Reflection and attenuation of a Rayleigh surface acoustic wave from a small stripe of viscous fluid spreading across the acoustic path have been observed. As the stripe spreads across the acoustic path, with an accompanying decrease in stripe height to conserve volume, the reflected and attenuated acoustic signals show a distinct pattern of resonances. This paper reports experimental results for the spreading of highly viscous poly(dimethyl)siloxane oils (100 000 cS) in the propagation path of a 169 MHz Rayleigh wave operating in pulse mode and develops a model to explain the observed resonances of the transmission coefficient. Simultaneous optical observations enable the time evolution of the shape of the stripe to be converted into changes in geometric parameters (contact width, spherical radius, height, and contact angle). To model the acoustic attenuation, an approach treating the fluid as a viscoelastic fluid with a single relaxation time (Maxwell model) is developed. This approach is able to explain the structure of the observed transmitted signal. Resonances in the transmission coefficient occur cyclically and correspond to the stripe height matching nλs/4, where n is odd and λs is the shear wavelength in the fluid.