THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

7.5 Radiation of Power from Open-Ended Pipes 135Equation (7.9) yields tan kL =−ψ in order for resonance frequencies to occur.With the assumption ψ ≪ 1, we obtaintan(nπ − k n L) = 8ka 8ka≈ tan (7.13)3π 3πwhere n = 1, 2, 3,.... Hencenπ = k n L + 8k na(7.14)3πThe resonance frequencies therefore aref n = n c2L + 8(7.15)3π aAll of the resonance frequencies are harmonics of the fundamentals. We also notethat the denominator L + 8a/(3π) constitutes the effective length of the pipe ratherthan actual length L.In the case of the unflanged pipe, the radiation impedance, indicated by boththeory and experiment is given approximately byZ nLρ 0 cS = (ka)2 + i(0.6ka) (7.16)4Here the end correction for the unflanged pipe equals 0.6a, with the effectivelength being L + 0.6a. We also note that the end corrections do not depend onthe frequency. Providing that λ n ≫ a, the resonance frequencies of flanged andunflanged pipes constitute harmonics of the fundamental. Hence, the driving frequencyof an open-ended organ pipe yields resonances that are harmonics of thedriving frequency. The above exposition so far has dealt with pipes of constantcross sections. If a pipe is flared at the open end, as is the case with many windinstruments such as the clarinet and the oboe, the results are modified, and theresonances may not necessarily be harmonics of the fundamental. Variations inthe flare design will emphasize or lessen certain harmonics present in the forcingfunction, thereby affecting the quality or timbre of the sound emanated by the pipe.7.5 Radiation of Power from Open-Ended PipesEquation (7.4) may be revised to readBA =Z nLρ 0 cS − 1Z nLρ 0 cS + 1 (7.17)When the termination impedance Z nL is known the power transmission coefficientT n can be established fromT n = 1 −B2∣A∣(7.18)