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The cross sections for one- and two-photon detachment of H- with excitation of the degenerate H(2s) and H(2p) levels are calculated within an adiabatic hyperspherical representation. Both the partial cross sections, σ(2s) and σ(2s), and the photoelectron angular distribution asymmetry parameters are obtained. Our one-photon detachment results are dominated by the 1P0 shape resonance feature above threshold, which is well known from prior theoretical and experimental work, with which our results are compared in detail. Our two-photon detachment cross sections and angular distribution asymmetry parameters exhibit a half-cycle of dipole-field-induced oscillation in the energy region from the H(n=2) threshold to approximately 34 meV above threshold. The occurrence of a half-cycle of oscillation with significant amplitude over a 34-meV energy region above the H(n=2) threshold is due to two circumstances unique to this two-photon process: first, in contrast to the strongly suppressed oscillations in the 1Se and 1P0 attractive dipole-field channels for the H(n=2)- e- system, the amplitude of oscillation in the channel having 1De symmetry is not strongly suppressed above threshold; second, electric dipole selection rules for the two-photon process do not permit population of the intense 1P0 shape resonance located above the H(n=2) threshold, which would otherwise obscure the half-cycle of dipole-field-induced oscillation. For these reasons, the process of two-photon detachment of H- is a favorable one for observing this dipole-field-induced oscillation above threshold, which has yet to be observed experimentally in any process.