This paper describes the propagation of three-dimensional symmetric waves localized near the traction-free edge of a semi-infinite elastic plate with either traction-free or fixed faces. For both types of boundary conditions, we present a variational proof of the existence of the low-order edge waves. In addition, for a plate with traction-free faces and zero Poisson ratio, the fundamental edge wave is described by a simple explicit formula, and the first-order edge wave is proved to exist. Qualitative variational predictions are compared with numerical results, which are obtained using expansions in threedimensional Rayleigh-Lamb and shear modes. It is also demonstrated numerically that for any non-zero Poisson ratio in a plate with traction-free faces, the eigenfrequencies related to the first-order wave are complex valued.

A semi-infinite elastic strip, subjected to traction free boundary conditions, is studied in the context of in-plane stationary vibrations. By using normal (RayleighLamb) mode expansion the problem of existence of the strip eigenmode is reformulated in terms of the linear dependence within infinite system of normal modes. The concept of Gram's determinant is used to introduce a generalized criterion of linear dependence, which is valid for infinite systems of modes and complex frequencies. Using this criterion, it is demonstrated numerically that in addition to the edge resonance for the Poisson ratio ν = 0, there exists another value of ν ≈ 0.22475 associated with an undamped resonance. This resonance is best explained physically by the orthogonality between the edge mode and the first Lamé mode. A semi-analytical proof for the existence of the edge resonance is then presented for both described cases with the help of the augmented scattering matrix formalism.

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The recently discovered undamped localized mode at the end of an elastic strip is demonstrated to be particularly relevant in the plane stress setting, where it exists for the Poisson ratio 0.29. This paper also emphasizes the difference between low-frequency edge modes, typically characterized by low variation across the plate (or shell) thickness, and high-frequency edge modes, whose natural frequencies are of the order of thickness resonance frequencies.