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Variants of the 1D Wave Equation

Variants of the 1D Wave Equation. Jason Batchelder 6/28/07. Overview. Objective Partial Differential Equations 1D Wave Equation with Damping 1D Wave Equation with Forcing Function Finite Difference Equations Results Lessons Learned What I Would do Differently. Objective.

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Variants of the 1D Wave Equation

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  1. Variants of the 1D Wave Equation Jason Batchelder 6/28/07

  2. Overview • Objective • Partial Differential Equations • 1D Wave Equation with Damping • 1D Wave Equation with Forcing Function • Finite Difference Equations • Results • Lessons Learned • What I Would do Differently

  3. Objective • Investigate Real World Variations on the 1D Wave Equation • Guitar String • Doesn’t oscillate forever as the 1D wave equation predicts • Is there a better way to model? • Straight Forward Damping • Aerodynamic Drag

  4. Free Body Diagram and Newton’s Law Taken from Mechanical Vibrations by Rao, pg 503

  5. 1D Wave Equation with Damping - PDE • Partial Differential Equation • Common Form of Wave Equation • Similar form to Spring-Damper System in Vibrations Acceleration Damping Tension

  6. 1D Wave Equation with Damping - FDE • 2nd Order Accurate in Time and Space, Explicit FDE • Used Central Difference Stencil on the 1st Derivative

  7. Assumption Used in Numerical Model • For the next time step, need to know current time step as well as previous time step • Due to 2nd Time Derivative • Also due to 2nd Order Accurate 1st Time Derivative • Assume that any time before the initial condition is the same at the initial condition • i.e. • FDE form: If initial condition is at i=1, then z(j,0)=z(j,1) • Unless Stated, assumes all coefficients are 1

  8. No Damping Case (k=0; CFL = 1) • Used to Check Model dx=0.01, dt=0.01

  9. No Damping Case (k=0; CFL = 1.001) • Not Stable for CFL>1 dx=0.01, dt=0.01001

  10. No Damping Case (k=0; CFL = 0.01) • Stable for CFL<1 First 100 time steps are so quick, little change occurs dx=0.01, dt=0.0001

  11. Damping Case (k=1; CFL = 1) • Stable for CFL<=1 dx=0.01, dt=0.01

  12. Damping Case (k=1; CFL = 1.001) • Unstable for CFL>1 Interestingly the model blows up near the same time step as the no damping case dx=0.01, dt=0.01001

  13. 1D Wave Equation with Forcing - PDE • Partial Differential Equation • Damping Function Replaced with Aerodynamic Drag • Aero Drag is a Non-Linear Term • Magnitude Function Used to Control Drag Direction Tension Aero Drag Acceleration

  14. 1D Wave Equation with Forcing - FDE • 2nd Order Accurate in Space, 1st Order Accurate in Time, Explicit FDE Originally tried to simplify this equation, but messed it up repeatedly, and difficult to do with the absolute value function in there

  15. Unforced Case (B=0; CFL = 1) • Stable for CFL=1 dx=0.01, dt=0.01

  16. Forced Case (B=1; CFL = 1) • Unstable for CFL=1 dx=0.01, dt=0.01

  17. Forced Case (B=1; CFL = 0.99) • Stable for CFL<1 dx=0.01, dt=0.0099

  18. Forced Case (B=2; CFL = 0.99) • Stable for CFL<1 dx=0.01, dt=0.0099

  19. Comparing Damped Case and Drag Case dx=0.01, dt=0.0099

  20. Comparing Effects of Drag Coefficient Increasing drag drops amplitude, but also changes frequency dx=0.01, dt=0.0099

  21. Lessons Learned • I can’t type • Sometimes it’s easier to enter the equation “As-Is” instead of trying to simplify it • Von Neumann stability analysis can’t always be solved (trial and error) • Non linear terms make life difficult • “Next Time” the difficulties would be in keeping track of the indices and simplifying the FDE

  22. What I Would Do Differently • Start earlier • More investigations on initial conditions • Simulate something more realistic like a guitar string • Get properties online • Ability to compare results to things like frequency

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