On the existence of a symmetric acoustic mode in a quadratic solid wedge

It is shown that in a solid truncated wedge whose local thickness depends quadratically on its height, along with the earlier predicted countable set of antisymmetric acoustic modes localized near the apex, there can also exist a single symmetric acoustic mode. This mode is localized in the vicinity of the wedge apex, and its characteristics are close to those of Rayleigh's surface acoustic mode. The phase velocity and the field structure of the mode are calculated. The lowest-frequency boundary for the region of its existence is determined.