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

2

2

2

2

2

2

,

sin,sin

t

ztxF

x

zP

t

zdxPdxtxFddPP

maFdirz

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

2

22

2

2

x

zc

t

zk

t

z

Acceleration Damping Tension

1D Wave Equation with Damping - FDE 2nd Order Accurate in Time and Space,

Explicit FDE Used Central Difference Stencil on the 1st

Derivative

2112

11

2

11 2

2

2

x

zzzc

t

zzk

t

zzz ij

ij

ij

ij

ij

ij

ij

ij

x

tP

x

tc

tk

tkzzzzz

ij

ij

ij

iji

j

21

2112

122

11

1

122

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

00

tt

z

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=0.01001

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=0.01001

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

2

2

2

2

t

z

t

z

t

zB

x

zP wire

Tension Aero Drag Acceleration

wireairD DCB 21

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

Accurate in Time, Explicit FDE

21111

2

11 22

t

zzz

t

zz

t

zzB

x

zzzP

ij

ij

ij

ij

ij

ij

ij

ij

ij

ij

111112

21 22

ij

ij

ij

ij

ij

ij

ij

ij

ij

ij zzzzzz

Bzzz

x

tPz

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