Elastic active matter or active solid consists of self-propelled units embedded in an elastic matrix. Active solid resists deformation; the shape-preserving property and the intrinsic non-equilibrium nature make active solids a superior component for self-driven devices. Nonetheless, the mechanical properties and emergent behavior of active solids are poorly understood. Using a biofilm-based bacterial active solid, here we discovered self-sustained elastic waves with unique wave properties not seen in passive solids, such as power-law scaling of wave speed with activity. Under isotropic confinement, the active solid develops two topologically distinct global motion modes that can be selectively excited, with a surprising step-like frequency jump at mode transition. Our findings reveal novel spatiotemporal order in elastic active matter and may guide the development of solid-state adaptive or living materials.