Spherical microphone arrays allow sound field extrapolation using spherical harmonic expansion. Orthogonality of spherical harmonic functions is a key property to perform sound field extrapolation. When measuring the surface impedance of plane materials, it can be convenient to use a hemispherical array whose equatorial plane is set on the material of interest. In that case, spherical harmonics cannot be used, as orthogonality of the spherical basis does not hold on a half-space. When the hemispherical array of microphones is located above a surface with non-trivial boundary conditions (not Neumann or Dirichlet) a closed-form, orthogonal solution of the wave equation in spherical coordinates does not exist .This paper presents a revisited version of a classical numerical approach to derive orthogonal, hemispherical solutions of the wave equation for non-trivial boundary conditions on the equatorial plane. This theory derives such solutions as a combination of spherical harmonics in the restricted hemispherical range, leading to the so-called spherical Slepian functions. It will be shown that such a functional basis makes it possible to accurately extrapolate the sound field of a source at an arbitrary position in free field using a direct frequency independent matrix. It is shown that the high frequency limitations of the method are related to the spatial sampling of the hemisphere, while measurement noise impacts accuracy in low frequency. Finally, simulation results are presented for a hemispherical array above an acoustically absorbing plane material, with the objective of identifying the surface impedance of the material.
