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Fourier Series. Dr. K.W. Chow Mechanical Engineering. Introduction. Conceptual question: While one can readily see that two vectors can be ‘perpendicular’ or ‘orthogonal’, how can we extend this concept to a sequence of functions?. Introduction.

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fourier series

Fourier Series

Dr. K.W. Chow

Mechanical Engineering

introduction
Introduction
  • Conceptual question: While one can readily see that two vectors can be ‘perpendicular’ or ‘orthogonal’, how can we extend this concept to a sequence of functions?
introduction1
Introduction
  • A general formulation: For a sequence of functions {φn} and

f(x) = Σcn φn

What is cn?

IF ∫ φm φn dx = 0 for m, n different, then

cn can be found fromthis ‘orthogonal’ property.

introduction2
Introduction
  • A general theory has been developed for linear, second order differential equations regarding these issues:

(a) Orthogonal ‘eigenfunctions’?

(b) Completeness in terms of expansion? (i.e. is it possible for any arbitrary function f(x) to be represented as a sum of φn(x)?)

introduction3
Introduction
  • Sin(x) and Cos(x) are the solutions of the simplest second order ordinary differential equations (ODEs)
  • d2y/dx2 + y = 0,
  • subject to certain boundary conditions.
introduction4
Introduction
  • Fourier series is an infinite series of sine and cosine
  • Dirichlet’s theorem (Sufficient, but not necessary):

If a function is 2L-periodic and piecewise continuous, then its Fourier series converges

  • At a point of discontinuity, the series converges to the mean value.
fourier series1
Fourier series
  • an and bn are calculated by the orthogonal properties of sines and cosines.
  • If one uses a0 as the constant term, two schemes for defining an, n = 0and n > 0.
  • If one uses a0/2, then one definition for all an .
introduction5
Introduction
  • Consider:
slide10

f(x)

n = 3

slide11

f(x)

n = 3

n = 7

slide12

f(x)

n = 3

n = 7

n = 15

introduction6
Introduction
  • The n = 15 series gives a very good approximation to the original function
introduction7
Introduction

Even function

Odd function

introduction8
Introduction
  • Even extension of a function results in Fourier cosine series
  • Odd extension of a function results in Fourier sine series

(Assuming function is given only in half the interval.)

introduction9
Introduction

Even extension

Odd extension

introduction10
Introduction

Even extension of f : Fourier series of f1will have cosine terms only

Odd extension of f : Fourier series of f2will have sine terms only

slide18
Consider

using odd extension

slide20

f(x)

n = 3

slide21

f(x)

n = 3

n = 7

slide22

f(x)

n = 3

n = 7

n = 50

introduction11
Introduction

The Fourier series converges to this value at the discontinuity.

introduction13
Introduction

Approximation using n = 1 series

introduction14
Introduction

Approximation using n = 2 series

introduction15
Introduction

Approximation using n = 3 series

introduction16
Introduction
  • Gibbs phenomenon – large oscillations of the series near a discontinuity.
introduction17
Introduction
  • Consider the square wave again:

5 terms

introduction18
Introduction

25 terms

introduction19
Introduction

25 terms

introduction20
Introduction
  • Animation to show the Gibbs phenomenon using different partial sums.
differentiation of series
Differentiation of series
  • Fourier coefficients are determined uniquely using orthogonality of trigonometric functions.

only when the series on the right is uniformly convergent. (d/dx = a local operator)

integration of series
Integration of series
  • Always permitted, but the resulting series is not a Fourier series, unless the constant term a0 is zero.
  • (Integration = a global operator).
1d heat conduction in finite domain
1D heat conduction in finite domain
  • Configuration:

Density of conductor:

Specific heat capacity: c

Heat conduction coefficient: k

Cross-sectional area: A

Temperature:

Temperature:

L

1d heat conduction in finite domain1
1D heat conduction in finite domain

Thermal conductivity defined by

  • Heat flux = - k A∂u/∂x

where A = cross sectional area,

u = temperature

(i.e. k is heat flux per unit area per unit temperature gradient).

1d heat conduction in finite domain2
1D heat conduction in finite domain

Assumptions:

  • Heat flow in the xdirection only
  • No external heat source
  • No heat loss
1d heat conduction in finite domain3
1D heat conduction in finite domain
  • Consider heat conduction across an infinitesimal element of the conductor:

Heat out

Heat in

1d heat conduction in finite domain4
1D heat conduction in finite domain
  • Heat flux at the left surface : - k A∂u(x,t)/∂x
  • Heat flux at the right surface:

- k A ∂u(x,t)/∂x - ∂[k A ∂u(x,t)/∂x]/∂x dx + …

  • Net heat flux INTO the element:
  • ∂[k A ∂u(x,t)/∂x]/∂x dx
1d heat conduction in finite domain5
1D heat conduction in finite domain
  • Net heat must be used to heat up the element (c = specific heat capacity):
1d heat conduction in finite domain6
1D heat conduction in finite domain
  • Therefore (if k and A not functions of x):
1d heat conduction in finite domain7
1D heat conduction in finite domain
  • When , the above equation becomes exact:

This is the 1D heat conduction equation in finite domain.

1d heat conduction in finite domain8
1D heat conduction in finite domain

Solution procedure

  • Separation of variables:
  • Substitute back into the heat equation:
1d heat conduction in finite domain10
1D heat conduction in finite domain
  • Assume both ends are kept at :
  • For non-trivial solution, choose:
1d heat conduction in finite domain11
1D heat conduction in finite domain
  • F satisfies the differential equation:
  • For non-trivial solutions:
  • The temporal part:
1d heat conduction in finite domain12
1D heat conduction in finite domain
  • Overall solution:
  • Using superposition principle, we obtain general solution:
1d heat conduction in finite domain13
1D heat conduction in finite domain
  • is the Fourier sine coefficient:
1d heat conduction in finite domain14
1D heat conduction in finite domain
  • is called the eigen-value
  • is called the

corresponding eigen-function

slide52

t = 0

t = 0.5

slide53

t = 0

t = 0.5

t = 1.2

slide54

t = 0

t = 0.5

t = 1.2

t = 5

slide55

t = 0

t = 0.5

t = 1.2

t = 5

t = 15

1d heat conduction in finite domain16
1D heat conduction in finite domain
  • The above procedure cannot be applied directly when the end points are not at
  • First find steady-state temperature distribution v:

Note that the steady-state temperature depends on x only.

1d heat conduction in finite domain17
1D heat conduction in finite domain
  • Introduce a new function (transient):
  • w satisfies the heat equation with homogeneous boundary conditions.
  • Solve for w with separation of variables and hence u can be found.
slide61

t = 0

t = 0.5

slide62

t = 0

t = 0.5

t = 1.2

slide63

t = 0

t = 0.5

t = 1.2

t = 5

slide64

t = 0

t = 0.5

t = 1.2

t = 5

t = 12

laplace s equation
Laplace’s equation
  • For heat conduction in higher dimensions,

where is the Laplacian.

laplace s equation1
Laplace’s equation
  • The steady-state solution in 2D satisfies:

which is the Laplace’s equation

  • In the presence of heat sources, u satisfies the Poisson’s equation:
laplace s equation2
Laplace’s equation

3 types of boundary conditions:

  • Dirichlet boundary condition
  • Neumann boundary condition
  • Robin boundary condition
laplace s equation3
Laplace’s equation
  • Dirichlet boundary condition
laplace s equation4
Laplace’s equation
  • Neumann boundary condition
laplace s equation5
Laplace’s equation
  • Robin boundary condition
laplace s equation6
Laplace’s equation

Consider (Dirichlet b.c.)

laplace s equation7
Laplace’s equation

u(x, y) = F(x) G(y)

F’’/F = – G’’/G = constant

F(0) = F(a) = 0 and thus the constant is

– n2π2 / L2 .

Hence

F ~ sin (n πx/L)

G ~ cosh

laplace s equation8
Laplace’s equation
  • Solution is obtained using separation of variables:
slide77

y

x

Isotherms (curves joining points with the same temperature) of the problem

1d wave equation in finite domain
1D wave equation in finite domain
  • Configuration:
  • Assumptions:

- CONSTANT TENSION and density,

- the slope of the vibration is small,

- gravity much smaller than tension.

L

1d wave equation in finite domain2
1D wave equation in finite domain
  • Vertical force at the left end:

Vertical force at the right end:

  • u = u(x, t) = displacement of the string
1d wave equation in finite domain3
1D wave equation in finite domain
  • Net vertical upward force on the element:
1d wave equation in finite domain4
1D wave equation in finite domain
  • Newton’s second law (ρ = linear density):
1d wave equation in finite domain5
1D wave equation in finite domain
  • When , the equation becomes exact.
  • If no external force is present:
1d wave equation in finite domain6
1D wave equation in finite domain
  • c = speed of the wave
  • Check the dimensions: Square root of (force/mass per unit length).
  • Mathematically, signs of two second derivatives same (contrast with Laplace equation).
1d wave equation in finite domain7
1D wave equation in finite domain
  • Consider a vibrating string with two ends fixed and initial position and velocity given;
  • i.e. initial and boundary conditions:
separation of variables
Separation of Variables

u(x, y) = F(x) G(t)

F’’/F = G’’/(c2 G) = constant

F(0) = F(a) = 0 and thus the constant is

– n2π2 / L2 .

Hence

F ~ sin (n πx/L)

G ~ C1sin (c n πt/L)+C2 cos (c n πt/L)

1d wave equation in finite domain8
1D wave equation in finite domain
  • Solution obtained by separation of variables:
1d wave equation in finite domain9
1D wave equation in finite domain
  • Modes of vibration are the profiles of the envelopes corresponding to different n

First mode

Second mode

Third mode

1d wave equation in finite domain10
1D wave equation in finite domain
  • The mode shapes will oscillate with time
1d wave equation in finite domain11
1D wave equation in finite domain
  • Another example – Vibration of a stretched string with a triangular initial profile.
  • Qualitatively, the shapes of the string at subsequent times are similar to the n = 1 mode previously. However, the shapes remain piecewise linear, and some ‘corners’ persist.
slide93

t = 0

Forward cycle

slide94

t = 0

Forward cycle

t = L/6c

slide95

t = 0

Forward cycle

t = L/6c

t = L/3c

slide96

t = 0

Forward cycle

t = L/6c

t = L/3c

t = L/2c

slide97

t = 0

Forward cycle

t = L/6c

t = L/3c

t = L/2c

t = 2L/3c

slide98

t = 0

Forward cycle

t = L/6c

t = L/3c

t = L/2c

t = 2L/3c

t = 5L/6c

slide100

Backward cycle

t = 7L/6c

t = L/c

slide101

Backward cycle

t = 4L/3c

t = 7L/6c

t = L/c

slide102

Backward cycle

t = 3L/2c

t = 4L/3c

t = 7L/6c

t = L/c

slide103

Backward cycle

t = 5L/3c

t = 3L/2c

t = 4L/3c

t = 7L/6c

t = L/c

slide104

Backward cycle

t = 11L/6c

t = 5L/3c

t = 3L/2c

t = 4L/3c

t = 7L/6c

t = L/c

slide105

Backward cycle

t = 2L/c

t = 11L/6c

t = 5L/3c

t = 3L/2c

t = 4L/3c

t = 7L/6c

t = L/c

1d wave equation in finite domain14
1D wave equation in finite domain
  • General solution (d’Alembert) to the wave equation:
  • Using trigonometric identities, the Fourier series solution can also be rewritten in the above form.
1d wave equation in finite domain15
1D wave equation in finite domain
  • x- ctrepresents a wave traveling to the right with speed c
1d wave equation in finite domain16
1D wave equation in finite domain
  • x+ ct represents a wave traveling to the left with speed c
1d wave equation in finite domain17
1D wave equation in finite domain
  • The lines are known as characteristic lines (or just characteristics).
  • The forms of functions Fand G depend on the initial conditions.
  • The initial profile splits into two waves of same amplitude traveling in opposite directions
1d wave equation in finite domain19
1D wave equation in finite domain
  • Method of characteristics is of tremendous theoretical significance but less practical interest.
  • Usually we just use separation of variables for finite domains and integral transform for infinite ones.
pdes in infinite domain
PDEs in infinite domain
  • Fourier series gives information in the interval
  • Fourier series will give the periodic extension outside this domain.
  • Fourier ‘fails’ if the given function is already defined along the whole real-axis.
from fourier series to fourier integrals
From Fourier series to Fourier integrals
  • To represent a function from minus infinity to plus infinity, we use a Fourier series over the interval (- L, L) and let L go to infinity.
  • Result (Fourier integral):
  • f = integral (integral f dξ) dx
from fourier series to fourier integrals1
From Fourier series to Fourier integrals
  • f(x) = sum over An cos (nπx/L)
  • An = 2/L integral f(ξ) cos (nπξ/L)dξ
  • Hence
  • f(x) = sum over n [ 2/L (integral of

f(ξ) cos (nπ (x – ξ)/L) dξ)]

Now convert ‘sum over n and (1/L)’ into another integral.

pdes in infinite domain1
PDEs in infinite domain
  • Separation of variables is usually not feasible or will fail.
  • Use integral transforms:

- Fourier transform

- Laplace transform

- …

pdes in infinite domain2
PDEs in infinite domain
  • Fourier integrals are analogous to Fourier series:
pdes in infinite domain3
PDEs in infinite domain
  • For functions defined in semi-infinite domain,

even extension Fourier cosine integral

odd extension Fourier sine integral

pdes in infinite domain4
PDEs in infinite domain
  • Fourier transform pair:
pdes in infinite domain5
PDEs in infinite domain
  • In some alternative versions, the + and - signs in the exponentials are interchanged.
  • There is no universally accepted format.
  • The constants in front of the integral signs are arbitrary, as long as their product is
applications of fourier transform
Applications of Fourier transform

Idea:

  • A PDE defined in an infinite domain is given.
  • Apply transform on each term in the equation, with respect to a certain independent variable, e.g. x
  • Derivatives in x become algebraic in ω.
applications of fourier transform1
Applications of Fourier transform
  • The transformed equation becomes an ODE in t (ω is a parameter, no derivatives in ω), rather than PDE in x and t.
  • Solve for the transformed function.
  • Apply inverse transform to obtain solution in the original coordinates.
applications of fourier transform2
Applications of Fourier transform

Common techniques:

  • Integration by parts
  • Exchange order of integrations
  • Contour integrals
  • Gaussian integrals
applications of fourier transform3
Applications of Fourier transform
  • Example : heat equation

Spatial conditions: u(x, t) decaying in far field.

This represents an initially concentrated source of unit intensity at the origin

slide126

t = 0.05

t = 0.1

slide127

t = 0.05

t = 0.1

t = 0.5

slide128

t = 0.05

t = 0.1

t = 0.5

t = 5

slide129

t = 0.05

t = 0.1

t = 0.5

t = 5

t = 50

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