Lecture 21 Revision session

1. Lecture 21Revision session Remember I’m 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/phy226-07-08.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 (they’re fun) then you’re 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. You’ll have the standard maths formulae and physical constants sheets (I’ll put a copy of it up on Phils Problems so you are sure what’s on it). You don’t 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. I’ll be in the department most days but email a question or tell me you want to meet up and I’ll make sure I’m in.

4. Concerned about what you need to know? Look through previous exam questions. 2008/2009 exam will be of very similar style. You don’t 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 don’t need to remember any equations or trial solutions, eg. Fourier and InhomoODE particular solutions. APART FROM TRIAL FOR COMPLEMENTARY 2ND ORDER EQUATION IS You don’t 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. Let’s 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 2πp bit added on to the argument. Step 2: say how many values of p you’ll 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 I’ll 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 m1=m2 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 m1and m2are 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

17. 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. (iii) Under-damped This will yield complex solutions due to presence of square root of a negative number. Let so thus As before general solution with complex roots can be written as The solution is the product of a sinusoidal term and an exponential decay term – so represents sinusoidal oscillations of decreasing amplitude. E.g. bed springs. 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 Step 1: Find the general solution to the related homogeneous equation and call it the complementary solution . Step 2: Find the particular solution of the equation by substituting an appropriate trial solution into the full original inhomogeneous ODE. e.g. If f(t) = t2try xp(t) = at2 + bt + c If f(t) = 5e3ttry xp(t) = ae3t If f(t) = 5eiωt try xp(t) =aeiωt If f(t) = sin2t try xp(t) = a(cos2t) + b(sin2t) If f(t) = cos wt try xp(t) =Re[aeiωt] see later for explanation If f(t) = sin wt try xp(t) =Im[aeiωt] If your trial solution has the correct form, substituting it into the differential equation will yield the values of the constants a, b, c, etc. Step 3: The complete general solution is then . Step 4: Apply boundary conditions to find the values of the constants

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 it’s 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. Let’s look again at example 4 of lecture 4 notes Let’s 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:- Let’s go through example 1 from notes…

24. 1 0 p 2p 3p x 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?

25. 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 don’t 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. We’re still only allowed to look at the function between x = 0 and x = d I’m 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. We’re still only allowed to look at the function between x = 0 and x = d I’m 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

35. Complexity, Symmetry and the Cosine Transform Since we say let’s 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 LET’S 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 Fouriercosine 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) is a spike centred at x = 0 d(x – x0) is a spike centred at x = x0

38. 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 ?

39. 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 < x < ∞ which I detect using a device with a Gaussian resolution function given by What is the frequency distribution of the detected signal function C(ω) given that ? Let’s find F(ω) first for the true signal … Let’s find G(ω) now for the resolution signal …

44. Example of convolution What is the frequency distribution of the detected signal function C(ω) given that ? Let’s find G(ω) now for the resolution signal … so We solved this in lecture 10 so let’s 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). Wave equation Elastic waves, sound waves, electromagnetic waves, etc. Schrödinger’s equation Quantum mechanics Diffusion equation Heat flow, chemical diffusion, etc. Laplace’s equation Electromagnetism, gravitation, hydrodynamics, heat flow. Poisson’s 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 V=IR.

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 Let’s 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 ….. If N is negative Linear harmonic oscillator 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 …..