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Maths 4 Page 1 of 2 Turn the page over Module code: PHYS238001 Module Title: Maths 4 ©UNIVERSITY OF LEEDS Resit Mid-Term Assessment School of Physics and Astronomy Semester Two 2020/21 Calculator instructions: You are allowed to use a calculator or a computer calculator in this assessment. Dictionary instructions: You are allowed to use your own dictionary in this assessment and/or the spell-checker facility on your computer. Assessment information: • This assessment is made up of 2 pages and is worth 30% of the module mark. • You have 48 hours to complete this open book online assessment. • You are recommended to take a maximum of 1 hour within the time available to complete the assessment. • You must answer all of the questions in this assessment. • You should indicate the final answer to each question by underlining it. At the end of each answer you should cite any websites or textbooks other than the course materials and recommended text books that you have used specifically to answer that question. You should always answer in your own words and not repeat material verbatim and you should explain each step of your working. • You must upload your answers via Minerva to GradeScope within the time allowed. You are advised to allow up to four hours to photograph your answers, and upload as a PDF to GradeScope. • When submitting your work, you must identify which questions are answered on which uploaded pages. You must also check that you have uploaded all the work you wish to be marked as part of this assessment and that the answers uploaded are clearly legible. Failure to do so may result in your work not being marked. • If there is anything that needs clarification or you have any problems, please email the module leader or
[email protected] and we will respond to you as quickly as possible within normal working hours UK time (9:00-17:00 hours, Monday-Friday). • This is a formal University assessment. You must not share or discuss any aspect of this assessment, your answers or the module more generally with anyone whether a student or not during the period the assessment is open, with the exception of the module leader and Physics exams team. Page 2 of 2 End. Module code: PHYS238001 SECTION A • You must answer all the questions from this section. • This section is worth 30 marks. • You are advised to spend 60 minutes on this section. A1. If y = e−3x is the particular solution of the ordinary differential equation y′′ + 8y′ + 17y = 2e−3x, find the specific solution that satisfies the boundary conditions y(0) = 2 and y(π/2) = 0. [5 Marks] A2. The function f(x) = eλx is an eigenfunction of the linear operator L = d dx . Show that it is also an eigenfunction of the linear operator L = d 3 dx3 − 6 d 2 dx2 − 4 d dx + 24. [6 Marks] A3. Assuming that a solution of the differential equation (x− 1)y′′ − (x− 3)y′ − y = 0, can be written as a power series y(x) = ∞∑ n=0 anx n, find the recursion formula for the coefficients an. [6 Marks] A4. Find the Fourier transform of the function f(x) = { 1 − x2 for |x| ≤ 1 0 for |x| > 1. [6 Marks] A5. A wavefunction φ(x, t) obeys the partial differential equation ∂2φ ∂t2 + (α+ β) ∂φ ∂t + αβφ = c2 ∂2φ ∂x2 , with α and β constants. Find the ordinary differential equation obeyed by φ̃(p, t) which is the Fourier transform of φ(x, t) with respect to x. [7 Marks] Maths 4 Page 1 of 5 Turn the page over Module code: PHYS238001 Module Title: Maths 4 ©UNIVERSITY OF LEEDS Resit End of Module Assessment School of Physics and Astronomy Semester Two 2020/21 Calculator instructions: You are allowed to use a calculator or a computer calculator in this assessment. Dictionary instructions: You are allowed to use your own dictionary in this assessment and/or the spell-checker facility on your computer. Assessment information: • This assessment is made up of 5 pages and is worth 70% of the module mark. • You have 48 hours to complete this open book online assessment. • You are recommended to take a maximum of 2 hours within the time available to complete the assessment. • You must answer all of the questions in this assessment. • You should indicate the final answer to each question by underlining it. At the end of each answer you should cite any websites or textbooks other than the course materials and recommended text books that you have used specifically to answer that question. You should always answer in your own words and not repeat material verbatim and you should explain each step of your working. • You must upload your answers via Minerva to GradeScope within the time allowed. You are advised to allow up to four hours to photograph your answers, and upload as a PDF to GradeScope. • When submitting your work, you must identify which questions are answered on which uploaded pages. You must also check that you have uploaded all the work you wish to be marked as part of this assessment and that the answers uploaded are clearly legible. Failure to do so may result in your work not being marked. • If there is anything that needs clarification or you have any problems, please email the module leader or
[email protected] and we will respond to you as quickly as possible within normal working hours UK time (9:00-17:00 hours, Monday-Friday). • This is a formal University assessment. You must not share or discuss any aspect of this assessment, your answers or the module more generally with anyone whether a student or not during the period the assessment is open, with the exception of the module leader and Physics exams team. Page 2 of 5 Turn the page over Module code: PHYS238001 SECTION B • You must answer all the questions from this section. • This section is worth 80 marks. • You are advised to spend 120 minutes on this section. B1. The evolution of small-amplitude water waves can be described by the partial differential equation, ∂φ ∂t (x, t) + c ∂φ ∂x (x, t) + β ∂3φ ∂x3 = 0, where c is the sound speed and β is a constant. (a) By writing φ(x, t) in terms of its Fourier transform with respect to position, φ̃(p, t), show that the solution for the Fourier transform is φ̃(p, t) = f̃(p)e−ip(c−βp 2)t, where f̃(p) is a function that specifies a specific state. Explain each step of the derivation. [6] (b) At time t = 0, the water wave can be described by the function g(x) = sin(kx) between x = −∞ and x = +∞. Here k = 2π λ is the wave number of a wave with wavelength λ. Find its Fourier transform with respect to position, g̃(p). [5] (c) Then, show that the evolution of a wave with wave number k is given by φ(x, t) = sin [ k ( x− (c− βk2)t )] . [6] (d) Small-amplitude water waves are dispersive, that is the wave speed of a wave de- pends on its wave number. Find the wavelength of the mode that propagates with a wave speed equal to 2c. [3] [20 Marks] Page 3 of 5 Turn the page over Module code: PHYS238001 B2. The equilibrium temperature T (x, y) on a flat metal sheet satisfies Laplace’s equation, which is ∂2T (x, y) ∂x2 + ∂2T (x, y) ∂y2 = 0. The metal sheet extends from x = 0 to x = L and y = 0 to y = L. On the boundaries x = 0 and x = L, T (x, y) = 0, while on the remaining boundaries, T (x, 0) = sin (4πx/L) and T (x, L) = sin (6πx/L). (a) Assuming solutions of the form T (x, t) = X(x)Y (y), use the method of separation of variables to find two ordinary differential equations satisfied by X(x) and Y (y). [5] (b) Solve the differential equations for X(x) and Y (y) and construct a general solution for T (x, y). [5] (c) Use the boundary conditions to find the specific solution for T (x, y). [7] (d) At y = L/2, the temperature can be approximated as T (x, L/2) = a sin(kx), where a and k are constants. Find this approximation and give expressions for a and k. [3] [20 Marks] Page 4 of 5 Turn the page over Module code: PHYS238001 B3. (a) The Fourier transform of f(x) is given by f̃(p). Show that the Fourier transform of eip0xf(x) is f̃(p− p0). [4] (b) Show that the Fourier transform of the Gaussian function f(x) = e−x 2/2σ, is itself a Gaussian function σe−σ 2k2/2. You may assume that ∫ ∞ −∞ e−u 2 du = √ π. [6] (c) The centre of a very long thin, narrow, straight pipe is at x = 0 and the pipe extends in the positive and negative x directions. It has a uniform circular cross section of area A. At time t = 0, a mass M of trace gas is released in the pipe at x = 0. The mass density ρ(x, t) of the trace gas is governed by the one dimensional diffusion equation ∂ρ(x, t) ∂t = D ∂2ρ(x, t) ∂x2 , where D is a positive, real constant. The mass density at t = 0 is ρ(x, 0) = M A δ(x), where δ(x) is the Dirac delta function. Show that ρ(x, t) = M A 1√ 4πDt e− x2 4Dt . [10] [20 Marks] Page 5 of 5 End. Module code: PHYS238001 B4. The temperature distribution on a flat disc with radius r = a is given by ∂T (r, θ, t) ∂t − 4∇2T (r, θ, t) = 0, with the temperature satisfying the boundary condition T (a, θ) = sin3 θ. Here α is a constant. (a) Write T (r, θ, t) as an appropriate Fourier series with respect to the angle θ. [3] (b) What partial differential equation governs the n-th coefficient? [3] (c) Use separation of variables to obtain two ordinary differential equations for the radial and time dependence of the n-th coefficient. [5] (d) Which value of the separation constant corresponds to the equilibrium temperature distribution? Solve the differential equation for the radial part using this value and find the general equilibrium solution for the temperature. [5] (e) Use the boundary conditions to find the specific solution for the equilibrium temper- ature distribution. [4] [20 Marks]