pa214 waves and fields fourier methods blue book new chapter 12 fourier sine series application to...
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PA214 Waves and Fields Fourier Methods
Blue book New
chapter 12
โข Fourier sine seriesโข Application to the wave equation
โข Fourier cosine seriesโข Fourier full range series
โข Complex form of Fourier seriesโข Introduction to Fourier transforms
and the convolution theorem
Fourier Methods
Dr Mervyn Roy (S6)www2.le.ac.uk/departments/physics/people/academic-staff/mr6
PA214 Waves and Fields Fourier Methods
Lecture notes www2.le.ac.uk/departments/physics/people/academic-staff/mr6
214 course texts โข Blue book, new chapter 12 available on Blackboard
Notes on Blackboardโข Notes on symmetry and on trigonometric identitiesโข Computing exercisesโข Exam tips
โข mock papersBooks
โข Mathematical Methods in the Physical Sciences (Mary L. Boas)โข Library!
Resources
PA214 Waves and Fields Fourier Methods
The wave equation
for a string fixed at and has harmonic solutions
Introduction
๐2 ๐ฆ๐ ๐ฅ2
= 1๐2๐2 ๐ฆ๐๐ก 2
,
๐ฆ (๐ฅ , ๐ก )=sin ๐๐ x๐ฟ (๐๐cos
๐๐๐๐ก๐ฟ
+๐๐sin๐๐๐๐ก๐ฟ ) .
Superposition tells us that sums of such terms must also be solutions,
๐ฆ (๐ฅ , ๐ก )=โ๐
sin๐๐ x๐ฟ (๐๐ cos
๐๐ ๐๐ก๐ฟ
+๐๐sin๐๐๐๐ก๐ฟ ) .
PA214 Waves and Fields Fourier Methods
๐ฆ (๐ฅ , ๐ก )=โ๐
sin๐๐ x๐ฟ (๐๐ cos
๐๐ ๐๐ก๐ฟ
+๐๐sin๐๐๐๐ก๐ฟ )
Set coefficients from initial conditions, e.g. string released from rest with then
PA214 Waves and Fields Fourier Methods
What happens if the initial shape of the string is something more complex?
In general can be any function
The implication is that we can represent any function as a sum of sines
โฆ and/or cosines or complex exponentials.
This is Fourierโs theorem
PA214 Waves and Fields Fourier Methods
We will find that a function in the range can be represented by the Fourier sine series
where
Fourier sine series (half-range)
๐๐=2๐ฟโซ0
๐ฟ
๐ (๐ฅ ) sin ๐๐ ๐ฅ๐ฟ
๐๐ฅ .
PA214 Waves and Fields Fourier Methods
How does this work ?Need a standard integral (new chapter 12 โ A.2)โฆ
Fourier sine series
โซ0
๐ฟ
sin๐๐ ๐ฅ๐ฟ
sin๐๐ ๐ฅ๐ฟ
๐๐ฅ= ๐ฟ2๐ฟ๐๐
PA214 Waves and Fields Fourier Methods
Square wave,
๐ (๐ฅ )=โ๐=1
โ
๐๐sin๐๐ ๐ฅ๐ฟ
,
๐๐=2๐ฟโซ0
๐ฟ
๐ (๐ฅ ) sin ๐๐ ๐ฅ๐ฟ
๐๐ฅ .
PA214 Waves and Fields Fourier Methods
Square wave,
๐ (๐ฅ )=โ๐=1
โ
๐๐sin๐๐ ๐ฅ๐ฟ
,
๐๐=2๐ฟโซ0
๐ฟ
๐ (๐ฅ ) sin ๐๐ ๐ฅ๐ฟ
๐๐ฅ .
๐ (๐ฅ )โ 4๐sin
๐ ๐ฅ๐ฟ
PA214 Waves and Fields Fourier Methods
Square wave,
๐ (๐ฅ )=โ๐=1
โ
๐๐sin๐๐ ๐ฅ๐ฟ
,
๐๐=2๐ฟโซ0
๐ฟ
๐ (๐ฅ ) sin ๐๐ ๐ฅ๐ฟ
๐๐ฅ .
๐ (๐ฅ )โ 4๐sin
๐ ๐ฅ๐ฟ
+43๐
sin3๐ x๐ฟ
PA214 Waves and Fields Fourier Methods
Square wave,
๐ (๐ฅ )=โ๐=1
โ
๐๐sin๐๐ ๐ฅ๐ฟ
,
๐๐=2๐ฟโซ0
๐ฟ
๐ (๐ฅ ) sin ๐๐ ๐ฅ๐ฟ
๐๐ฅ .
๐ (๐ฅ )โ 4๐sin
๐ ๐ฅ๐ฟ
+43๐
sin3๐ x๐ฟ
+45๐
sin5๐ ๐ฅ๐ฟ
PA214 Waves and Fields Fourier Methods
Square wave,
๐ (๐ฅ )=โ๐=1
โ
๐๐sin๐๐ ๐ฅ๐ฟ
,
๐๐=2๐ฟโซ0
๐ฟ
๐ (๐ฅ ) sin ๐๐ ๐ฅ๐ฟ
๐๐ฅ .
๐ (๐ฅ )โ 4๐sin
๐ ๐ฅ๐ฟ
+43๐
sin3๐ x๐ฟ
+45๐
sin5๐ ๐ฅ๐ฟ
+47๐
sin7๐ ๐ฅ๐ฟ
PA214 Waves and Fields Fourier Methods
Square wave,
๐ (๐ฅ )=โ๐=1
โ
๐๐sin๐๐ ๐ฅ๐ฟ
,
๐๐=2๐ฟโซ0
๐ฟ
๐ (๐ฅ ) sin ๐๐ ๐ฅ๐ฟ
๐๐ฅ .
๐ (๐ฅ )โ 4๐sin
๐ ๐ฅ๐ฟ
+43๐
sin3๐ x๐ฟ
+โฆ+419๐
sin19๐ ๐ฅ๐ฟ
PA214 Waves and Fields Fourier Methods
Periodic extension of Fourier sine series
๐ (๐ฅ )=โ๐=1
โ
๐๐sin๐๐ ๐ฅ๐ฟ
sin๐๐ ๐ฅ๐ฟ
=โ sinโ๐๐ ๐ฅ
๐ฟ,
sin๐๐ ๐ฅ๐ฟ
=sin๐๐ (๐ฅ+2๐ฟ)
๐ฟ.
and that
We know that sine waves have odd symmetry,
PA214 Waves and Fields Fourier Methods
Within can expand any function as a sum of sine waves,
๐ (๐ฅ )=โ๐=1
โ
๐๐sin๐๐ ๐ฅ๐ฟ
.
How does this expansion behave outside of the range ?
PA214 Waves and Fields Fourier Methods
String fixed at and
The wave equation
Initial conditions and
๐ฆ (๐ฅ , ๐ก )=โ๐
sin๐๐ x๐ฟ (๐ต๐cos
๐๐๐๐ก๐ฟ
+๐ด๐sin๐๐๐๐ก๐ฟ ).
๐ต๐=2๐ฟโซ0
๐ฟ
๐ (๐ฅ ) sin ๐๐ ๐ฅ๐ฟ
๐๐ฅ
๐๐ ๐๐ด๐
๐ฟ= 2๐ฟโซ0
๐ฟ
๐ (๐ฅ ) sin ๐๐ ๐ฅ๐ฟ
๐๐ฅ
PA214 Waves and Fields Fourier Methods
Can go through the same procedure with the solutions to other PDEse.g. Laplace equation (see workshop 1 exercise 3),
๐2๐๐๐ฅ2
+ ๐2๐๐ ๐ฆ2
=0.
Imagine the boundary conditions are and then
๐ (๐ฅ , ๐ฆ )=โ๐=1
โ
๐ต๐sin๐๐ x๐ฟ
๐โ๐๐ ๐ฆ /๐ฟ
and can find coefficients from boundary condition for