The Sakuma-Nishiguchi equation describes surface acoustic waves in a semi-infinite layered medium with a free surface. We show that while this equation does not support ${\mathrm{sech}}^{2}$-like solitary waves, it does support a cos-like traveling-wave solution, which for certain values of the parameters is similar to a periodic train of ${\mathrm{sech}}^{2}$-like pulses. We also determine the relation between the velocity of this periodic traveling wave and its amplitude and period of oscillation. \textcopyright{} 1996 The American Physical Society.