Figure 7.16.Comparison of measured spectra and simulated quantities. Experimental pipe #2,hS= 20 mm, wS= 12 mm, andLS= 117 mm. Left: Measurement at the mouth opening and the simulated input admit-tance. Right: Spectrum measured and simulated at the slot.
As it can be seen, a very good match of the spectral baselines and simulated functions is achieved in the whole frequency range. The calculated input admittance closely follows the baseline of the spectrum measured at the mouth opening. Higher eigenfrequencies (from the sixth mode) are also clearly followable in the measured spectrum. The abrupt inharmonicity appearing near the 5th–6thmode can clearly be observed both in the measured spectrum and in the simulated input admittance function.
As far as the spectrum at the slot is concerned, it can be seen that the baseline is predicted with good accuracy using the numerical tuning slot model. The quasi-periodic envelope structure is attained with good precision. Similarly good match of the calculated functions and measured spectra has been found using different slot geometries on both pipes.
7.7 Conclusions
The design parameters of tuning slots were shown to have a great effect on the eigenfrequency-structure and the resulting sound quality in the previous chapter. The purpose of this chapter was to find a suitable acoustic model of the tuning slot, by which the effects observed in the measurements can be foretold.
Therefore, woodwind tonehole models were reviewed, with attention paid on the geometri-cal dissimilarities and the limitations of the common T-circuit model. By adapting the simulation technique proposed by Lefebvre & Scavone [91, 92], a numerical tuning slot model was set up utilizing the finite element and perfectly matched layer methods. The analytical tonehole and numerical tuning slot models were applied in the one-dimensional waveguide model of various pipes. Results from different models were directly compared to measurement data, examining the fundamental and the eigenfrequency-structure. The presented measurements and compar-isons cover a wide range of slot parameters, regarding the effective radius and the oblongness of the slot.
The observed discrepancies between the models were explained by the examination of the equivalent lengths. It was shown that there is a remarkable deviation in the prediction of the series length correctionta for large (δ ≈ 1) and oblong (hS/wS ≥ 2) slots. Also the frequency dependence of the shunt length correction differs greatly in the numerical and analytical models.
The latter difference is explained by the very low ratio of wall thickness to effective slot radius (t/b), where limitations of traditional analytical tonehole models are already exceeded.
It was found that both models can predict the fundamental frequency with good accuracy, except for the case of highly oblong slots (hS/wS ≈8), where the analytical tonehole model un-derestimates the fundamental to a significant extent. The eigenfrequency-structure of the pipe is also predicted reliably by means of the finite element slot model, whereas significant dissimilari-ties of the two models were observed especially in the middle frequency range.
Finally, it can be stated that the proposed hybrid (waveguide/finite element) model can suc-cessfully be applied for the simulation of labial organ pipes with tuning slots. The model is capable of foretelling the most important acoustic properties of the resonator with good accu-racy. Using the database created from the more than fifty finite element models, simulation of further tuning slot pipes can readily be performed. With the reliable prediction of the fundamen-tal frequency and the eigenfrequency-structure, better control over the sound characteristics of tuning slot pipes could be attained in the future, exceeding the limitations of slot design rules currently applied in organ building practice. The latter goals could be achieved by incorporating the results of the acoustic model presented in this chapter into the design approach proposed in the previous chapter.
Chapter 8
Modeling the resonators of reed organ pipes
This chapter presents a hybrid technique for the acoustical modeling of resonators and shallots of lingual organ pipes. The proposed approach combines the one-dimensional model of an axisym-metric resonator and the three-dimensional finite element method extended by infinite elements for the simulation of the radiation impedance. From the FE/IE simulations a scalable database is created which facilitates the incorporation of the numerical results into the one-dimensional model. A simple low frequency approximation is applied to model the acoustic radiation from the shallot into the pipe boot. The introduced technique is validated by means of comparisons to transfer function measurements performed on resonators of different forms. The results indicate that the incorporation of the finite element results into the model leads to a remarkable improve-ment in the prediction of the natural resonance frequencies of the resonator. This chapter is a revised and extended version of the conference paper [C11].
8.1 Introduction
Unlike labial organ pipes discussed in Chapters 5–7, the pitch of lingual organ pipes is deter-mined by the frequency of the vibrating tongue, rather than the acoustic modes of the resonator.
Nevertheless, the resonator has an essential role in forming the timbre of pipe. Furthermore, due to the acoustic feedback mechanism, the resonator can also affect the frequency of the tongue
vi-bration in a smaller or greater extent, depending on the strength of their coupling. Although symmetric resonators the
resonators of lingual pipes can have various forms, a great number of stops (such asCrumhorns, Trumpets,Oboesetc.) consist of axisymmetric pipes having conical and cylindrical parts. To be able to characterize such resonators acoustically, the one-dimensional transmission line model of cylindrical ducts—introduced in Section 3.3.2—is extended in this chapter to conical and com-posite waveguides.
Discussions with organ builder partners in the framework of a European project [3] have re-vealed that there are no common rules for designing the resonators of reed pipes. Measurements also prove that the design traditions applied currently in practice do not exploit the capabilities
of the resonator to a full capacity. Toarrive at a better efficiency of resonator design—i.e., to be objective able to scale the resonator based on the desired timbre—a first step is the establishment of an
acoustical model capable of foretelling the acoustic behavior of the pipe accurately.
This chapterintroduces a novel hybrid technique for the prediction of the input admittance structure and input impedance functions of the shallot–resonator acoustic system. First, the theory of the
propagation of acoustic waves in axisymmetric ducts is reviewed in Section 8.2. A simulation 101
z dz
ζ
ζ+ dzgradζ Horn wall
Figure 8.1.Motion of a curved shell of gas in a flaring horn, after [26]
technique for the radiation impedance from a conical pipe end is presented next in Section 8.3.
Section 8.4 discusses a simple low frequency acoustic model of the shallot. Finally, the proposed method is validated by comparing the predicted frequencies of natural resonance to that of the same obtained by measurements in Section 8.5. An application based on the modeling approach presented in this chapter is introduced in Appendix C.