Post on 19-Dec-2015
transcript
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
Calculus Prerequisites
T = [ ]1
1 + dydx( )²( ) i + (
dydx
1 + dydx( )²) j |T|
y = f(x) Angle of Inclination
T
y
x
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
Solution to the Wave Equation
• Partial Differential equations
• Multivariate Chain rule
• D’Alemberts Solution
• Infinite String Case
• Finite String Case
• Connections with Fourier Analysis
Boundary Value Problem
• Finite String Problem
• Fixed Ends with 0 < x < l
• [u] = 0 and [u] = 0X = 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
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
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
)(
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,(
xt
u
)()0,( xxu
)(')(')0,( xgxfxt
u
)(')(' txgtxft
u
Which is a cauchy problem
Reasonable initial conditions
0)(')(')0,(
xgxfxt
u
So we have
)()()( xxgxf
0)(')(' xgxf
And we have to solve for f and g
)()()()0,( xxgxfxu
)()(),( txgtxftxu
Finite Solution
)()0,( xxu
0)0,(
xt
u
0),0( tu
0),( tLu
02
)()(),0(
tttu
02
)()(),(
tLtLtLu
Boundary Value Problem
Boundary conditions
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
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