On unsteady surface forces, and sound produced by the normal chopping of a rectilinear vortex

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Abstract
An investigation is made of the sound produced when a rectilinear vortex is cut at right angles to its axis by a non-lifting airfoil of symmetric section. The motions are at sufficiently low Mach number that the wavelength of the sound is large relative to the chord of the airfoil. In these circumstances the airfoil experiences no fluctuating lift during the interaction, and the radiation may be ascribed to an acoustic source of dipole type whose strength is equal to the unsteady drag. It is argued that previous analyses of the related problem of ‘unsteady thickness noise’ have ignored certain terms whose inclusion greatly reduces the predicted intensity of the radiation. A general formula for the surface forces (derived in an appendix) is applied to deduce that the dipole strength is proportional to the square of the circulation of the vortex, and depends on the spanwise acceleration of the vortex induced by images in the airfoil. Numerical results are presented for typical airfoil sections, and a comparison is made with the unsteady lifting noise generated when the axis of the vortex is inclined at a small angle to the normal to the median plane of the airfoil.