Lecture 21 Revision session

1
Lecture 21Revision session
Remember Im available for questions all through
Christmas
Remember Phils Problems and your notes
everything
http//www.hep.shef.ac.uk/Phil/PHY226.htm
2
Revision for the exam
http//www.shef.ac.uk/physics/exampapers/2007-08/p
hy226-07-08.pdf
Above is a sample exam paper for this course
There are 5 questions. You have to answer Q1 but
then choose any 2 others
Previous years maths question papers are up on
Phils Problems very soon
Q1 Basic questions to test elementary concepts.
Looking at previous years you can expect complex
number manipulation, integration, solving ODEs,
applying boundary conditions, plotting functions,
showing x is solution of PDE. Easy stuff.
Q2-5 More detailed questions usually centred
about specific topics InhomoODE, damped SHM
equation, Fourier series, Half range Fourier
series, Fourier transforms, convolution, partial
differential equation solving (including applying
an initial condition to general solution for a
specific case), Cartesian 3D systems, Spherical
polar 3D systems, Spherical harmonics
The notes are the source of examinable material
NOT the lecture presentations
I wont be asking specific questions about Quantum
mechanics outside of the notes
3
Revision for the exam
The notes are the source of examinable material
NOT the lecture presentations
Things to do now
Read through the notes using the lecture
presentations to help where required.
At the end of each section in the notes try Phils
problem questions, then try the tutorial
questions, then look at your problem and homework
questions.
If you can do these questions (theyre fun) then
youre in excellent shape for getting over 80 in
the exam.
Look at the past exam papers for the style of
questions and the depth to which you need to know
stuff.
Youll have the standard maths formulae and
physical constants sheets (Ill put a copy of it
up on Phils Problems so you are sure whats on
it). You dont need to know any equations e.g.
Fourier series or transforms, wave equation,
polars.
Any problems see me in my office or email me
Same applies over holidays. Ill be in the
department most days but email a question or tell
me you want to meet up and Ill make sure Im in.
4
Concerned about what you need to know?
Look through previous exam questions. 2008/2009
exam will be of very similar style.
You dont need to remember any proofs or
solutions (e.g. Parseval, Fourier series, Complex
Fourier series) apart from damped SHM which you
should be able to do.
You dont need to remember any equations or trial
solutions, eg. Fourier and InhomoODE particular
solutions. APART FROM TRIAL FOR COMPLEMENTARY 2ND
ORDER EQUATION IS
You dont need to remember solutions to any PDE
or for example the Fourier transform of a
Gaussian and its key widths, etc. However you
should understand how to solve any PDE from start
to finish and how to generate a Fourier transform.
Things you need to be able to do
Everything with complex numbers solve ODEs and
InhomoODEs, apply boundary conditions integrate
and differentiate general stuff know even and
odd functions understand damped SHM, how to
derive its solutions depending on damping
coefficient and how to draw them how to
represent an infinitely repeating pattern as a
Fourier series, how to represent a pulse as a
sine or cosine half range Fourier series how to
calculate a Fourier transform how to
(de)convolve two functions the steps needed to
solve any PDE and apply boundary conditions and
initial conditions (usually using Fourier
series) how to integrate and manipulate
equations in 3D cartesian coordinates how to do
the same in spherical polar coordinates how to
prove an expression is a solution of a spherical
polar equation.
5
Lets take a quick look through the course and
then do the exam from last year
6
Binomial and Taylor expansions
7
Integrals
Try these integrals using the hints provided
8
More integrals
Summary


Previous page
Remember odd x even function
Previous page
9
Even and odd functions
So even x even even even x odd odd
odd x odd even
An even function is f(x)f(-x) and an odd
function is f(x) -f(-x),
10
Complex numbers
Argand diagram
Cartesian a ib
Imaginary
r
b
Real
q
a
Polar
so
where
11
Working with complex numbers
Add / subtract
Multiply / divide
Powers
12
Working with complex numbers
Roots
Example Step 1 write down z in polars with
the 2pp bit added on to the argument. Step 2
say how many values of p youll need (as many as
n) and write out the rooted expression ..
Step 3 Work it out for each value of p.
If what is z½?
here n 2 so Ill need 2 values of p p 0 and
p 1.
p 0
p 1
13
1st order homogeneous ODE
e.g. radioactive decay
1st method Separation of variables
gives
2nd method Trial solution
Guess that trial solution looks like
Substitute the trial solution into the ODE
Comparison shows that
so write
14
2nd order homogeneous ODE
Solving
Step 1 Let the trial solution be
Now substitute this back into the ODE
remembering that
and This is now called the auxiliary equation
Step 2 Solve the auxiliary equation for
and
Step 3 General solution is
or
if m1m2 For complex roots
solution is
which is same as
or
Step 4 Particular solution is found now by
applying boundary conditions
15
2nd order homogeneous ODE
Example 3 Linear harmonic oscillator with damping
Step 1 Let the trial solution be
So and
Step 2 The auxiliary is then
with roots
Step 3 General solution is then. HANG ON!!!!!
In the last lecture we determined the
relationship between x and t when
meaning that will always be real
What if or ???????????????????
16
2nd order homogeneous ODE
Example 3 Damped harmonic oscillator
Auxiliary is
roots are
BE CAREFUL THERE ARE THREE DIFFERENT CASES!!!!!
(i) Over-damped gives two real
roots
Both m1 and m2 are negative so x(t) is the sum of
two exponential decay terms and so tends pretty
quickly, to zero. The effect of the spring has
been made of secondary importance to the huge
damping, e.g. aircraft suspension
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2nd order homogeneous ODE
Example 3 Damped harmonic oscillator
Auxiliary is
roots are
BE CAREFUL THERE ARE THREE DIFFERENT CASES!!!!!
(ii) Critically damped gives a
single root
Here the damping has been reduced a little so the
spring can act to change the displacement
quicker. However the damping is still high enough
that the displacement does not pass through the
equilibrium position, e.g. car suspension.
18
2nd order homogeneous ODE
Example 3 Damped harmonic oscillator
Auxiliary is
roots are
BE CAREFUL THERE ARE THREE DIFFERENT CASES!!!!!
We do this so that W is real
19
2nd order homogeneous ODE
Example 3 Damped harmonic oscillator
Auxiliary is
roots are
BE CAREFUL THERE ARE THREE DIFFERENT CASES!!!!!

20
Inhomogeneous ordinary differential equations

21
Extra example of inhomo ODE
Solve
Step 1 With trial solution find
auxiliary is
Step 2 So treating it as a homoODE
Step 3 Complementary solution is
Step 4 Use the trial solution
and substitute it
in FULL expression.
so
cancelling
Comparing sides gives.
Solving gives
Step 5 General solution is
22
Finding partial solution to inhomogeneous ODE
using complex form
Sometimes its easier to use complex numbers
rather than messy algebra
Since we can write
then we can also say that
and
where Re
and Im refer to the real and imaginary
coefficients of the complex function.
Lets look again at example 4 of lecture 4 notes
Lets solve the DIFFERENT inhomo ODE
If we solve for X(t) and take only the real
coefficient then this will be a solution for
x(t)
Sustituting
so
Therefore since
take real part
23
Finding coefficients of the Fourier Series
Summary
The Fourier series can be written with period L as
The Fourier series coefficients can be found by-
Lets go through example 1 from notes
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Finding coefficients of the Fourier Series
Find Fourier series to represent this repeat
pattern.
Steps to calculate coefficients of Fourier series
1. Write down the function f(x) in terms of x.
What is period?
Period is 2p
2. Use equation to find a0?
3. Use equation to find an?
4. Use equation to find bn?
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Finding coefficients of the Fourier Series
4. Use equation to find bn?
5. Write out values of bn for n 1, 2, 3, 4,
5, .
6. Write out Fourier series with period L, an,
bn in the generic form replaced with values for
our example
26
Fourier Series applied to pulses
If the only condition is that the pretend
function be periodic, and since we know that even
functions contain only cosine terms and odd
functions only sine terms, why dont we draw it
either like this or this?
Odd function (only sine terms)
Even function (only cosine terms)
What is period of repeating pattern now?
27
Fourier Series applied to pulses
Half-range sine series
We saw earlier that for a function with period L
the Fourier series is-
where
In this case we have a function of period 2d
which is odd and so contains only sine terms, so
the formulae become-
where
Remember, this is all to simplify the Fourier
series. Were still only allowed to look at the
function between x 0 and x d
Im looking at top diagram
28
Fourier Series applied to pulses
Half-range cosine series
Again, for a function with period L the Fourier
series is-
where
Again we have a function of period 2d but this
time it is even and so contains only cosine
terms, so the formulae become-
where
Remember, this is all to simplify the Fourier
series. Were still only allowed to look at the
function between x 0 and x d
Im looking at top diagram
29
Fourier Series applied to pulses
Summary of half-range sine and cosine series
The Fourier series for a pulse such as can be
written as either a half range sine or cosine
series. However the series is only valid between
0 and d
Half range sine series
where
Half range cosine series
where
30
Fourier Transforms
where
where
The functions f(x) and F(k) (similarly f(t) and
F(w)) are called a pair of Fourier transforms
k is the wavenumber, (compare
with ).


31
Fourier Transforms
Example 1 A rectangular (top hat) function
Find the Fourier transform of the function
given that
This function occurs so often it has a name it
is called a sinc function.


32
Can you plot exponential functions?
The one-sided exponential function
What does this function look like?
The function
For any real number a the absolute value or
modulus of a is denoted by ?a? and is defined
as

What does this function look like?

33
Fourier Transforms
Example 4 The one-sided exponential function
Show that the function
has Fourier transform


34
Complexity, Symmetry and the Cosine Transform
We know that the Fourier transform from x space
into k space can be written as-
We also know that we can write
So we can say-
What is the symmetry of an odd function x an even
function ?
Odd
If f(x) is real and even what can we say about
the second integral above ? Will F(k) be real or
complex ?
2nd integral is odd (disappears) and F(k) is
real
If f(x) is real and odd what can we say about
the first integral above ? Will F(k) be real or
complex ?

1st integral is odd (disappears), F(k) is
complex
What will happen when f(x) is neither odd nor
even ?

Neither 1st nor 2nd integral disappears, and
F(k) is usually complex
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Complexity, Symmetry and the Cosine Transform
Since we say
lets see if we can shorten the amount of maths
required for a specific case
f(x) is real and even
As before the 2nd integral is odd, disappears,
and F(k) is real
so
But remember that
So

LETS GO BACKWARDS
Now since F(k) is real and even it must be true
that were we to then find the Fourier transform
of F(K) , this can be written-

36
Complexity, Symmetry and the Cosine Transform
Fourier cosine transform
Here is the pair of Fourier transforms which may
be used when f(x) is real and even only
Example 5 Repeat Example 1 using Fourier cosine
transform formula above.
Find F(k) for this function


37
Dirac Delta Function
The delta function d(x) has the value zero
everywhere except at x 0 where its value is
infinitely large in such a way that its total
integral is 1.
The Dirac delta function d(x) is very useful in
many areas of physics. It is not an ordinary
function, in fact properly speaking it can only
live inside an integral.
d(x x0) is a spike centred at x x0
d(x) is a spike centred at x 0
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Dirac Delta Function
The product of the delta function d(x x0) with
any function f(x) is zero except where x x0.
Formally, for any function f(x)
Example What is
?
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Dirac Delta Function
The product of the delta function d(x x0) with
any function f(x) is zero except where x x0.
Formally, for any function f(x)
Examples
(a) find
(b) find
(c) find the FT of
40
Convolutions
If the true signal is itself a broad line then
what we detect will be a convolution of the
signal with the resolution function
Resolution function
Convolved signal

True signal

We see that the convolution is broader then
either of the starting functions. Convolutions
are involved in almost all measurements. If the
resolution function g(t) is similar to the true
signal f(t), the output function c(t) can
effectively mask the true signal.
http//www.jhu.edu/signals/convolve/index.html


41
Deconvolutions
We have a problem! We can measure the resolution
function (by studying what we believe to be a
point source or a sharp line. We can measure the
convolution. What we want to know is the true
signal!
This happens so often that there is a word for it
we want to deconvolve our signal.

There is however an important result called the
Convolution Theorem which allows us to gain an
insight into the convolution process. The
convolution theorem states that-

i.e. the FT of a convolution is the product of
the FTs of the original functions.
We therefore find the FT of the observed signal,
c(x), and of the resolution function, g(x), and
use the result that
in order to find f(x).


If
then taking the inverse transform,
42
Deconvolutions
Of course the Convolution theorem is valid for
any other pair of Fourier transforms so not only
does ..
and therefore

allowing f(x) to be determined from the FT

but also
and therefore

allowing f(t) to be determined from the FT

43
Example of convolution
I have a true signal
between 0 lt x lt 8 which I detect using a device
with a Gaussian resolution function given by

What is the frequency distribution of the
detected signal function C(?) given that
?
Lets find F(?) first for the true signal
Lets find G(?) now for the resolution signal
44
Example of convolution
What is the frequency distribution of the
detected signal function C(?) given that
?
Lets find G(?) now for the resolution signal
so
We solved this in lecture 10 so lets go straight
to the answer
So if
then ..
and ..
45
Introduction to PDEs
In many physical situations we encounter
quantities which depend on two or more variables,
for example the displacement of a string varies
with space and time y(x, t). Handing such
functions mathematically involves partial
differentiation and partial differential
equations (PDEs).
Elastic waves, sound waves, electromagnetic
waves, etc.
Wave equation
Quantum mechanics
Schrödingers equation
Heat flow, chemical diffusion, etc.
Diffusion equation
Electromagnetism, gravitation, hydrodynamics,
heat flow.
Laplaces equation

Poissons equation
As Laplace but in regions containing mass,
charge, sources of heat, etc.
46
The principle of superposition
The wave equation (and all PDEs which we will
consider) is a linear equation, meaning that the
dependent variable only appears to the 1st power.
i.e. In
x never appears as x2 or x3 etc.
For such equations there is a fundamental theorem
called the superposition principle, which states
that if and are solutions of the
equation then
is also a solution, for any constants c1, c2.
Can you think when you used this principle last
year??
Waves and Quanta The net amplitude caused by two
or more waves traversing the same space
(constructive or destructive interference), is
the sum of the amplitudes which would have been
produced by the individual waves separately. All
are solutions to the wave equation.
Electricity and Magnetism Net voltage within a
circuit is the sum of all smaller voltages, and
both independently and combined they obey VIR.
47
The One-Dimensional Wave Equation
A guitarist plucks a string of length L such that
it is displaced from the equilibrium position as
shown at t 0 and then released.

Find the solution to the wave equation to predict
the displacement of the guitar string at any
later time t

Lets go thorugh the steps to solve the PDE for
our specific case ..
48
The One-Dimensional Wave Equation
Find the solution to the wave equation to predict
the displacement of a guitar string of length L
at any time t
Step 1 Separation of the Variables Since Y(x,t)
is a function of both x and t, and x and t are
independent of each other then the solutions will
be of the form

where the big X and T are functions of x and
t respectively. Substituting this into the wave
equation gives

Step 2 Rearrange equation Rearrange the equation
so all the terms in x are on one side and all the
terms in t are on the other

49
The One-Dimensional Wave Equation
Find the solution to the wave equation to predict
the displacement of a guitar string of length L
at any time t
Step 3 Equate to a constant Since we know that
X(x) and T(t) are independent of each other, the
only way this can be satisfied for all x and t is
if both sides are equal to a constant
Suppose we call the constant N. Then we have

and
(i) (ii)
which rearrange to

and
(i)
(ii)
50
The One-Dimensional Wave Equation
Find the solution to the wave equation to predict
the displacement of a guitar string of length L
at any time t
Step 4 Decide based on situation if N is
positive or negative We have ordinary
differential equations for X(x) and T(t) which
we can solve but the polarity of N affects the
solution ..
Linear harmonic oscillator
If N is negative

If N is positive
Unstable equilibrium
Which case we have depends on whether our
constant N is positive or negative. We need to
make an appropriate choice for N by considering
the physical situation, particularly the boundary
conditions.

Decide now whether we expect solutions of X(x)
and T(t) to be exponential or trigonometric ..
51
The One-Dimensional Wave Equation
Find the solution to the wave equation to predict
the displacement of a guitar string of length L
at any time t
From before
and
Step 4 Continued ..
Remember that if N is negative, solutions will
pass through zero displacement many times, whilst
if N is positive solutions only tend to zero
once.
From this we deduce N must be negative. Lets
write

So (i) becomes

From lecture 3, this has general solution
and in the same way
52
The One-Dimensional Wave Equation
Find the solution to the wave equation to predict
the displacement of a guitar string of length L
at any time t
Step 5 Solve for the boundary conditions for
X(x) In our case the boundary conditions are
Y(0, t) Y(L, t) 0. This means X(0) X(L)
0, i.e. X(x) is equal to zero at two different
points. (This was crucial in determining the
sign of N.)
Now we apply the boundary conditions X(0) 0
gives A 0. Saying B ? 0 then X(L) 0
requires sin kL 0, i.e. kL np. So k can
only take certain values where n
is an integer


So we have
for n 1, 2, 3, .
53
The One-Dimensional Wave Equation
Find the solution to the wave equation to predict
the displacement of a guitar string of length L
at any time t
Step 6 Solve for the boundary conditions for
T(t)
From previous page
By standard trigonometric manipulation we can
rewrite this as

54
The One-Dimensional Wave Equation
Find the solution to the wave equation to predict
the displacement of a guitar string of length L
at any time t
Step 7 Write down the special solution for
Y(x,t)
Hence we have special solutions
We see that each Yn represents harmonic motion
with a different wavelength (different
frequency). In the diagram below of course time
is fixed constant (as its a photo not a movie!!)

(Mistake in notes please correct harmonic
numbers in diagram below)
55
The One-Dimensional Wave Equation
Find the solution to the wave equation to predict
the displacement of a guitar string of length L
at any time t
Step 8 Constructing the general solution for
Y(x,t)
We have special solutions
Bearing in mind the superposition principle, the
general solution of our equation is the sum of
all special solutions


This is the most general answer to the problem.
For example, if a skipping rope was oscillated at
both its fundamental frequency and its 3rd
harmonic, then the rope would look like the
dashed line at some specific point in time and
its displacement could be described just by the
equation - (mistake in notes at top of page 4,
3rd not 2nd harmonic)
NB. The Fourier series is a further example of
the superposition principle.
56
The One-Dimensional Wave Equation
Find the solution to the wave equation to predict
the displacement of a guitar string of length L
at any time t
Step 8 continued Constructing the general
solution for Y(x,t)
Since
, if we differentiate we find
At t 0 the string is at rest, i.e.


For this to be true
for all n and x, and this is only true if
So the general solution becomes
57
The One-Dimensional Wave Equation
Find the solution to the wave equation to predict
the displacement of a guitar string of length L
at any time t
Step 9 Use Fourier analysis to find values of Bn
The guitarist plucked the string of length L such
that it was displaced from the equilibrium
position as shown and then released at t 0.
This shape can therefore be represented by the
half range sine (or cosine) series.
Half-range sine series


where
It can be shown (see Phils Problems 5.10) that
this shape can be represented by
58
The One-Dimensional Wave Equation
Find the solution to the wave equation to predict
the displacement of a guitar string of length L
at any time t
Step 9 continued Use Fourier series to find
values of Bn
Since Fourier series at t 0 is
and the general solution is
then at t 0 the
general solution is


and we see above that the coefficients Bn are the
coefficients of the Fourier series for the given
initial configuration at t 0. Therefore we
can write the general solution at t 0 as ..

for
59
The One-Dimensional Wave Equation
Find the solution to the wave equation to predict
the displacement of a guitar string of length L
at any time t
Step 10 Finding the full solution for all times
The solution at t 0 is
But we also know that the general solution at all
times is


Hence, by trusting the superposition principle
treating each harmonic as a separate oscillating
sinusoidal waveform which is then summed together
like a Fourier series to get the resulting shape,
we deduce that at later times the configuration
of the string will be-
60
The One-Dimensional Wave Equation
Find the solution to the wave equation to predict
the displacement of a guitar string of length L
at any time t
What does this all mean ????
This means that if you know the initial
conditions and the PDE that defines the
relationships between all variables, the full
solution can be found which describes the shape
at any later time.


Want to know how heat passes down a rod, how
light waves attenuate and interfere through a
prism, how to define time dependent Schrodinger
eigenfunctions, or how anything else that is a
linear function with multiple variables interacts
???? Then this is what you should use.
61
SUMMARY of the procedure used to solve PDEs
1. We have an equation
with supplied boundary conditions
2. We look for a solution of the form
3. We find that the variables separate
4. We use the boundary conditions to deduce the
polarity of N. e.g.
5. We use the boundary conditions further to
find allowed values of k and hence X(x).
so
6. We find the corresponding solution of the
equation for T(t).
7. We hence write down the special solutions.

• By the principle of superposition, the general
  solution is the sum of all special
• solutions..

9. The Fourier series can be used to find the
full solution at all times.
62
3D Coordinate Systems
2. Integrals in 3D Cartesian Coordinates
We have dV dx dy dz, and must perform a triple
integral over x, y and z. Normally we will only
work in Cartesians if the region over which we
are to integrate is cuboid.

Example 1 Find the 3D Fourier transform,

if
and
and
The integral is just the product of three 1D
integrals, and is thus easily evaluated
Just integrating over x gives
Mistake in notes
This is therefore a product of three sinc
functions, i.e.
63
Polar Coordinate Systems
1. Spherical Polar Coordinates
Spherical polars are the coordinate system of
choice in almost all 3D problems. This is because
most 3D objects are shaped more like spheres than
cubes, e.g. atoms, nuclei, planets, etc. And many
potentials (Coulomb, gravitational, etc.) depend
on radius.

Physicists define r, q, f as shown in the
figure. They are related to Cartesian
coordinates by
.
2. 3D Integrals in Spherical Polars
The volume element is
(given on data sheet).
To cover over all space, we take
Example 1 Show that a sphere of radius R has
volume 4pR3/3. So
64
Polar Coordinate Systems
3. ?2 in Spherical Polars Spherical Solutions

As given on the data sheet,
Example 3 Find spherically symmetric solutions
of Laplaces Equation ?2V(r) 0.
(Spherically symmetric means that V is a
function of r but not of q or f.)
Therefore we can say

Really useful bit!!!!
If (as in the homework) we were given an
expression for V(r) and had to prove that it was
a solution to the Laplace equation, then wed
just stick it here and start working outwards
until we found the LHS was zero.
If on the other hand we have to find V(r) then we
have to integrate out the expression.
65
Polar Coordinate Systems

Multiplying both sides by r2 gives
Integrating both sides gives
where A is a constant.
and so .
This rearranges to
.

Integrating we get the general solution
Weve just done Q3(i) of the homework
backwards!!! (see earlier note in red)
66
Extra tips for the exam
When we write
and say

We mean that y 5 when t 0
When we say prove that
is a solution of

then to answer the question STICK IT IN BOTH SIDES
When you solve the complementary solution of a
2nd order differential equation, you need to know
that the trial solution is ALWAYS
You also need to know the different forms of the
complementary solutions