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 ...

The propagation of low-frequency noise outdoors was studied using the noise of a large (75-m-diam) 4-MW downwind wind turbine. Acoustic measurements were made with low-frequency microphone systems placed on the ground at five downwind sites ranging from 300 to 10 000 m (6.3 mile) away from the wind turbine. Despite high background noise associated with the inherently hostile wind environment found around operating wind turbines, certain wind turbine harmonics were measured with signal to noise ratios exceeding 7 dB at the 6.3-mile site. The wind turbine fundamental was 1 Hz and the wind speed was generally 30 knots at the hub height (75 m) during the acoustic testing. The harmonic levels when plotted versus propagation distance exhibit a 3 dB per doubling of distance divergence. In addition, the lower harmonics were attenuated less than the higher harmonics. The cylindrical spreading and frequency dependence seen in the measured results are indicative of surface wave propagation. Theoretical predictions of a source above a finite impedance boundary are compared to the measured results.

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A unique set of experiments is described in which simultaneous acoustic measurements were made at 18 locations around a large horizontal axis wind turbine in a circular array having a radius of 200 m. Results are presented for frequencies from 3–3000 Hz, for power output values from 0.1–2.1 MW and for several rotational speeds and skew angles. Additional data for simultaneous measurements using a linear array will be presented to illustrate sound propagation in both the upwind and downwind directions to distances of several kilometers. [Work supported by DOE.]

The Obukhov quasipotential function for the acoustic field in a boundary layer of exponential profile is used to obtain a modal description of low-frequency sound propagation. As the wind speed approaches zero, the governing equations approach the Helmholtz equation with an impedance boundary condition. The solutions for the acoustic field with a boundary layer can be given as a continuous plane-wave spectrum with variable amplitudes given by generalized hypergeometric functions. An analysis with the hypergeometric functions gives one or more acoustic modes, depending on frequency. The acoustic modes propagate as cylindrical waves, with amplitude varying inversely with the square root of distance. An estimate of the wavenumber of the fundamental mode shows that its attenuation is proportional to the product of wind speed and boundary layer displacement thickness. The propagation theory is compared to data from a wind turbine at Medicine Bow, Wyoming. A microphone array was used to measure low-frequency sound at ground level at distances from 200 to 20 00(3 m from the turbine. Atmospheric temperature and wind speed profiles were measured, as was ground impedance, so that the theory may be compared without ambiguity to the data.