Sound Propagation in a Tube of Arbitrary Cross-Sectional Shape

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  • 任意の形の断面をもつ管の中の音の伝搬
  • ニンイ ノ カタチ ノ ダンメン オ モツ カン ノ ナカ ノ オト ノ デンパン

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Taking the viscosity and heat conduction of air into account, a theoretical study is made on the propagation of a palne sound wave at high frequencies in cylindrical tubes of arbitrary cross-sectional shape. The mean complex density of air ρ and the mean complex bulk modulus of air K may be expressed respectively as, ρ=ρ_0/{1-W(φ_1)}, K=γP_0/{1+(γ-1)W(φ_2)}, where φ_1=(S/L)√<ωρ_0/2μ>, φ_2=(S/L)√<ωc_pρ_0/2λ>, ρ_0 is equilibrium density of air, μ coefficient of viscosity, c_p specific heat at constant presure, γ ratio of specific heats, λ thermal conductivity, P_0 equilibrium presure, ω circular frequency, L and S are perimeter and cross-sectional area of tube respectively, and W(φ) is a function depending upon cross-sectional shape. Solving the fundamental equations, approximate formulae for W(φ) are obtained as follows; i)for a tube having cross section without corners, W(φ)=(1-j)/(2)1/(φ)+(jπδS)/(2L^2)1/(φ^2)+. . . , where δ is 1 or 0 for single or double tubes respectively, and j=√<-1>, ii)for a tube having polygonal cross section with N corners, W(φ)=(1-j)/(2)1/(φ)-(jS)/(L^2)Σ^N_(m=1)K(α_m)1/(φ^2), where K(α) is a constant depending upon the angle of corner α. Thus, in the first approximation, it is found that ρ and K depend only on L/S but are independent of the cross-sectional shape. In the second appoximation, it is found that ρ and K depend on L, S and δ or K(α) but are independent of other quantities concerning the cross-sectional shape. Exact formulae for W(φ) of four simple cross-sectional shapes are compared with the approximate formulae. After numerical calculation, the values of the first approximation are found to be in fairly good agreement with the exact ones, and the valeus of the second approximation are found to be in very good agreement with the exact ones. It is ascertained that the first approximation of the present theory agrees fairly good with the experimental results by Takeuchi and Nakamura for a tube composed of mutually circumscribed circular rods.

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