Finite Difference Methods to Solve the Wave Equation To develop the governing equation, Sum the Forces The Wave Equation Equations of Motion

Explicit Method Formulation Must first start with Initial Conditions Fixed Ends Initial Displacement Initial Velocity Finite Difference Equation to Predict U(x,t 1 ) Finite Difference Equation to Predict U(x,t 2 ), etc.

Explicit Method Results Pick f(x) and g(x) such that exact solution can be found. Summary of Results 1) Error related to number Time Steps a) Individual step error decreases b) Total error increases 2) Error decreases with Length Steps 3) Error decreases to zero as approaches 1 4) Explicit Method is unstable for > 1

Implicit Method Results The following implicit equation is used to find values of U(x,t 2 ), etc Equation provides M-2 equations with M Unknowns. System of linear equations easily solved. Provides stability for values of higher than 1.

Conclusions Explicit method is stable for  1 Error will decrease to zero as approaches 1 Total error increases with number of time steps, N Total error decreases with number of length steps, M The implicit method will converge for 