  Wave turbulence in a thin elastic plate is experimentally investigated. By
using a Fourier transform profilometry technique, the deformation field of the
plate surface is measured simultaneously in time and space. This enables us to
compute the wavevector-frequency Fourier ($\mathbf k, \omega$) spectrum of the
full space-time deformation velocity. In the 3D ($\mathbf k, \omega$) space, we
show that the energy of the motion is concentrated on a 2D surface that
represents a nonlinear dispersion relation. This nonlinear dispersion relation
is close to the linear dispersion relation. This validates the usual
wavenumber-frequency change of variables used in many experimental studies of
wave turbulence. The deviation from the linear dispersion, which increases with
the input power of the forcing, is attributed to weak non linear effects. Our
technique opens the way for many new extensive quantitative comparisons between
theory and experiments of wave turbulence.
