The amplitude attenuation with depth of the surface waves both Love and Rayleigh waves, in elastic earth model is of exponential nature (Stonley, 1958). Studies on the effect of viscoelasticity on the surface wave attenuation with depth have so far not been reported. The applicability of the results for elastic earth models to the actual problems may be limited since the stress wave propagation in most solids is associated with the process of energy dessipation, [Kolsky, (1960), Knopoff (1969)]. This process, for many of the earth materials can be represented by linear viscoelastic models. Therefore, for present investigations, a simple model with single surface layer exhibiting Voigt type of visco- elasticity overlying an elastic half space was taken for the study of amplitude attenution of Love waves of varying wave lengths, ranging between 0.3 to 2.4 times the thickness of surface layer. Amplitude attenuation has been studied upto depths, from free surface, which are whithin two times the thickness of surface layer. Similar studies for models approximating, more to the real earth, may permit a better understanding of the attenuation of earthquake caused ground motions with depth.