Imperial College London BSc/MSci EXAMINATION June 2008 This paper is also taken for the relevant Examination for the Associateship MATHS & STATISTICS OF MEASUREMENT For Second-Year Physics Students Tuesday, 10th June 2008: 10:00 to 12:00 Answer ALL parts of Section A and TWO parts of section B. Marks shown on this paper are indicative of those the Examiners anticipate assigning. General Instructions Write your CANDIDATE NUMBER clearly on each of the 4 answer books provided. If an electronic calculator is used, write its serial number in the box at the top right hand corner of the front cover of each answer book. USE ONE ANSWER BOOK FOR EACH QUESTION. Enter the number of each question attempted in the horizontal box on the front cover of its corresponding answer book. Hand in 4 answer books even if they have not all been used. You are reminded that Examiners attach great importance to legibility, accuracy and clarity of expression. c Imperial College London 2008 xxxx/y/zzz 1 Turn over for questions
SECTION A 1. (i) In class we showed that the square wave function { 0 1/2 < x < 0 f (x) = 1 0 < x < +1/2 can be represented by the Fourier Series f (x) = 1 2 + 2 ( sin(2πx) + sin(6πx) π 1 3 Consider the function g(x) = + sin(10πx) 5 { +1 π < x < 0 1 0 < x < +π. ) +. By expressing g(x) in terms of f (x) (or otherwise), find its Fourier Series representation. To what values does the Fourier Series converge at x = 0, π,π? [3 marks] (ii) By means of the substitution y = u(x)x determine the general solution of the following differential equation: dy dx = y x + 3y2 x 2. Verify your answer by substituting your solution back into the differential equation. (iii) Determine the general solution of the following differential equation: d 2 y dx 2 6dy + 9y = 0. dx [3 marks] [Total 10 marks] xxxx/y/zzz 2 Please turn over
2. (i) A variable f (x,y) = ylnx is to be determined by measurements of independent variables x and y. Write down the expected error in f if the measurement errors in x and y are σ x and σ y respectively. [2 marks] (ii) On a rainy day it is observed that, on average, one drop falls every second in an area of one cm 2 of the surface of a pond. What is the probability that in any given second one or zero drops will fall in an area of 4 cm 2 of the pond s surface? [5 marks] (iii) A measurement of a variable y gave the following results: 1.30961 1.30485 1.29016 1.29553 1.30524 Give an estimate of the average of y and its error. [3 marks] [Total 10 marks] xxxx/y/zzz 3 Please turn over
SECTION B 3. Laguerre s differential equation is: x d2 y + (1 x)dy + my = 0, dx2 dx where m is a constant, not necessarily integer. (i) By substituting into the equation, determine the coefficients C 1, C 2, C 3 of the power series solution: y = C 0 +C 1 x +C 2 x 2 +C 3 x 3 +... in terms of the single arbitrary constant C 0. Verify that your answers agree with the general expression for the x n coefficient: n m(m 1)...(m n + 1) C n = ( 1) n!n! (ii) Determine the radius of convergence of the series. C 0 [10 marks] (iii) When the constant m is an integer the power series terminates. In this case, writing the solution of Laguerre s equation as y = C 0 L m (x), the term L m (x) is the Laguerre polynomial. Write down complete expressions for the first three polynomials L 0 (x), L 1 (x), L 2 (x). Verify that L 0, L 1 are orthogonal on the interval [0, ] with respect to the weight function e x, by evaluating the integral: 0 e x L 0 L 1 dx [6 marks] xxxx/y/zzz 4 Please turn over
4. A taut wire is fixed at points x = 0, x = L. The displacement u(x,t) is governed by the wave equation 2 u x 2 = 1 2 u v 2 t 2, where v is the speed of propagation. (i) By assuming a separable solution of the form u(x,t) = X(x)T (t), derive ordinary differential equations to be solved for each of the functions X(x), T (t). [6 marks] (ii) Show that the boundary conditions impose a general solution of the form u = sin n=1 ( nπx )[ A n sin L ( nπvt ) + B n cos L ( nπvt )] L [6 marks] (iii) At t = 0 the displacement is everywhere zero, at which time the wire is hit at the mid-point such that u t (x,0) = 2a 0x/L 0 < x L/2 u t (x,0) = 2a 0(1 x/l) L/2 < x L Determine expressions for the coefficients A n, B n of the solution. [8 marks] xxxx/y/zzz 5 Please turn over
5. The Gaussian distribution function is proportional to [ g(x) = exp 1 x 2 ] 2 σ 2. The Fourier Transform of g(x) is given by g(ω) = 2πσ 2 exp [ 12 ] ω2 σ 2. (i) Use this and the definition of the Fourier Transform below to calculate the normalization factor K such that G(x) = Kg(x) is a properly-normalized probability distribution (i.e., G(x) dx = 1). (ii) The convolution of f (x) with h(x) is given by [ f h](x) = + f (x x)h(x ) dx. Write down the convolution theorem, relating the Fourier Transform of [ f h](x) to the transforms of f (x) and h(x) [2 marks] (iii) Show that the convolution of a Gaussian of width σ 1 with another Gaussian of width σ 2 is a Gaussian, too. What is its width? (iv) What is the Fourier Transform of the delta function, δ(x)? [2 marks] (v) (a) Sketch the normalized Gaussian function G(x) for σ = 2,1, and 0.5 and roughly indicate its behaviour for σ = 0.01. From your sketches, what is the limiting value for G(x), x 0, as σ 0? (b) Show that the σ 0 limit of the normalized Gaussian function, G(x), satisfies the following properties of the delta function: normalization, δ(x) dx = 1 ; the sifting property, δ(x x 0 ) f (x) dx = f (x 0 ), for the case when f (x) is a Gaussian function of width σ; and that it has the same Fourier transform as δ(x). [8 marks] If needed, the inverse and forward Fourier Transforms are given by f (x) = + dω 2π f + (ω)e iωx and F [ f ](ω) = f (ω) = dx f (x)e iωx. xxxx/y/zzz 6 Please turn over
6. (i) The probability of obtaining a result x to be in the infinitesimal interval between x and x + dx is proportional to x. The result can lie in the range 0 x l. (a) Derive an expression for the probability distribution function P(x) for x and sketch the function. (b) Calculate the variance σx 2 of x and indicate the expected 1-σ range in your sketch. [5 marks] (c) Sketch the distribution if the probability is proportional to x over the range l x l instead and indicate the 1-σ range for this case in a separate sketch. (ii) A set of independent samples x 1, x 2, x 3,..., x N are drawn randomly. (a) Write down the estimators ˆµ and ˆσ 2 you would use to obtain unbiased estimates of the parent distribution mean µ and variance σ 2 respectively. [3 marks] (b) Hence show that an estimate of the error in ˆµ is given by σ ˆµ = ˆσ N. xxxx/y/zzz 7 End of examination paper