Downwind sound propagation in an atmospheric boundary layer

Abstract

A theoretical analysis is given for an acoustic monopole in an atmospheric boundary layer. The analysis is based on the Obukhov quasipotential function (which defines both acoustic pressure and velocity) and assumes an isothermal atmosphere, an exponential boundary-layer flow profile, and a ground impedance function. It is shown that acoustic waves in the boundary layer can be represented by plane waves with variable amplitude. The wave amplitudes are given by the generalized hypergeometric function oFj. The present work is an extension of previous work by Wenzel, who studied surface waves associated with a ground plane without flow, and by Chunchuzov, who identified a discrete mode spectrum in an exponential boundary layer over a hard surface. It is shown that downwind propagation of low-frequency sound can be represented by these discrete modes, which spread as cylindrical waves. The downwind attenuation of the fundamental mode is proportional to frequency squared, wind speed, boundary-layer displacement thickness, and the real part of the ground admittance. The analysis is supported by acoustic data from a wind turbine at Medicine Bow, Wyoming. Nomenclaturetraveling wave amplitude c = speed of sound / = frequency F = upward-traveling wave, Ae + ikzZ &Fi = generalized hypergeometric function G = downward-traveling wave, Be~i kzZ J v -Bessel function of the first kind k -plane wave number co/c k x ,k y ,k z = wave numbers in x, y, and z directions L = boundary condition operator M = Mach number p -acoustic pressure p v = first Bessel function cross product q v = second Bessel function cross product Q = volumetric source strength r, 6,z = cylindrical coordinates R = spherical radius SPL = sound pressure level t = time u t v, w = acoustic velocities in x, y, and z directions, respectively U -wind velocity W[F,G} = Wronskian of F and G x,y,z = rectangular coordinates, downwind, cross wind, and vertical, respectively Y v = Bessel function of the second kind a. = plane wave attenuation coefficient or downwind attenuation j8 = specific ground admittance T(z) = complex gamma function d\ -boundary-layer displacement thickness e = strip width parameter f = waveguide coordinate 9 = standing wave function v = Wronskian p = density £ = strip coordinatê = Obukhov quasipotential co = circular frequency toy = boundary-layer vorticity Subscripts m n 5 x,r 0 oo-mode index = harmonic number = source position = in the respective coordinate direction = on the ground, z = 0 = above the boundary layer, z^°°I ntroduction T HIS paper will analyze the acoustic field of a source in an atmospheric boundary layer and compare the analysis to the downwind propagation of noise from a wind turbine. The dependent acoustic variable used is the Obukhov quasipotential function, 1 an extension of the conventional acoustic potential 2 which accounts for the effect of steady flow vorticity in the acoustic momentum equation. The analytical model is that of an acoustic monopole above a ground plane with a finite acoustic impedance. 2 The wind boundary ...

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