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We analytically predict and numerically demonstrate the existence of a family of bound modes in the continuum (BICs) in bi-layered spring mass chains. A coupled array of such chains is then used to illustrate transversely bound waves propagating along a channel in a lattice. We start by considering the compact region formed by coupling two spring mass chains with defects and predict bound modes arising due to reflection symmetries in this region. Dispersion analysis of a waveguide consisting of an array of appropriately coupled bi-layered chains reveals the presence of a branch having bound modes in the passband. Finally, detailed numerical analyses verify the existence of a BIC and its propagation through the waveguide at passband frequencies without energy leakage. The framework allows to achieve BICs and their propagation, for any arbitrary size and location of the compact region. Such BICs open avenues for novel classes of resonators with extremely high $Q$ factors due to zero energy leakage and allow for guiding confined waves in structures without requiring bandgaps.