaki, 1995) Valimaki, V. (1995). Discrete-Time Modeling of Acoustic Tubes Using Fractional Delay Filters. PhD thesis, Report no. 37, Helsinki University of Technology Faculty of Elec. Eng., Lab. of Acoustic and Audio Signal Processing, Espoo, Finland... 

Speech spectra have important features called formants which are the three five gross peaks in the spectral shape located between 200 and 4000 Hz. These correspond to the resonances of the acoustic tube of the vocal tract. We will discuss formants further in Chapter 8 (Subtractive Synthesis). For now, we will note that the formant locations for the ahh vowel are radically different from those for the eee vowel, even though the harmonic spacing (and thus, the perceived pitch) is the same for the two vowels. We know this, because a singer can sing the same pitch on many vowels (different spectral shapes, but same harmonic spacings), or the same vowel on many pitches (same spectral shape and formant locations, but different harmonic spacings). 

This chapter extends our modeling techniques from solids to tubes and chambers of air (or other gasses). First we will look at the simple ideal acoustic tube. Next, we will enhance the model to create musical instruments like a clarinet. Then we ll look at our first three-dimensional acoustical systems— air cavities— but we ll discover that many such systems can be treated as simpler (essentially lumped resonance) models. 

Figure 11.1 shows a simple cylindrical acoustic tube of cross-sectional area a. The force acts to move air in the x direction along the tube. 

We noted that the ideal string equation and the ideal acoustic tube equation are essentially identical. Just as there are many refinements possible to the plucked-string model to make it more realistic, there are many possible improvements for the clarinet model. Replacing the simple reed model with a variable mass/spring/damper allows the modeling of a lip reed as is found in... 

Just as we proved the D Alembert solution to the ideal string (Section B. 1.2), we know that a solution to Equation C.9 for the acoustic tube is ... 

We can relate pressure and velocity in the acoustic tube by going back to Equation C.5 ... 

C.4 The Acoustic Tube fith Varying Cross-Sectional Area... 

Figure C.2. Junction of two acoustic tubes with different areas.

The variable k is called the scattering coefficient, (from wave scattering theory) and the equations are called scattering equations. They express the behavior of the wave equation at the boimdary between acoustic tube segments of different characteristic impedances, where part of the incoming wave is... 

Figure C.3. Network of acoustic tube segments, and ladder filter Implementation using chain of scattering junctions.

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