Since surface acoustic waves (SAW) are often generated on substrates to which nanostrips are periodically attached, it is very important to consider the effect of interface between the deposited strip and the substrate surface, which is an unavoidable issue in manufacturing. In this paper, we propose a theoretical model that takes into account the interface damping and calculate the dispersion relationships both for frequency and attenuation of SAW resonance. This results show that the interface damping has an insignificant effect on resonance frequency, but, interestingly, attenuation of the SAW can decrease significantly in the high frequency region as the interface damping increases. Using picosecond ultrasound spectroscopy, we confirm the validity of our theory; the experimental results show similar trends both for resonant frequency and attenuation in the SAW resonance. Furthermore, the resonant behavior of the SAW is simulated using the finite element method, and the intrinsic cause of interface damping on the vibrating system is discussed. These findings strongly indicate the necessity of considering interfacial damping in the design of SAW devices.