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Silly String

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 .

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Silly String

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  1. Silly String Starring

  2. Zeb

  3. Jeff

  4. Mike

  5. Bubba

  6. Introduction • Descriptions • Statements

  7. A string can be defined as a rigidbody whose dimensions are small when compared with its length.

  8. The string in our model will be stretched between two fixed pegs that are separated by a distance of length L. Peg 1 Peg 2 L

  9. Tension (T0) will be the force of the two pegs pulling on the string. For our model, we will assume near constant tension.

  10. 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.

  11. Derivation of the Wave Equation • Basic modeling assumptions • Review of Newton’s Law • Calculus prereqs • Equational Derivation

  12. Model Assumptions L Transverse: Vibration perpendicular to the X-axis

  13. Model Assumptions • Density is assumed constant  = 1 • Initial Deformation is small

  14. Model Assumptions L T = 1 Tension T is constant and tangent to the curve of the string

  15. Newton’s Second Law F = ma

  16. y Angle of Inclination j i |T| T  x [ ] ( ) ( ) T 1 = dy dy dy + ( ) ² ( ) ² 1 + dx dx dx 1 + Calculus Prerequisites y = f(x)

  17. u s x x x+x Equational Derivation

  18. u Horizontal Forces s s x x x+x Vertical Forces Equational Derivation

  19. Vertical Forces: [ ] (x + x, t) - T u u u u x x x x ( ) 2 1 + ( ) 2 (x + x, t) (x, t) 1 + (x, t) Get smaller and go to zero

  20. Vertical Forces: [ ] - (x + x, t) T 1 1   u u x x (x, t) ²u = (s) (x,t) t²

  21. Vertical Forces: Net Force - u u (x + x, t) (x, t) x x ²u Mass = s (x,t) t² Acceleration

  22. Vertical Forces = s (x,t) x - u u (x + x, t) (x, t) x x x ²u t²

  23. Vertical Forces ²u = (x,t) (x, t) x² ²u t² One dimensional wave equation

  24. Solution to the Wave Equation • Partial Differential equations • Multivariate Chain rule • D’Alemberts Solution • Infinite String Case • Finite String Case • Connections with Fourier Analysis

  25.  ²y  ²y y y  ²y 0 A + B + C + D + E + F y = x²  t² x xx t 2nd Order Homogeneous Partial Differential Equation

  26. Classification of P.D.E. types •  = B² - AC • Hyperbolic  > 0 • Parabolic  = 0 • Elliptic  < 0

  27. Boundary Value Problem • Finite String Problem • Fixed Ends with 0 < x < l • [u] = 0 and [u] = 0 X = 0 X = l

  28. Cauchy Problem • Infinite String Problem • Initial Conditions • [u] =  (x) and [du/dt]  (x) = t=0 0 t=0 l

  29. 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

  30. Let u = g(x,y) v = h(x) So F = f(u,v) = uv² + u²

  31. Variable Dependency Diagram g x f u x u g F y f h v x v y

  32. F f g f u = + x u x v x x = ((v² + 2u)(y cos(x)) + (2uv)e ) = x (e )² + 2y sin(x) (y cos(x)) + 2(y sin(x) e e ) x x

  33. Multi-Variable Chain Rule for Second Derivative Very Similar to that of the first derivative

  34. 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

  35. Using Multi-Variable Chain Rule u u u = + t ξ η u [ ] - u ²u  = + η t² t ξ

  36. Using Algebra to reduce the equation ²u ²u ²u 2 ²u - + = η² t² ξ² ηξ

  37. u u u = + x ξ η u u ²u  [ ] = + x² x ξ η

  38. Using Algebra to simplify ²u 2 ²u ²u ²u = + + x² ξ² ηξ η²

  39. Substitute What we just found ²u ²u = t² x²

  40. ²u 2²u ²u ²u 2 ²u ²u + = + + ξ² ηξ η² ξ² ηξ η² 2 ²u 2 ²u = ηξ ηξ 4 ²u = 0 ηξ

  41. We finally come up with ²u = 0 ηξ When u = u(ξ, η) ξ = x - t η = x + t

  42. Intermission

  43. Sorry, we don’t sell to strings here Can I get a Beer? A String walks into a bar

  44. Again we can’t serve you because you are a string Can I get a beer? I’m afraid not!

  45. And Now Back to the Models Presentation

  46. D’Alemberts Solution

  47. Integrating with respect to Ada Next integrating with respect to Xi Relabeling in more conventional notation gives Then unsubstituting

  48. Infinite String Solution Which is a cauchy problem Reasonable initial conditions

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