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We study the band structure and scattering of in-plane coupled longitudinal and shear stress waves in linear layered media and observe that exceptional points (EP) appear for elastic (lossless) media, when parameterized with real-valued frequency and tangential wave vector component. The occurrence of these EP pairs is not limited to the original stop bands. They could also appear in all mode pass bands, leading to the formation of new stop bands. The scattered energy near these locations is studied along with the associated polarization patterns. The broken phase symmetry is observed inside the frequency bands book-ended by these EP pairs. This is especially manifested by the chirality of the trajectory of the particle velocity, which gets selected by a ‘‘direction’’ of the wave, e.g. the imaginary part of normal component of the wave vector, or the energy flux direction just outside the band. Additionally, EP pairs also appear in the spectrum of the (modified) scattering matrix when mechanical gain is theoretically included to balance the loss in a parity-time symmetric finite structure. These EP pairs lead to amplification of transmission to above 1 and single-sided reflectivity, both phenomena associated with broken phase symmetry, with intriguing potential applications. 

The source angle localization problem is studied based on scattering of elastic waves in two dimensions by a phononic array and the exceptional points of its band structure. Exceptional points are complex singularities of a parameterized eigen-spectrum, where two modes coalesce with identical mode shapes. These special points mark the qualitative transitions in the system behavior and have been proposed for sensing applications. The equi-frequency band structures are analyzed with focus on the angle-dependent modal behaviors. At the exceptional points and critical angles, the eigen-modes switch their energy characteristics and symmetry, leading to enhanced sensitivity as the scattering response of the medium is inherently angle-dependent. An artificial neural network is trained with randomly weighted and superposed eigen-modes to achieve deep learning of the angle-dependent dynamics. The trained algorithm can accurately classify the incident angle of an unknown scattering signal, with minimal sidelobe levels and suppressed main lobewidth. The neural network approach shows superior localization performance compared with standard delay-and-sum technique. The proposed application of the phononic array highlights the physical relevance of band topology and eigen-modes to a technological application, adds extra strength to the existing localization methods, and can be easily enhanced with the fast-growing data-driven techniques.