The propagation of surface acoustic waves has been investigated for a layered system (water-isotropic layer-isotropic substrate) where the layer is faster than the substrate. It has been found that when the velocity of the surface wave reaches the velocity of the shear wave of the substrate (cutoff) the behaviour of the attenuation has an anomalous character. Attenuation drops down towards zero below cutoff and increases abruptly beyond cutoff. Acoustic microscopy measurements on aluminum coated with oxalic oxide film were in coincidence with theory.©1997 American Institute of Physics.