Ultrasonic attenuation in d-wave superconductors

The longitudinal ultrasonic attenuation at low temperatures for a $d$-wave superconducting order parameter is examined. The ratio of the superconducting to normal state attenuation, $\alpha_{s}/\alpha_{n}$, depends strongly on the direction of propagation of ultrasound with maxima corresponding to nodes of the order parameter on the Fermi surface. We propose that measurements of ultrasonic attenuation in a single crystal of the high-$T_{c}$ superconductors can give the information regarding the position of nodes.