Vibrational Edge Modes for Wedges with Arbitrary Interior Angles

Abstract

We present a theory of long-wavelength acoustic phonons localized at the apex of a variable-angle semi-infinite wedge made up of an isotropic cubic elastic medium. Stress-free boundary conditions are incorporated into the calculation by assuming position-dependent elastic constants. The equations of motion are solved numerically by first performing a linear mapping of the wedge into a right-angle wedge, and then expanding each displacement component in a double series of Laguerre functions. When the Cauchy relation is satisfied and when the interior angle of the wedge is between 125\ifmmode^\circ\else\textdegree\fi{} and 180\ifmmode^\circ\else\textdegree\fi{}, the speed of the lowest-frequency edge mode, which is of ${\ensuremath{\Gamma}}_{1}$ symmetry, is very nearly equal to the speed of Rayleigh surface waves. For wedge angles less than 100\ifmmode^\circ\else\textdegree\fi{}, the speed of the lowest-frequency edge mode, which is now of ${\ensuremath{\Gamma}}_{2}$ symmetry, decreases rapidly with angle and appears to vanish in the limit as the angle approaches 0\ifmmode^\circ\else\textdegree\fi{}. For these acute angles, additional edge modes of ${\ensuremath{\Gamma}}_{2}$ symmetry appear with speeds below the Rayleigh value.