silly string
DESCRIPTION
Silly String. Starring. Zeb. Jeff. Mike. Bubba. Introduction. Descriptions Statements. A string can be defined as a rigid body whose dimensions are small when compared with its length . - PowerPoint PPT PresentationTRANSCRIPT
Introduction• Descriptions
• Statements
A string can be defined as a rigid body whose dimensions are small when compared with its length.
The string in our model will be stretched between two fixed pegs that are separated by a distance of length L.
L
Peg 1 Peg 2
Tension (T0) will be the force of the two pegs pulling on the string.
For our model, we will assume near constant tension.
Density can be defined as the ratio of the mass of an object to its volume.
For a string, density is mass per unit length.
In our model, we will also assume near constant density for the string.
Derivation of the Wave Equation
• Basic modeling assumptions• Review of Newton’s Law• Calculus prereqs• Equational Derivation
Transverse:
Vibration perpendicular to the X-axis
Model Assumptions
L
Model Assumptions
•Density is assumed constant
= 1
•Initial Deformation is small
Model Assumptions
Tension T is constant and tangent to the curve of the string
T = 1
L
Newton’s Second Law
F = ma
Calculus Prerequisites
T = [ ]1
1 + dydx( )²( ) i + (
dydx
1 + dydx( )²) j |T|
y = f(x) Angle of Inclination
T
y
x
Equational Derivation
x x+x
s
u
x
Equational Derivation
x x+x
s
u
x
Vertical Forces
Horizontal Forces
s
T[ ux (x + x, t)
1 +ux (x, t)( )2]-ux (x, t)
1 +ux (x + x, t)( )2
Vertical Forces:
Get smaller and go to zero
Vertical Forces:
T[ ux (x + x, t)
]-ux (x, t)
1 1
= (s) ²ut² (x,t)
Vertical Forces:
ux
(x + x, t) - ux
(x, t) = s²ut² (x,t)
MassAcceleration
Net Force
ux
(x + x, t) - ux
(x, t)
x
Vertical Forces
= s²ut² (x,t)
x
²ux²
(x, t) = ²ut² (x,t)
Vertical Forces
One dimensional wave equation
Solution to the Wave Equation
• Partial Differential equations
• Multivariate Chain rule
• D’Alemberts Solution
• Infinite String Case
• Finite String Case
• Connections with Fourier Analysis
2nd Order Homogeneous Partial Differential Equation
A B C FED 0 ²y ²y ²yx² xx ++ +++ t² t
y yx y =
Classification of P.D.E. types
= B² - AC • Hyperbolic > 0• Parabolic = 0• Elliptic < 0
Boundary Value Problem
• Finite String Problem • Fixed Ends with 0 < x < l• [u] = 0 and [u] = 0
X = 0 X = l
Cauchy Problem
• Infinite String Problem• Initial Conditions• [u] = (x) and [du/dt] (x) t=0 t=00 l=
Multi-Variable Chain Rule example
f(x,y) = xy² + x²
g(x,y) = y sin(x)
h(x) = e
F(x,y) = f(g(x,y),h(x))
x
Let u = g(x,y)
v = h(x)
So F = f(u,v) = uv² + u²
F
u
v
x
y
fu
fv
g
g
x
xh
y
Variable Dependency Diagram
fu
Fx =
gx +
fv
ux
= ((v² + 2u)(y cos(x)) + (2uv)e )x
= (e )² + 2y sin(x) (y cos(x))
+ 2(y sin(x) e e )
x
xx
Multi-Variable Chain Rule for Second Derivative
Very Similar to that of the first derivative
Our Partial Differential Equation
ξ = x – t
η = x + t
So ξ + η = 2x x = (ξ + η)/2
And - ξ + η = 2t t = (η – ξ)/2
Using Multi-Variable Chain Rule
uξ
u t η
u= +
²ut²
t
=ξ
uη
- u +[ ]
Using Algebra to reduce the equation
²ut² ξ²
2 ²uηξ
²u -= ²u
η²+
ηuu
ξ
x
= +
²uηuu
ξx²= +
ux
[ ]
Using Algebra to simplify
²uη²²u²u
ξ²x²= + ηξ
2 ²u+
Substitute What we just found
²ut²
²ux²
=
η²²u²u
ξ²+ ηξ
2²uη²²u²u
ξ²+ ηξ
2 ²u+=
ηξ2 ²u
= ηξ2 ²u
ηξ4 ²u
= 0
We finally come up with
ηξ ²u
= 0
When u = u(ξ, η)
η = x + t
ξ = x - t
Intermission
Can I get a Beer?
Sorry, we don’t
sell to strings here
A String walks into a bar
Can I get a beer?
Again we can’t serve you
because you are a string
I’m afraid not!
And Now Back to the Models Presentation
D’Alemberts Solution
2
2
2
2
xu
tu
tx tx
2
22
2
2
2
2
2nu
nduu
tu
2
22
2
2
2
2
2nu
nduu
xu
2
22
2
2
2
22
2
2
22nu
nduu
nu
nduu
042
u
02
u
)(
ku
)()( cdku
)()(),( cku
)()(),( gfu
)()(),( txgtxftxu
Then unsubstituting
Relabeling in more conventional notation gives
Integrating with respect to Ada
Next integrating with respect to Xi
Infinite String Solution
)()(),( txgtxftxu
0)0,( x
tu
)()0,( xxu
)(')(')0,( xgxfxtu
)(')(' txgtxftu
Which is a cauchy problem
Reasonable initial conditions
0)(')(')0,( xgxfx
tu
So we have
)()()( xxgxf
0)(')(' xgxf
And we have to solve for f and g
)()()()0,( xxgxfxu
)()(),( txgtxftxu
2)()( xcxf
2)()( cxxg
When we solved for f and g, we found
Then when we plug those into U
)()(),( txgtxftxu
2)(
2)( ctxtxc
2)()( txtx
Finite Solution
)()0,( xxu
0)0,( x
tu
0),0( tu0),( tLu
02
)()(),0(
tttu
02
)()(),(
tLtLtLu
Boundary Value Problem
Boundary conditions
0)()(),0( tttu
0)()(),( tLtLtLu
tLb
0)()( bb
)()( bb
)()( LttL
0)()( LttL
bt
This is a periodic function with period 2L. If the boundary conditions hold this above is true. This equation relates to the sin and cos functions.
0)()2( xLx
NEED A CONCLUSION
A Special thanks To
• Dr. Steve Deckelman for all your help and support
• S.L. Sobolev “Partial Differential Equations of Mathematical Physics
• Scott A. Banaszynski for use of his wonderful guitar
?
Thank you for coming, enjoy the rest of the presentations.