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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' and 180', the speed of the lowest-frequency edge mode, which is of I, symmetry, is very nearly equal to the speed of Rayleigh surface waves. For wedge angles less than 100', the speed of the lowest-frequency edge mode, which is now of I', symmetry, decreases rapidly with angle and appears to vanish in the limit as the angle approaches O'. For these acute angles, additional edge modes of I, symmetry appear with speeds below the Rayleigh value.

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