Effective Gol’dberg number for diverging waves

Abstract

Interest in characterizing nonlinearity in jet noise has motivated consideration of an effective Gol’dberg number for diverging waves [Baars and Tinney, Bull. Am. Phys. Soc. 57, 17 (2012)]. Fenlon [J. Acoust. Soc. Am. 50, 1299 (1971)] developed expressions for the minimum value of Γ, the Gol’dberg number as defined for plane waves, for which shock formation occurs in diverging spherical and cylindrical waves. The conditions were deduced from a generalized Khokhlov solution and depend on the ratio xsh/r0, where r0 is source radius, and xsh the plane-wave shock formation distance for Γ=∞. Alternatively, by taking the ratio of the nonlinear and thermoviscous terms in Fenlon’s Eq. (2), it is proposed here that effective Gol’dberg numbers may be identified for spherical and cylindrical waves: Λ=Γexp(-πxsh/2r0) and Λ=Γ/(1 + πxsh/4r0), respectively. For a given value of Λ, the diverging waves achieve approximately the same degree of nonlinear distortion as a plane wave for which the value of Γ is the same. Conversely, to achieve the same degree of nonlinear distortion as a plane wave with a given value of Γ, the value of Γ for, e.g., a spherical wave must be larger by a factor of exp(πxsh/2r0). Extensions to other spreading laws are presented.