We used Brillouin scattering in p-s polarization to study shear horizontal (SH) acoustic surface phonons in a silicon on insulator structure made of a Si/${\mathrm{SiO}}_{2}$ bilayer on a Si substrate. Two main peaks were measured, the first belonging to the discrete part and the second to the continuous part of the spectrum. The latter feature originates from a type of pseudosurface wave, or quasiresonance, typical of the investigated structure [Bottani et al., J. Phys. Condens. Matter 6, L85 (1994)]. Because of the SH polarization of the phonons considered Brillouin scattering takes place only via the elasto-optic mechanism, that is, by modulation of the bulk dielectric function of the material. To fully explain the experimental data we performed a computation of the p-s Brillouin cross section. The SH phonon spectrum, both discrete and continuous, was computed numerically using a slab approximation. The guided-wave nature of the discrete mode and the strong surface localization of the pseudomode are illustrated by means of the layer projected phonon density of states. We computed the incident p electric field transmitted in the medium and the fluctuating polarization vector radiating Brillouin light. The p-s cross section was evaluated using a Green-function method introduced some time ago by Laks and Mills [Phys. Rev. B 20, 4962 (1979)] [2]. The computed intensity of the scattered field is in good agreement with the experimental finding.