The computer model used in BIAS is founded upon electronic analogies found in acoustics research. An instrument is treated as a dissipative transmission-line. The sound pressure is analogous to the voltage and flow of the current. Thermoviscouse losses are accounted for by appropriate (complex) resistance. The circuit is terminated with a radiation impedance [Zt], that results in partially reflected waves, as in a real instrument.

The computation of the instrument over its length is done through a series of conical and cylindrical sections. The acoustical transmission properties of these single elements may be exactly determined through frequency-dependent matrices (Ai(f))- see [5] for exact matrices.

*Fig. 1: Sectioning of trumpet tube length (mensur) into smaller elements*

Pressure (p) and velocity (u) before and after the i-th element overlap.

The impedance [Zi] at i results from

The transmission characteristics of the complete instrument are achieved by calculating the product of the partial matrices

The computation of the input impedance is completed in reverse order, beginning with the radiation impedance.

*Fig. 2: An impedance curve simulated using BIAS*

Possible sources of error are:

- the possible erroneous measurement of the inner dimensions (mensur) of the tube
- the assumption of a circular cross-section of the mensur
- that only the fundamental mode is observed.

In the case of a cylindrical tube with a known geometry, these limitations lead to a reliable simulation in the frequency range

d is the maximum tube diameter [m] and c is the sound speed [m/s]. With complicated tube dimensions as they occur in real brass instruments, estimated accuracy of 60-80% is more realistic.

Diameter of the bell |
ideal cut-off frequency |
80% of the cut-off frequency |
60% of the cut-off frequency |

10,0 cm |
2001,0 Hz |
1600,8 Hz |
1200,6 Hz |

12,0 cm |
1667,5 Hz |
1334,0 Hz |
1000,5 Hz |

14,0 cm |
1429,3 Hz |
1143,4 Hz |
857,6 Hz |

16,0 cm |
1250,6 Hz |
1000,5 Hz |
750,4 Hz |

18,0 cm |
1111,7 Hz |
889,3 Hz |
667,0 Hz |

20,0 cm |
1000,5 Hz |
800,4 Hz |
600,3 Hz |

22,0 cm |
909,5 Hz |
727,6 Hz |
545,7 Hz |

24,0 cm |
833,8 Hz |
667,0 Hz |
500,3 Hz |

26,0 cm |
769,6 Hz |
615,7 Hz |
461,8 Hz |

28,0 cm |
714,6 Hz |
571,7 Hz |
428,8 Hz |

30,0 cm |
667,0 Hz |
533,6 Hz |
400,2 Hz |

*Table 1: Validity range according to the formula for variable diameter with c=345m/s*

Within the aforementioned limitations, the position of the resonance peaks may deviate on an average of less than 0.6% (10 cent) in calculations.

The height of the resonance peaks deviates with an error of 15-25%.

Comparisons of simulations with measurements lead to the following observations:

- The computation assumes a sound speed of c=345m/s, which represents a temperature of 23.6 degrees Celsius. If an experiment is performed at another temperature, the basic tuning of the instrument is no longer constant.
- The measurement system of BIAS replaces the lips of the musician with a rubber disc with protrudes into the mouthpiece and reduces its volume. The mensur must be accordingly shortened by 2mm.

[1] Caussé, René; Kergomard, J.; Lurton, X. (**1984**)

"**Input Impedance of Brass Musical Instruments -- Comparison between Experiment and Numerical Models**,"

Journal of the Acoustical Society of America (JASA)**75**, 241-254.

"

Journal of the Acoustical Society of America (JASA)

[2] Keefe, Douglas H. (**1984**)

"**Acoustical wave propagation in cylindrical ducts: Transmission line parameter approximations for isothermal and nonisothermal boundary conditions**,"

Journal of the Acoustical Society of America (JASA)**75**, 58-62.

"

Journal of the Acoustical Society of America (JASA)

[3] Lampton, M (**1978**)

"**Transmission Matrices in Electroacoustics**,"

Acustica**39**, 239 ff.

"

Acustica

[4] Levine, H.; Schwinger, J. (**1948**)

"**On the radiation of sound from an unflanged circular pipe**,"

Phys. Rev.**73**, 383-406.

"

Phys. Rev.

[5] Mapes-Riordan, Dan (**1993**)

"**Horn Modeling with Conical and Cylindrical Transmission-Line Elements**,"

Journal of the Audio Engineering Society**41**, 471-483.

"

Journal of the Audio Engineering Society

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