Flute acoustics: measurement, modelling and design - School of ...

CHAPTER 2. THEORY AND LITERATURE REVIEW 21
0.9
0.85
Dalmont et al. 2001
Benade and Murday 1967
0.8
δ cyl
/ a
0.75
0.7
0.65
0.6
2a
0.55
0.5
2b
0.45
0.4
0 0.2 0.4 0.6 0.8 1
a/b
Figure 2.5: The dimensionless end correction δ cyl /a for a cylindrical pipe flanged by a cylinder
as a function of radius ratio.
have negligible effect. Equation (2.30) is applicable over the ranges 0.02 ≤ h/a ≤ 1, 0.025 ≤
w/a ≤ 0.25 and 1.3 ≤ d/a ≤ 1.65. Experiments by Dalmont et al. involving resonance analysis
covering a subset of parameters was in agreement with the simulation.
Benade & Murday (1967) also measured the end correction for a disk placed over the end
of a tube. Using resonance analysis, Benade & Murday found (using the same nomenclature as
Dalmont et al.)
δ disk − δ circ = a[0.61(d/a) 0.18 (a/h) 0.39 ]. (2.31)
The results of Benade & Murday and Dalmont et al. are compared in Figure 2.6 for varying
values of h/a. The curves in this figure are for d/a = 1.375 and w/a = 0.1. Note that Benade
& Murday do not include the wall thickness w as a parameter in their formula. Significant
disagreement exists between these results, particularly for large h (where the effect is small).
Dalmont et al. (2001) also provide a fit-formula for the end correction of a perforated disk
poised over a cylindrical flange (perforated keypads are found on e.g. modern flutes).
δ perf − δ circ =
δ disk − δ circ
1 + 5(δ e /a) −1.35 , (2.32a)
(h/a)
−0.2
where
δ e /a = [1.64a/q − 0.15a/d − 1.1 + ea/q 2 ],
(2.32b)
e is the thickness of the disk and q is the radius of the perforation.