A theoretical analysis of acoustic waves refracted by a spherical boundary across which velocity and density increase abruptly and below which velocity and density may either increase or decrease continuously with depth is formulated in terms of waves generated at a harmonic point source and scattered by a radially heterogeneous spherical body. Through the application of an Earth-flattening transformation on the radial solution and the Watson transform on the sum over eigenfunctions, the solution to the spherical problem for high frequencies is expressed as an integral for the corresponding half-space problem in which the effect of boundary curvature maps into an effective positive velocity gradient with depth. The results of both analytical and numerical evaluation of this integral can be summarized as follows for body waves in the crust and upper mantle:

(1) In the special case of a critical velocity gradient (a gradient equal and opposite to the effective curvature gradient), waves interacting with the boundary at the critical angle of incidence have the same form as the classical head wave for flat, homogeneous layers.

(2) For gradients more negative than critical, the amplitude of waves incident at the critical angle decay more rapidly with distance than the classical head wave.

(3) For gradients that are positive, null, and less negative than critical, the amplitude of waves near the critical angle decays less rapidly with distance than the classical head wave, and at sufficiently large distances, the refracted wave field can be adequately described in terms of ray-theoretical diving waves. At intermediate distances from the critical point, the spectral amplitude of the refracted wave is scalloped due to multiple diving wave interference.