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 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

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

1D Wave Equation with Damping - PDE Partial Differential Equation Common Form of Wave Equation Similar form to Spring-Damper System in Vibrations AccelerationDampingTension

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

Assumption Used in Numerical Model For the next time step, need to know current time step as well as previous time step  Due to 2 nd Time Derivative  Also due to 2 nd Order Accurate 1 st 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

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

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

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

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

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=

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 TensionAero DragAcceleration

1D Wave Equation with Forcing - FDE 2 nd Order Accurate in Space, 1 st 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

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

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

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

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

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

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

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

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