Morse's equation for the radiation impedance of a vibrating rigid piston is extended to give the generalized radiation impedances Z m, 1 of all modes in circular and annular ducts which have arbitrary wall admittance /3 and which terminate in an infinite baffle. The equation for Z m, 1 is reduced to a single infinite integral which is a function of the mode radiation directivity factors. An infinite matrix equation is derived which relates the generalized mode reflection coefficients R • • 1 to the radiation impedances. Results are presented which show that the duct-wall admittance is a significant parameter in the effect of duct-termination reflections and that mode-coupling reflection effects may be more important than direct-mode reflection effects.

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Eigenvalue equations of the form D(λ)=0 are studied for acoustic modes in circular and annular ducts without flow. The ducts have locally reacting walls with arbitrary wall admittance. It is shown that circular duets may have double eigenvalues, or wall admittances where both D(λ) and D′(λ) are zero. These double eigenvalues are formed by the coalescence of the lowest-order eigenfunction with some higher-order eigenfunction. The eigenfunctions, or acoustic modes, associated with the double eigenvalues are found to be orthogonal to themselves so that the standard expansion formula for the acoustic field, which is obtained from a separation of variables analysis, is invalid. An alternate expansion formula for the acoustic field is derived. This alternate formula shows that the coalesced acoustic mode is linearly amplified by the transmission distance and exponentially attenuated at the expected rate. The expansion formula also reveals a new eigenfunction, or acoustic mode, which is exponentially attenuated the same rate as the coalesced eigenfunction. It is shown that annular ducts may have double eigenvalues for arbitrary values of one of the wall admittances. Annular ducts may also have triple eigenvalues, or selected values of the wall admittances, where D(λ), D′(λ), and D″(λ) are all zero. The associated triple eigenfunctions are shown to be self-orthogonal. The expansion formula for the sound field in an annular duct with a triple eigenvalue shows a quadratically amplified acoustic mode, a new linearly amplified mode, and another new mode, all of which are exponentially attenuated at the same rate. Methods of calculating single eigenvalues by means of numerically integrating ordinary differential eigenvalue equations are given and discussed with regard to multiple eigenvalues. Results are presented showing double eigenvalues of circular and annular ducts and triple eigenvalues of annular ducts.

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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 ...