The response of an infinite Bernoulli‐Euler plate placed on the surface of a randomly vibrating viscoelastic half‐space is calculated, allowing for the presence of shear stresses between the plate and the half‐space. The shear stresses arise from the condition that the relative motion between the plate and the half‐space vanishes at the interface. The three components of displacement of the free surface of the half‐space before the plate is added are assumed to be stationary homogeneous random functions of position and time. From the wavenumber‐frequency spectra of these displacements the frequency spectra of the three components of displacement of the plate half‐space interface are calculated. As an example, the frequency spectrum of the vertical interface displacement is calculated for two assumed forms of the wavenumber spectra of the free‐surface displacements.