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

20 CHAPTER 2. THEORY AND LITERATURE REVIEW
Figure 2.4: The acoustic reactance X and the acoustic resistance R as functions of ka for an
infinitely flanged pipe (a) and an unflanged pipe (b). X and R are both in units of ρc/πa 2
(Fletcher & Rossing 1998, after Beranek (1954)).
The impedance (including radiation loss) of an unflanged pipe was calculated by Levine &
Schwinger (1948). Dalmont et al. (2001) give simple fit formulae for the frequency-dependent
end correction and the magnitude of the reflection coefficient:
[ 1 + 0.044(ka)
2
]
˜δ open = δ open
1 + 0.19(ka) 2 − 0.02sin2 (2ka) for ka < 1.5 (2.21)
and
1 + 0.2ka− 0.084(ka) 2
|R open |=
1 + 0.2ka+ ( 1 for ka < 3.5. (2.22)
2
− 0.084)(ka)2
2.2.8.2 Infinitely flanged pipe
The low-frequency, lossless end correction for an infinitely flanged pipe is greater than for
a unflanged pipe, since the solid angle for radiation is reduced (Dalmont et al. 2001):
δ flanged = 0.8216a. (2.23)
Rayleigh (1894) and Nomura et al. (1960) give the impedance for an infinitely flanged pipe. Fit
formulae by Norris & Sheng (1989) are accurate for ka < 3.5:
[
] −1
˜δ flanged = δ flanged 1 + (0.77ka)2
(2.24)
1 + 0.77ka
and
|R flanged |=
1 + 0.323ka− 0.077(ka)2
1 + 0.323ka+ (1 − 0.077)(ka) 2 . (2.25)