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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