The Differential Equation for a Vibrating String Bernd Schr oder - - PowerPoint PPT Presentation

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the differential equation for a vibrating string
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The Differential Equation for a Vibrating String Bernd Schr oder - - PowerPoint PPT Presentation

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Model Forces The Equation The Differential Equation for a Vibrating String Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String Model

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SLIDE 1

logo1 Model Forces The Equation

The Differential Equation for a Vibrating String

Bernd Schr¨

  • der

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

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SLIDE 2

logo1 Model Forces The Equation

Modeling Assumptions

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

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SLIDE 3

logo1 Model Forces The Equation

Modeling Assumptions

  • 1. The string is made up of individual particles that move

vertically.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

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SLIDE 4

logo1 Model Forces The Equation

Modeling Assumptions

  • 1. 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.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

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SLIDE 5

logo1 Model Forces The Equation

Modeling Assumptions

  • 1. 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 Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

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SLIDE 6

logo1 Model Forces The Equation

Decomposing the Tensile Force

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

slide-7
SLIDE 7

logo1 Model Forces The Equation

Decomposing the Tensile Force

✲

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

slide-8
SLIDE 8

logo1 Model Forces The Equation

Decomposing the Tensile Force

x

✲

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

slide-9
SLIDE 9

logo1 Model Forces The Equation

Decomposing the Tensile Force

x

✲ ✰

Ft Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

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SLIDE 10

logo1 Model Forces The Equation

Decomposing the Tensile Force

x

✲ ✰ ✛

  • Ft

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

slide-11
SLIDE 11

logo1 Model Forces The Equation

Decomposing the Tensile Force

x

✲ ✰ ✛ ❄

  • Fv
  • Ft

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

slide-12
SLIDE 12

logo1 Model Forces The Equation

Decomposing the Tensile Force

x

✲ ✰ ✛ ❄

α

  • Fv
  • Ft

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

slide-13
SLIDE 13

logo1 Model Forces The Equation

Decomposing the Tensile Force

x x+∆x

✲ ✰ ✛ ❄

α

  • Fv
  • Ft

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

slide-14
SLIDE 14

logo1 Model Forces The Equation

Decomposing the Tensile Force

x x+∆x

✲ ✰ ✛ ❄ ✿

α

  • Fv
  • Ft
  • Ft

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

slide-15
SLIDE 15

logo1 Model Forces The Equation

Decomposing the Tensile Force

x x+∆x

✲ ✰ ✛ ❄ ✿ ✻

α

  • Fv
  • Fv
  • Ft
  • Ft

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

slide-16
SLIDE 16

logo1 Model Forces The Equation

Decomposing the Tensile Force

x x+∆x

✲ ✰ ✛ ❄ ✿ ✻ ✲

α

  • Fv
  • Fv
  • Ft
  • Ft

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

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SLIDE 17

logo1 Model Forces The Equation

Decomposing the Tensile Force

x x+∆x

✲ ✰ ✛ ❄ ✿ ✻ ✲

α ˜ α

  • Fv
  • Fv
  • Ft
  • Ft

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

slide-18
SLIDE 18

logo1 Model Forces The Equation

Decomposing the Tensile Force

x x+∆x

✲ ✰ ✛ ❄ ✿ ✻ ✲

α ˜ α

  • Fv
  • Fv
  • Ft
  • Ft

F(x) ≈ Fv(x+∆x)−Fv(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

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SLIDE 19

logo1 Model Forces The Equation

The Vertical Force at a Point

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

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SLIDE 20

logo1 Model Forces The Equation

The Vertical Force at a Point

F(x) ≈ Fv(x+∆x)−Fv(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

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SLIDE 21

logo1 Model Forces The Equation

The Vertical Force at a Point

F(x) ≈ Fv(x+∆x)−Fv(x) = Ft sin( ˜ α)−Ft sin(α)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

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SLIDE 22

logo1 Model Forces The Equation

The Vertical Force at a Point

F(x) ≈ Fv(x+∆x)−Fv(x) = Ft sin( ˜ α)−Ft sin(α)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

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SLIDE 23

logo1 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 Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

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SLIDE 24

logo1 Model Forces The Equation

The Vertical Force at a Point

F(x) ≈ Fv(x+∆x)−Fv(x) = Ft sin( ˜ α)−Ft sin(α)

  • sin(θ) ≈ tan(θ),θ small
  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

slide-25
SLIDE 25

logo1 Model Forces The Equation

The Vertical Force at a Point

F(x) ≈ Fv(x+∆x)−Fv(x) = Ft sin( ˜ α)−Ft sin(α)

  • sin(θ) ≈ tan(θ),θ small
  • ≈

Ft tan( ˜ α)−Ft tan(α)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

slide-26
SLIDE 26

logo1 Model Forces The Equation

The Vertical Force at a Point

F(x) ≈ Fv(x+∆x)−Fv(x) = Ft sin( ˜ α)−Ft sin(α)

  • sin(θ) ≈ tan(θ),θ small
  • ≈

Ft tan( ˜ α)−Ft tan(α)

✲ ✻

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

slide-27
SLIDE 27

logo1 Model Forces The Equation

The Vertical Force at a Point

F(x) ≈ Fv(x+∆x)−Fv(x) = Ft sin( ˜ α)−Ft sin(α)

  • sin(θ) ≈ tan(θ),θ small
  • ≈

Ft tan( ˜ α)−Ft tan(α)

✲ ✻ x

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

slide-28
SLIDE 28

logo1 Model Forces The Equation

The Vertical Force at a Point

F(x) ≈ Fv(x+∆x)−Fv(x) = Ft sin( ˜ α)−Ft sin(α)

  • sin(θ) ≈ tan(θ),θ small
  • ≈

Ft tan( ˜ α)−Ft tan(α)

✲ ✻ x

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

slide-29
SLIDE 29

logo1 Model Forces The Equation

The Vertical Force at a Point

F(x) ≈ Fv(x+∆x)−Fv(x) = Ft sin( ˜ α)−Ft sin(α)

  • sin(θ) ≈ tan(θ),θ small
  • ≈

Ft tan( ˜ α)−Ft tan(α)

✲ ✻ x

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

slide-30
SLIDE 30

logo1 Model Forces The Equation

The Vertical Force at a Point

F(x) ≈ Fv(x+∆x)−Fv(x) = Ft sin( ˜ α)−Ft sin(α)

  • sin(θ) ≈ tan(θ),θ small
  • ≈

Ft tan( ˜ α)−Ft tan(α)

✲ ✻ x 1

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

slide-31
SLIDE 31

logo1 Model Forces The Equation

The Vertical Force at a Point

F(x) ≈ Fv(x+∆x)−Fv(x) = Ft sin( ˜ α)−Ft sin(α)

  • sin(θ) ≈ tan(θ),θ small
  • ≈

Ft tan( ˜ α)−Ft tan(α)

✲ ✻ x 1

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

slide-32
SLIDE 32

logo1 Model Forces The Equation

The Vertical Force at a Point

F(x) ≈ Fv(x+∆x)−Fv(x) = Ft sin( ˜ α)−Ft sin(α)

  • sin(θ) ≈ tan(θ),θ small
  • ≈

Ft tan( ˜ α)−Ft tan(α)

✲ ✻ x f ′(x) 1

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

slide-33
SLIDE 33

logo1 Model Forces The Equation

The Vertical Force at a Point

F(x) ≈ Fv(x+∆x)−Fv(x) = Ft sin( ˜ α)−Ft sin(α)

  • sin(θ) ≈ tan(θ),θ small
  • ≈

Ft tan( ˜ α)−Ft tan(α)

✲ ✻ x f ′(x) 1 θ

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

slide-34
SLIDE 34

logo1 Model Forces The Equation

The Vertical Force at a Point

F(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)
  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

slide-35
SLIDE 35

logo1 Model Forces The Equation

The Vertical Force at a Point

F(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 d dxu(x+∆x)− d dxu(x)

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

slide-36
SLIDE 36

logo1 Model Forces The Equation

The Vertical Force at a Point

F(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 d dxu(x+∆x)− d dxu(x)

  • f(x+∆x) ≈ f(x)+f ′(x)∆x
  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

slide-37
SLIDE 37

logo1 Model Forces The Equation

The Vertical Force at a Point

F(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 d dxu(x+∆x)− d dxu(x)

  • f(x+∆x) ≈ f(x)+f ′(x)∆x
  • ≈

Ft

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

slide-38
SLIDE 38

logo1 Model Forces The Equation

The Vertical Force at a Point

F(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 d dxu(x+∆x)− d dxu(x)

  • f(x+∆x) ≈ f(x)+f ′(x)∆x
  • ≈

Ft d dxu(x)+∆x· d2 dx2u(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

slide-39
SLIDE 39

logo1 Model Forces The Equation

The Vertical Force at a Point

F(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 d dxu(x+∆x)− d dxu(x)

  • f(x+∆x) ≈ f(x)+f ′(x)∆x
  • ≈

Ft d dxu(x)+∆x· d2 dx2u(x)− d dxu(x)

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

slide-40
SLIDE 40

logo1 Model Forces The Equation

The Vertical Force at a Point

F(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 d dxu(x+∆x)− d dxu(x)

  • f(x+∆x) ≈ f(x)+f ′(x)∆x
  • ≈

Ft d dxu(x)+∆x· d2 dx2u(x)− d dxu(x)

  • =

Ft∆x d2 dx2u(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

slide-41
SLIDE 41

logo1 Model Forces The Equation

Using Newton’s Second Law

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

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SLIDE 42

logo1 Model Forces The Equation

Using Newton’s Second Law

ma

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

slide-43
SLIDE 43

logo1 Model Forces The Equation

Using Newton’s Second Law

ma = F(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

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SLIDE 44

logo1 Model Forces The Equation

Using Newton’s Second Law

ma = F(x) = Ft∆x ∂ 2 ∂x2u(x,t)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

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SLIDE 45

logo1 Model Forces The Equation

Using Newton’s Second Law

ma = F(x) = Ft∆x ∂ 2 ∂x2u(x,t) ρl∆x

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

slide-46
SLIDE 46

logo1 Model Forces The Equation

Using Newton’s Second Law

ma = F(x) = Ft∆x ∂ 2 ∂x2u(x,t) ρl∆x ∂ 2 ∂t2u(x,t)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

slide-47
SLIDE 47

logo1 Model Forces The Equation

Using Newton’s Second Law

ma = F(x) = Ft∆x ∂ 2 ∂x2u(x,t) ρl∆x ∂ 2 ∂t2u(x,t) = Ft∆x ∂ 2 ∂x2u(x,t)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

slide-48
SLIDE 48

logo1 Model Forces The Equation

Using Newton’s Second Law

ma = F(x) = Ft∆x ∂ 2 ∂x2u(x,t) ρl∆x ∂ 2 ∂t2u(x,t) = Ft∆x ∂ 2 ∂x2u(x,t) ρl Ft ∂ 2 ∂t2u(x,t) = ∂ 2 ∂x2u(x,t)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

slide-49
SLIDE 49

logo1 Model Forces The Equation

The One-Dimensional Wave Equation

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

slide-50
SLIDE 50

logo1 Model Forces The Equation

The One-Dimensional Wave Equation

The equation of motion for small oscillations of a frictionless string is ∂ 2 ∂x2u(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.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

slide-51
SLIDE 51

logo1 Model Forces The Equation

The One-Dimensional Wave Equation

The equation of motion for small oscillations of a frictionless string is ∂ 2 ∂x2u(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 wave equation.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

slide-52
SLIDE 52

logo1 Model Forces The Equation

The One-Dimensional Wave Equation

The equation of motion for small oscillations of a frictionless string is ∂ 2 ∂x2u(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 wave equation. Our derivation is valid for small oscillations and negligible friction.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

slide-53
SLIDE 53

logo1 Model Forces The Equation

The One-Dimensional Wave Equation

The equation of motion for small oscillations of a frictionless string is ∂ 2 ∂x2u(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 wave equation. Our derivation is valid for small oscillations and negligible friction. The cancellation of the ∆x was “clean”.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String