MATHEMATICAL METHODS FOR PHYSICS III (3)

SPH 203: MATHEMATICAL METHODS FOR PHYSICS III INSTRUCTION:  Answer question one [30 marks] and attempt any other two questions [20 marks each]. QUESTION ONE: 30 MARKS a) Show that the Legendre polynomial given below can be recognized as Sturm-Liouville equation. [1 mark] 𝑑 𝑑 (1 − 𝑥 2 ) 𝑦𝑙 (𝑥) = −𝑙(𝑙 + 1)𝑦𝑙 (𝑥) 𝑑𝑥 𝑑𝑥 b) State the convolution theorem. [1 mark] 2 𝑛 c) Find the series solutions, about 𝑧 = 0, of 𝑦 ′′ − (1−𝑧)2 𝑦 = 0 if 𝑦 = ∑∞ 𝑛=0 𝑎𝑛 𝑧 . [3 marks] d) Let 𝑢(𝑟, 𝜃) = 𝑎0 + 𝑎1 𝑟 cos 𝜃 + 𝑏1 𝑟 sin 𝜃 + 𝑎2 𝑟 2 cos 2𝜃 + 𝑏2 𝑟 2 sin 2𝜃 + ⋯ be the Laplace’s equation in a circle. Show that the given equation is exactly the Fourier series for 𝑢0 . [2 marks] e) Consider a one-dimensional wave equation of a string given as 𝜕2 𝑦 𝜕𝑥 2 = 1 𝜕2 𝑦 𝑐 2 𝜕𝑡 2 . If the string is fixed at 𝑥 = 0 and 𝑥 = 𝐿 under the boundary conditions 𝑦(0, 𝑡) = 𝑦(𝐿, 𝑡) = 0 for 𝑡 ≥ 0, use separation of variable method to find the solution 𝑦(𝑥, 𝑡). Bring out the following concepts in your working: Fundamental mode, first harmonic, second harmonic, higher harmonics, eigensolutions, eigenfrequencies. [6 marks] f) Suppose 𝑢(𝑟, 𝜃) = 1 1−𝑟 2 ( ) is temperature equation inside a circle. What happens to 2𝜋 1+𝑟 2 −2𝑟 cos 𝜃 the equation when 𝑟 = 0, 1 and 𝜃 = 0? [3 marks] g) Write down the wave equation in three dimension. [1 mark] h) The Delta series function 𝛿(𝑥) = 1 2𝜋 1 + [cos 𝑥 + cos 2𝑥 + cos 3𝑥 + ⋯ ] cannot truly 𝜋 converge. However, the graph in Figure 1.1 shows that the partial sum after cos 5𝑥 and cos 10𝑥 approaches 𝛿(𝑥). They oscillate faster and faster away from 𝑥 = 0. Study it and answer the question that follows. Figure 1.1: The sums 𝜹𝑵 (𝒙) → 𝜹(𝒙) mulamaustine@gmail.com Find a neat formula for the partial sum 𝛿𝑁 (𝑥) that stops at cos 𝑁𝑥. Start by writing each term 2 cos 𝜃 as 𝑒 𝑖𝜃 + 𝑒 −𝑖𝜃 . [3 marks] i) For the function 𝑓(𝑥) = 1 − 𝑥, 0 ≤ 𝑥 ≤ 1, find the Fourier sine series. [3 marks] j) Prove that the cross-correlation 𝐶(𝑧) of the Gaussian and Lorentzian distributions 𝑓(𝑡) = 𝐶̃ (𝑧) = 1 𝜏√2𝜋 1 2𝜋 𝑒 𝑒 (−𝑡 2 ⁄2𝜏 2 ) 𝑎 , 𝑔(𝑡) = ( ) (−𝜏2 𝜔2 ⁄2) (−𝑎|𝜔|) 𝑒 𝜋 1 𝑡 2 +𝑎2 , has as its Fourier transform the function . Hence show that 𝐶(𝑧) = 1 𝜏√2𝜋 𝑒 𝑎2 −𝑧2 ) 2𝜏2 ( 𝑎𝑧 cos ( 2 ). [3 marks] 𝜏 k) A hot metal bar is moved into a freezer (zero temperature). The sides of the bar are coated so that heat only escapes at the ends. What is the temperature 𝑢(𝑥, 𝑡) along the bar at time 𝑡? [2 marks] l) By considering the real and imaginary parts of the product 𝑒 𝑖𝜃 𝑒 𝑖𝜙 prove the standard formulae for cos(𝜃 + 𝜙) and sin(𝜃 + 𝜙). [2 marks] QUESTION TWO: 20 MARKS a) Express sin4 𝜃 entirely in terms of the trigonometric functions of multiple angles and deduce that its average value over a complete cycle is 3⁄8. [5 marks] b) Systems that can be modelled as damped harmonic oscillators are wide-spread; pendulum clocks, car shock absorbers, tuning circuits in television sets and radios, and collective electron motions in plasmas and metals are just a few examples. In all these cases, one or more variables describing the system obey(s) an equation of the form 𝑥̈ + 2𝛾𝑥̇ + 𝜔02 𝑥 = 𝑃 cos 𝜔𝑡, where 𝑥̇ = 𝑑𝑥⁄𝑑𝑡 and the inclusion of the factor 2 is conventional. In the steady state (i.e. after the effects of any initial displacement or velocity have been damped out) the solution of the equation takes the form 𝑥(𝑡) = 𝐴 cos(𝜔𝑡 + 𝜙). By expressing each term in the form 𝐵cos(𝜔𝑡 + 𝜖) and representing it by a vector of magnitude 𝐵 making an angle 𝜖 with the 𝑥 − axis, draw a closed vector diagram, at 𝑡 = 0, say, that is equivalent to the equation. Further, i. ii. Show that 𝜙 = tan−1 ( −2𝛾𝜔 𝜔02 −𝜔2 ). [13 marks] Obtain an expression for A in terms of P, 𝜔0 and 𝜔. [2 marks] mulamaustine@gmail.com QUESTION THREE: 20 MARKS a) Find the temperature, 𝑇(𝜌, 𝑡) inside a spherical turkey, initially at 40 °F, which is placed in a 350 ℉ oven. Assume that the turkey is of constant density and that the surface of the turkey is maintained at the oven temperature [Neglect convection and radiation processes inside the oven.]. The problem to be solved is formulated as follows; 𝑘 𝜕 (𝑟 2 𝜕𝑇 𝑘 𝜕 (𝑟 2 𝜕𝑢 𝑇𝑡 = 𝑟 2 𝜕𝑟 𝑢𝑡 = 𝑟 2 𝜕𝑟 𝜕𝑟 ), 0 < 𝜌 < 𝑎, 𝑡 > 0, 𝑇(𝑎, 𝑡) = 350, 𝑇𝑖 = 40, 𝑇(𝜌, 𝑡) finite at 𝜌 = 0, 𝑡 > 0, However, we notice that the boundary condition is not homogeneous. This can be fixed by introducing an auxiliary function (the difference between the turkey and oven temperatures) 𝑢(𝜌, 𝑡) = 𝑇𝑖 − 𝑇𝑎 , where 𝑇𝑎 = 350. Then, the problem to be solved becomes; 𝜕𝑟 ), 0 < 𝜌 < 𝑎, 𝑡 > 0, 𝑢(𝑎, 𝑡) = 0, 𝑢(𝜌, 𝑡) finite at 𝜌 = 0, 𝑡 > 0, 𝑢(𝜌, 0) = 𝑇𝑖 − 𝑇𝑎 = −310, where 𝑇i = 40. [12 marks] b) A possible equation of state for a gas takes the form 𝑃𝑉 = 𝑅𝑇𝑒 (−𝛼⁄𝑉𝑅𝑇) , in which 𝛼 and 𝑅 𝜕𝑃 𝜕𝑉 𝜕𝑇 are constants. Find expressions for ( ) , ( ) , ( ) , and show that 𝜕𝑉 𝜕𝑃 𝜕𝑉 𝜕𝑇 ( ) (𝜕𝑇 ) (𝜕𝑃) = −1. [8 marks] 𝜕𝑉 𝑇 𝑃 𝑉 𝑇 𝜕𝑇 𝑃 𝜕𝑃 𝑉 QUESTION FOUR: 20 MARKS a) Convert the finite real Fourier series 7 + 4 cos 𝑥 + 6 sin 𝑥 − 8 sin(2𝑥) + 10 cos(24𝑥) to a finite complex Fourier series. [7 marks] b) In the 𝑥𝑦 − plane new coordinates 𝑠 and 𝑡 are defined by 𝑠 = the equation 𝜕2 𝜙 𝜕𝑥 2 − 𝜕2 𝜙 𝜕𝑦 2 𝑥+𝑦 2 and 𝑡 = 𝑥−𝑦 2 . Transform = 0 into the new coordinates and deduce that its general solution can be written as 𝜙(𝑥, 𝑦) = 𝑓(𝑥 + 𝑦) + 𝑔(𝑥 − 𝑦), where 𝑓(𝑢) and 𝑔(𝑣) are arbitrary functions of 𝑢 and 𝑣 respectively. [13 marks] mulamaustine@gmail.com QUESTION FIVE: 20 MARKS a) The displacement of a damped harmonic oscillator as a function of time is given by 0, 𝑡<0 𝑓(𝑡) = { −𝑡/𝜏 𝑒 sin 𝜔0 𝑡, 𝑡 ≥ 0 Find the Fourier transform of this function and give the physical interpretation of Parseval’s theorem. [8 marks] b) In order to make a focussing mirror that concentrates parallel axial rays to one spot (or conversely forms a parallel beam from a point source) a parabolic shape should be adopted. If a mirror that is part of a circular cylinder or sphere were used, the light would be spread out along a curve. This curve is known as a caustic and is the envelope of the rays reflected from the mirror. Denoting by 𝜃, the angle which a typical incident axial ray makes with the normal to the mirror at the place where it is reflected, the geometry of reflection (the angle of incidence equals the angle of reflection) is shown in Figure 5.1. Figure 5.1: Reflecting mirror 1 Show that a parametric specification of the caustic is 𝑥 = 𝑅 cos 𝜃 ( + sin2 𝜃), 𝑦 = 𝑅 sin3 𝜃, 2 where R is the radius of curvature of the mirror. The curve is in fact part of epicycloid. [12 marks] mulamaustine@gmail.com