An account is given of the three-dimensional structure of the sound field produced by a spinning sinusoidal distribution of thickness or loading sources. Particular attention is paid to the creeping evanescent waves in the near field and their physical interpretation as trapped edge waves; an exponentially small amount of energy leaks or tunnels through them to emerge as far-field acoustic radiation (as it does through the inhomogeneous waves carried by a waveguide in a cylindrically layered medium). The dependence of the structure on three parameters is investigated in detail: the Mach number M at the outermost radius of the source; the harmonic number n, defined so that the source strength is a function of n$\theta $, where $\theta $ denotes azimuthal angle; and the type of source, i.e. thickness or loading. Parameter values considered include those for subsonic, sonic and supersonic motion, and for high and low harmonics. The field is calculated by reducing a special case of Rayleigh's double integral to a single integral containing a function related to the Chebyshev polynomials, then integrating numerically to give contour plots of pressure as a function of position on various plane and cylindrical sections. These show that the evanescent waves occupy a spherical or ellipsoidal region, and consist of crescents of alternating high and low pressure, shaped and arranged like the segments of an orange; its `peel' marks the transition to the propagating spiral waves of the far field, i.e. the radiation zone. Contour plots on meridional sections are similar to those for the oscillating hertzian electric dipole, suggesting that the field is approximately that produced by a suitably phased arrangement of its acoustic counterpart. When M > 1, the source distribution straddles both the evanescent and the radiation zone; at high supersonic M, the meridional contour plots display an intense beaming pattern, with side-lobes between the main beam and source plane. The results of the paper agree with previous work on propeller acoustics, especially the asymptotic theory.