Explain the Method of Characteristics for transient flow analysis.

Method of Characteristics (MOC) solves hyperbolic partial differential equations governing unsteady pipe flow by transforming them into ordinary differential equations along characteristic lines. The governing equations (continuity and momentum) propagate as waves at speed a (acoustic velocity) relative to fluid velocity. Characteristic lines: C+ with slope dx/dt = V+a carries Riemann invariant J+ = V + (g/a)H + (fV|V|/2D)Δt, C- with slope dx/dt = V-a carries J- = V - (g/a)H - (fV|V|/2D)Δt. Solution marches in time, computing pressure and velocity at grid intersections from upstream conditions. Boundary conditions handled by combining characteristic equations with device-specific relationships (valves, pumps, reservoirs). Essential for water hammer analysis, surge protection design, and pipeline integrity assessment.