The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a...
Transcript of The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a...
![Page 1: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/1.jpg)
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Model Forces The Equation
The Differential Equation for a VibratingString
Bernd Schroder
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 2: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/2.jpg)
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Model Forces The Equation
Modeling Assumptions
1. The string is made up of individual particles that movevertically.
2. u(x, t) is the vertical displacement from equilibrium of theparticle at horizontal position x and at time t.
����������
����������u > 0
u < 0
u = 0
- x
��
�
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 3: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/3.jpg)
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Model Forces The Equation
Modeling Assumptions1. The string is made up of individual particles that move
vertically.
2. u(x, t) is the vertical displacement from equilibrium of theparticle at horizontal position x and at time t.
����������
����������u > 0
u < 0
u = 0
- x
��
�
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 4: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/4.jpg)
logo1
Model Forces The Equation
Modeling Assumptions1. The string is made up of individual particles that move
vertically.2. u(x, t) is the vertical displacement from equilibrium of the
particle at horizontal position x and at time t.
����������
����������u > 0
u < 0
u = 0
- x
��
�
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 5: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/5.jpg)
logo1
Model Forces The Equation
Modeling Assumptions1. The string is made up of individual particles that move
vertically.2. u(x, t) is the vertical displacement from equilibrium of the
particle at horizontal position x and at time t.
����������
����������u > 0
u < 0
u = 0
- x
��
�
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 6: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/6.jpg)
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Model Forces The Equation
Decomposing the Tensile Force
-
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 7: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/7.jpg)
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Model Forces The Equation
Decomposing the Tensile Force
-
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 8: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/8.jpg)
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Model Forces The Equation
Decomposing the Tensile Force
x
-
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 9: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/9.jpg)
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Model Forces The Equation
Decomposing the Tensile Force
x
-
+ ~Ft
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 10: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/10.jpg)
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Model Forces The Equation
Decomposing the Tensile Force
x
-
+
�
~Ft
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 11: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/11.jpg)
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Model Forces The Equation
Decomposing the Tensile Force
x
-
+
�
?
~Fv
~Ft
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 12: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/12.jpg)
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Model Forces The Equation
Decomposing the Tensile Force
x
-
+
�
?
α
~Fv
~Ft
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 13: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/13.jpg)
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Model Forces The Equation
Decomposing the Tensile Force
x x+∆x
-
+
�
?
α
~Fv
~Ft
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 14: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/14.jpg)
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Model Forces The Equation
Decomposing the Tensile Force
x x+∆x
-
+
�
?
:
α
~Fv
~Ft
~Ft
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 15: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/15.jpg)
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Model Forces The Equation
Decomposing the Tensile Force
x x+∆x
-
+
�
?
:6
α
~Fv
~Fv ~Ft
~Ft
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 16: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/16.jpg)
logo1
Model Forces The Equation
Decomposing the Tensile Force
x x+∆x
-
+
�
?
:6
-
α
~Fv
~Fv ~Ft
~Ft
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 17: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/17.jpg)
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Model Forces The Equation
Decomposing the Tensile Force
x x+∆x
-
+
�
?
:6
-
α
α
~Fv
~Fv ~Ft
~Ft
F(x) ≈ Fv(x+∆x)−Fv(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 18: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/18.jpg)
logo1
Model Forces The Equation
Decomposing the Tensile Force
x x+∆x
-
+
�
?
:6
-
α
α
~Fv
~Fv ~Ft
~Ft
F(x) ≈ Fv(x+∆x)−Fv(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 19: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/19.jpg)
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Model Forces The Equation
The Vertical Force at a Point
F(x) ≈ Fv(x+∆x)−Fv(x)= Ft sin(α)−Ft sin(α)
0.25 ≈ 14.3◦
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 20: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/20.jpg)
logo1
Model Forces The Equation
The Vertical Force at a PointF(x) ≈ Fv(x+∆x)−Fv(x)
= Ft sin(α)−Ft sin(α)
0.25 ≈ 14.3◦
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 21: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/21.jpg)
logo1
Model Forces The Equation
The Vertical Force at a PointF(x) ≈ Fv(x+∆x)−Fv(x)
= Ft sin(α)−Ft sin(α)
0.25 ≈ 14.3◦
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 22: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/22.jpg)
logo1
Model Forces The Equation
The Vertical Force at a PointF(x) ≈ Fv(x+∆x)−Fv(x)
= Ft sin(α)−Ft sin(α)
0.25 ≈ 14.3◦
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 23: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/23.jpg)
logo1
Model Forces The Equation
The Vertical Force at a PointF(x) ≈ Fv(x+∆x)−Fv(x)
= Ft sin(α)−Ft sin(α)
0.25 ≈ 14.3◦
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 24: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/24.jpg)
logo1
Model Forces The Equation
The Vertical Force at a PointF(x) ≈ Fv(x+∆x)−Fv(x)
= Ft sin(α)−Ft sin(α)(
sin(θ) ≈ tan(θ),θ small)
≈ Ft tan(α)−Ft tan(α)
-
6
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 25: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/25.jpg)
logo1
Model Forces The Equation
The Vertical Force at a PointF(x) ≈ Fv(x+∆x)−Fv(x)
= Ft sin(α)−Ft sin(α)(
sin(θ) ≈ tan(θ),θ small)
≈ Ft tan(α)−Ft tan(α)
-
6
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 26: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/26.jpg)
logo1
Model Forces The Equation
The Vertical Force at a PointF(x) ≈ Fv(x+∆x)−Fv(x)
= Ft sin(α)−Ft sin(α)(
sin(θ) ≈ tan(θ),θ small)
≈ Ft tan(α)−Ft tan(α)
-
6
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 27: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/27.jpg)
logo1
Model Forces The Equation
The Vertical Force at a PointF(x) ≈ Fv(x+∆x)−Fv(x)
= Ft sin(α)−Ft sin(α)(
sin(θ) ≈ tan(θ),θ small)
≈ Ft tan(α)−Ft tan(α)
-
6
x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 28: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/28.jpg)
logo1
Model Forces The Equation
The Vertical Force at a PointF(x) ≈ Fv(x+∆x)−Fv(x)
= Ft sin(α)−Ft sin(α)(
sin(θ) ≈ tan(θ),θ small)
≈ Ft tan(α)−Ft tan(α)
-
6
x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 29: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/29.jpg)
logo1
Model Forces The Equation
The Vertical Force at a PointF(x) ≈ Fv(x+∆x)−Fv(x)
= Ft sin(α)−Ft sin(α)(
sin(θ) ≈ tan(θ),θ small)
≈ Ft tan(α)−Ft tan(α)
-
6
x
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 30: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/30.jpg)
logo1
Model Forces The Equation
The Vertical Force at a PointF(x) ≈ Fv(x+∆x)−Fv(x)
= Ft sin(α)−Ft sin(α)(
sin(θ) ≈ tan(θ),θ small)
≈ Ft tan(α)−Ft tan(α)
-
6
x
1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 31: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/31.jpg)
logo1
Model Forces The Equation
The Vertical Force at a PointF(x) ≈ Fv(x+∆x)−Fv(x)
= Ft sin(α)−Ft sin(α)(
sin(θ) ≈ tan(θ),θ small)
≈ Ft tan(α)−Ft tan(α)
-
6
x
1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 32: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/32.jpg)
logo1
Model Forces The Equation
The Vertical Force at a PointF(x) ≈ Fv(x+∆x)−Fv(x)
= Ft sin(α)−Ft sin(α)(
sin(θ) ≈ tan(θ),θ small)
≈ Ft tan(α)−Ft tan(α)
-
6
x
f ′(x)
1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 33: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/33.jpg)
logo1
Model Forces The Equation
The Vertical Force at a PointF(x) ≈ Fv(x+∆x)−Fv(x)
= Ft sin(α)−Ft sin(α)(
sin(θ) ≈ tan(θ),θ small)
≈ Ft tan(α)−Ft tan(α)
-
6
x
f ′(x)
1θ
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 34: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/34.jpg)
logo1
Model Forces The Equation
The Vertical Force at a PointF(x) ≈ Fv(x+∆x)−Fv(x)
= Ft sin(α)−Ft sin(α)(
sin(θ) ≈ tan(θ),θ small)
≈ Ft tan(α)−Ft tan(α)(tan(θ) = f ′(x)
)
= Ft
(ddx
u(x+∆x)− ddx
u(x)) (
f (x+∆x) ≈ f (x)+ f ′(x)∆x)
≈ Ft
(ddx
u(x)+∆x · d2
dx2 u(x)− ddx
u(x))
= Ft∆xd2
dx2 u(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 35: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/35.jpg)
logo1
Model Forces The Equation
The Vertical Force at a PointF(x) ≈ Fv(x+∆x)−Fv(x)
= Ft sin(α)−Ft sin(α)(
sin(θ) ≈ tan(θ),θ small)
≈ Ft tan(α)−Ft tan(α)(tan(θ) = f ′(x)
)= Ft
(ddx
u(x+∆x)− ddx
u(x))
(f (x+∆x) ≈ f (x)+ f ′(x)∆x
)≈ Ft
(ddx
u(x)+∆x · d2
dx2 u(x)− ddx
u(x))
= Ft∆xd2
dx2 u(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 36: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/36.jpg)
logo1
Model Forces The Equation
The Vertical Force at a PointF(x) ≈ Fv(x+∆x)−Fv(x)
= Ft sin(α)−Ft sin(α)(
sin(θ) ≈ tan(θ),θ small)
≈ Ft tan(α)−Ft tan(α)(tan(θ) = f ′(x)
)= Ft
(ddx
u(x+∆x)− ddx
u(x)) (
f (x+∆x) ≈ f (x)+ f ′(x)∆x)
≈ Ft
(ddx
u(x)+∆x · d2
dx2 u(x)− ddx
u(x))
= Ft∆xd2
dx2 u(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 37: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/37.jpg)
logo1
Model Forces The Equation
The Vertical Force at a PointF(x) ≈ Fv(x+∆x)−Fv(x)
= Ft sin(α)−Ft sin(α)(
sin(θ) ≈ tan(θ),θ small)
≈ Ft tan(α)−Ft tan(α)(tan(θ) = f ′(x)
)= Ft
(ddx
u(x+∆x)− ddx
u(x)) (
f (x+∆x) ≈ f (x)+ f ′(x)∆x)
≈ Ft
(
ddx
u(x)+∆x · d2
dx2 u(x)− ddx
u(x))
= Ft∆xd2
dx2 u(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 38: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/38.jpg)
logo1
Model Forces The Equation
The Vertical Force at a PointF(x) ≈ Fv(x+∆x)−Fv(x)
= Ft sin(α)−Ft sin(α)(
sin(θ) ≈ tan(θ),θ small)
≈ Ft tan(α)−Ft tan(α)(tan(θ) = f ′(x)
)= Ft
(ddx
u(x+∆x)− ddx
u(x)) (
f (x+∆x) ≈ f (x)+ f ′(x)∆x)
≈ Ft
(ddx
u(x)+∆x · d2
dx2 u(x)
− ddx
u(x))
= Ft∆xd2
dx2 u(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 39: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/39.jpg)
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Model Forces The Equation
The Vertical Force at a PointF(x) ≈ Fv(x+∆x)−Fv(x)
= Ft sin(α)−Ft sin(α)(
sin(θ) ≈ tan(θ),θ small)
≈ Ft tan(α)−Ft tan(α)(tan(θ) = f ′(x)
)= Ft
(ddx
u(x+∆x)− ddx
u(x)) (
f (x+∆x) ≈ f (x)+ f ′(x)∆x)
≈ Ft
(ddx
u(x)+∆x · d2
dx2 u(x)− ddx
u(x))
= Ft∆xd2
dx2 u(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 40: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/40.jpg)
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Model Forces The Equation
The Vertical Force at a PointF(x) ≈ Fv(x+∆x)−Fv(x)
= Ft sin(α)−Ft sin(α)(
sin(θ) ≈ tan(θ),θ small)
≈ Ft tan(α)−Ft tan(α)(tan(θ) = f ′(x)
)= Ft
(ddx
u(x+∆x)− ddx
u(x)) (
f (x+∆x) ≈ f (x)+ f ′(x)∆x)
≈ Ft
(ddx
u(x)+∆x · d2
dx2 u(x)− ddx
u(x))
= Ft∆xd2
dx2 u(x)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 41: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/41.jpg)
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Model Forces The Equation
Using Newton’s Second Law
ma = F(x) = Ft∆x∂ 2
∂x2 u(x, t)
ρl∆x∂ 2
∂ t2u(x, t) = Ft∆x
∂ 2
∂x2 u(x, t)
ρl
Ft
∂ 2
∂ t2u(x, t) =
∂ 2
∂x2 u(x, t)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 42: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/42.jpg)
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Model Forces The Equation
Using Newton’s Second Law
ma
= F(x) = Ft∆x∂ 2
∂x2 u(x, t)
ρl∆x∂ 2
∂ t2u(x, t) = Ft∆x
∂ 2
∂x2 u(x, t)
ρl
Ft
∂ 2
∂ t2u(x, t) =
∂ 2
∂x2 u(x, t)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 43: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/43.jpg)
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Model Forces The Equation
Using Newton’s Second Law
ma = F(x)
= Ft∆x∂ 2
∂x2 u(x, t)
ρl∆x∂ 2
∂ t2u(x, t) = Ft∆x
∂ 2
∂x2 u(x, t)
ρl
Ft
∂ 2
∂ t2u(x, t) =
∂ 2
∂x2 u(x, t)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 44: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/44.jpg)
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Model Forces The Equation
Using Newton’s Second Law
ma = F(x) = Ft∆x∂ 2
∂x2 u(x, t)
ρl∆x∂ 2
∂ t2u(x, t) = Ft∆x
∂ 2
∂x2 u(x, t)
ρl
Ft
∂ 2
∂ t2u(x, t) =
∂ 2
∂x2 u(x, t)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 45: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/45.jpg)
logo1
Model Forces The Equation
Using Newton’s Second Law
ma = F(x) = Ft∆x∂ 2
∂x2 u(x, t)
ρl∆x
∂ 2
∂ t2u(x, t) = Ft∆x
∂ 2
∂x2 u(x, t)
ρl
Ft
∂ 2
∂ t2u(x, t) =
∂ 2
∂x2 u(x, t)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 46: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/46.jpg)
logo1
Model Forces The Equation
Using Newton’s Second Law
ma = F(x) = Ft∆x∂ 2
∂x2 u(x, t)
ρl∆x∂ 2
∂ t2u(x, t)
= Ft∆x∂ 2
∂x2 u(x, t)
ρl
Ft
∂ 2
∂ t2u(x, t) =
∂ 2
∂x2 u(x, t)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 47: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/47.jpg)
logo1
Model Forces The Equation
Using Newton’s Second Law
ma = F(x) = Ft∆x∂ 2
∂x2 u(x, t)
ρl∆x∂ 2
∂ t2u(x, t) = Ft∆x
∂ 2
∂x2 u(x, t)
ρl
Ft
∂ 2
∂ t2u(x, t) =
∂ 2
∂x2 u(x, t)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 48: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/48.jpg)
logo1
Model Forces The Equation
Using Newton’s Second Law
ma = F(x) = Ft∆x∂ 2
∂x2 u(x, t)
ρl∆x∂ 2
∂ t2u(x, t) = Ft∆x
∂ 2
∂x2 u(x, t)
ρl
Ft
∂ 2
∂ t2u(x, t) =
∂ 2
∂x2 u(x, t)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 49: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/49.jpg)
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Model Forces The Equation
The One-Dimensional Wave Equation
The equation of motion for small oscillations of a frictionlessstring is
∂ 2
∂x2 u(x, t) = k∂ 2
∂ t2u(x, t),
where k =ρl
Ft> 0, with ρl being the linear density of the string
and Ft being the tensile force.This equation is also called the one-dimensional waveequation.Our derivation is valid for small oscillations and negligiblefriction.The cancellation of the ∆x was “clean”.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 50: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/50.jpg)
logo1
Model Forces The Equation
The One-Dimensional Wave EquationThe equation of motion for small oscillations of a frictionlessstring is
∂ 2
∂x2 u(x, t) = k∂ 2
∂ t2u(x, t),
where k =ρl
Ft> 0, with ρl being the linear density of the string
and Ft being the tensile force.
This equation is also called the one-dimensional waveequation.Our derivation is valid for small oscillations and negligiblefriction.The cancellation of the ∆x was “clean”.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 51: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/51.jpg)
logo1
Model Forces The Equation
The One-Dimensional Wave EquationThe equation of motion for small oscillations of a frictionlessstring is
∂ 2
∂x2 u(x, t) = k∂ 2
∂ t2u(x, t),
where k =ρl
Ft> 0, with ρl being the linear density of the string
and Ft being the tensile force.This equation is also called the one-dimensional waveequation.
Our derivation is valid for small oscillations and negligiblefriction.The cancellation of the ∆x was “clean”.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 52: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/52.jpg)
logo1
Model Forces The Equation
The One-Dimensional Wave EquationThe equation of motion for small oscillations of a frictionlessstring is
∂ 2
∂x2 u(x, t) = k∂ 2
∂ t2u(x, t),
where k =ρl
Ft> 0, with ρl being the linear density of the string
and Ft being the tensile force.This equation is also called the one-dimensional waveequation.Our derivation is valid for small oscillations and negligiblefriction.
The cancellation of the ∆x was “clean”.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String
![Page 53: The Differential Equation for a Vibrating String · 2008-11-06 · The Differential Equation for a Vibrating String. logo1 Model Forces The Equation Modeling Assumptions 1. The string](https://reader034.fdocuments.in/reader034/viewer/2022042016/5e74686b3408f62ce057f3ba/html5/thumbnails/53.jpg)
logo1
Model Forces The Equation
The One-Dimensional Wave EquationThe equation of motion for small oscillations of a frictionlessstring is
∂ 2
∂x2 u(x, t) = k∂ 2
∂ t2u(x, t),
where k =ρl
Ft> 0, with ρl being the linear density of the string
and Ft being the tensile force.This equation is also called the one-dimensional waveequation.Our derivation is valid for small oscillations and negligiblefriction.The cancellation of the ∆x was “clean”.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Differential Equation for a Vibrating String