T E H U N I V E R S I T Y O H F R G E D I N B U Fourier Analysis Workshop : Fourier Series Professor John A. Peacock School of Physics and Astronomy jap@roe.ac.uk Session: 203/4 24th & 27th September 203. By writing sin A and cos B in terms of exponentials, prove that 2sinA cos B =sin(a + B)+sin(A B). 2. If f(x) andg(x) areperiodicwithfundamentalperiodx, showthatthefollowingarealso periodic with the same period: (a) h(x) =af(x)+bg(x) (b) j(x) =cf(x) g(x) where a, b, c are constants. 3. Find the fundamental periods for the following functions: (a) cos 2x (b) 3 cos 3x +2cos2x (c) cos 2 x (d) cos x (e) sin 3 x. 4. Show that Z L L m x n x 0 dx sin sin = L L L m 6= n m = n Printed: September 22, 203
5. (a) Sketch f(x) =(+sinx) 2 and determine its fundamental period. (b) Using a trigonometric identity for sin 2 x in terms of cos 2x, writedownthefourierseries for f(x) (don tdoanyintegralstoobtainthecoe cients). 6. Show that the Fourier Series expansion of the periodic function <x<0 f(x) = + 0 <x< is f(x) = 4 X k=0 sin[(2k +)x]. 2k + 7. (a) Show that the Fourier Series for f(x) =x in the range <x< is f(x) =2 X ( ) m+ sin mx. m m= (b) Hence, by carefully choosing a value of x, show that 3 + 5 7...= 4. 8. Using a trigonometric identity, or otherwise, compute the Fourier Series for f(x) = x sinx for <x<,andhenceshowthat 4 = 2 + 3 3 5 + 5 7... Printed: September 22, 203 2
Fourier Analysis Workshop 2: More on Fourier Series Professor John A. Peacock School of Physics and Astronomy jap@roe.ac.uk Session: 203/4 st & 4th October 203 Handin Deadline: 4 p.m. Friday th October 203. Consider the function f(x) = cos x. (a) What is its fundamental period? (b) Sketch the function for 2 <x<2 (c) Show that the Fourier Series expansion for f(x) is f(x) = 2 + 4 X ( ) m+ 4m 2 cos(2mx). m= 2. Let f(x) =+cos 2 ( x). (a) Sketch f(x) and determine its fundamental period. (b) Using a trigonometric identity, and without doing any calculations, write down a Fourier Series for f(x). 3. If f(x) =sinx for 0 apple x apple, (a) compute the fundamental period for a sine series expansion (b) compute its Fourier sine series (c) sketch the function in (b) for 2 <x<2. (d) compute the fundamental period for a cosine series expansion (e) compute its cosine series and show it is sin x = 2 4 X 4m 2 m= cos(2mx) (f) sketch the function in (e) for 2 <x<2. 4. Consider f(x) =e x,definedin0<x<. Expand it as a Fourier sine series, and sketch Printed: September 22, 203 3
the function for 2 <x<2. 5. If f(x) =x for apple x apple, (a) show that its Fourier Series is f(x) = X n= ( ) n+ 2 n sin(n x). (b) Hence show that X k=0 ( ) k 2k + = 4. 6. Compute the complex Fourier Series for f(x) = x, < x <. 7. If f(x) = x for apple x apple, (a) show that its Fourier Series is f(x) = 2 4 X n=0 cos[(2n +)x] (2n +) 2. (b) Hence show that X n=0 (2n +) 2 = 2 8. Printed: September 22, 203 4
Fourier Analysis Workshop 3: Parseval, and ODEs by Fourier Series Professor John A. Peacock School of Physics and Astronomy jap@roe.ac.uk Session: 203/4 8th & th October 203. Compute the complex Fourier Series for f(x) = and show it is c n =(cos(k n ) )i/k n. <x<0 + 0 <x< 2. Prove Parseval s theorem for the (sine and cosine) Fourier Series. 3. You are given that the Fourier Series of f(x) = x (defined for apple x apple ) is f(x) = 2 4 X n=0 cos[(2n +)x] (2n +) 2. State Parseval s theorem, and prove that X (2n +) = 4 4 96. n=0 4. You are given that the Fourier Series of f(x) =x (defined for apple x apple ) is X f(x) = ( ) n+ 2 n sin(n x). Using Parseval s theorem, show that n= X n= n 2 = 2 6. 5. (a) By expanding both sides as Fourier Sin Series, show that the solution to the equation d 2 y dx 2 + y =2x Printed: September 22, 203 5
with boundary conditions y(x =0)=0,y(x =)=0is y(x) = 4 X ( ) n+ n( n 2 2 ) sin(n x). n= (b) Show that the r.m.s. value of y(x) is v p hy2 (x)i = 4 u t 2 X n= n 2 ( n 2 2 ) 2 6. If the function f(x)isperiodicwithperiod2 and has a complex Fourier Series representation X f(x) = f n e inx n= then show that the solution of the di erential equation is y(x) = dy + ay = f(x) dx X n= f n a + in einx. 7. An RLC series circuit has a sinusoidal voltage V 0 sin!t imposed, so the current I obeys: L d2 I dt 2 + RdI dt + CI =!V 0 cos!t. (a) What is the fundamental period of the voltage? (b) Write I(t) asafourierseries, and show that a n and b n satisfy C a 0 2 + X n= I(t) = a 0 2 + X [a n cos(n!t)+b n sin(n!t)] n= a n Ln 2! 2 cos(n!t) Rn! sin(n!t)+c cos(n!t) + b n Ln 2! 2 sin(n!t)+rn! cos(n!t)+c sin(n!t) =!V 0 cos!t. (c) Hence show that only a and b survive, with amplitudes a = b =!V 0 ( L! 2 + C) (C L! 2 ) 2 + R 2! 2! 2 V 0 R (C L! 2 ) 2 + R 2!. 2 Printed: September 22, 203 6
8. Asimpleharmonicoscillatorwithnaturalfrequency! 0 and no damping is driven by a driving acceleration term f(t) =sint +sin2t. (a) Write down the di erential equation which the displacement y(t) obeys. (b) Compute the fundamental period of the driving terms on the right hand side, and hence T (where the solution is assumed periodic on T < t < T). (c) Assuming the solution is periodic with the same fundamental period as the driving term, find the resultant motion. (d) Calculate the r.m.s. displacement of the oscillator. Printed: September 22, 203 7
Fourier Analysis Workshop 4: Fourier Transforms Professor John A. Peacock School of Physics and Astronomy jap@roe.ac.uk Session: 203/4 5th & 8th October 203 Handin Deadline: 4 p.m. Friday 25th October 203. Prove that, for a real function f(x), its Fourier Transform satisfies f( k) = f (k). 2. In terms of f(k), the Fourier Transform of f(x), what are the Fourier Transforms of the following? (a) g(x) =f( x) (b) g(x) =f(2x) (c) g(x) =f(x + a) (d) g(x) =df /dx. (e) g(x) =xf(x) 3. Consider a Gaussian quantum mechanical wavefunction x 2 (x) =A exp, where A is a normalisation constant, and the width of the Gaussian is. (a) Compute the Fourier Transform (k) and show that it is also a Gaussian. (b) Noting that the probability density function is (x) 2,showbyinspectionthatthe uncertainty in x (by which we mean the width of the Gaussian) is x = / p 2. (c) Then, using de Broglie s relation between the wavenumber k and the momentum, p = hk, compute the uncertainty in p, and demonstrate Heisenberg s Uncertainty Principle, p x = h 2. 2 2 4. Express the Fourier Transform of g(x) e iax f(x) in terms of the FT of f(x). 5. The function f(x) isdefinedby f(x) = e x x>0 0 x<0 Printed: September 22, 203 8
(a) Calculate the FT of f(x), and, using Q4, of e ix f(x) andofe ix f(x). (b) Hence show that the FT of g(x) =f(x)sinx is ( + ik) 2 + (c) Finally, calculate the FT of h(x) =f(x)cosx. 6. (a) Show that the FT of f(x) =e a x is f(k) =2a/(a 2 + k 2 ), if a>0. (b) Sketch the FT of the cases a =anda =3onthesamegraph,andcommentonthe widths. (c) Using the result of question 4, show that the FT of g(x) =e x sin x is g(k) = 4ik 4+k 4. 7. Let (a) Show that the FT of h a (x) is h a (x) e ax x 0 0 x<0. h a (k) = a + ik. (b) Take the FT of the equation and show that df dx (x)+2f(x) =h (x) f(k) = +ik 2+ik. (c) Hence show that f(x) =e x e 2x is a solution to the equation (for x>0). (d) Verify your answer by solving the equation using an integrating factor. (e) Comment on any di erence in the solutions. 8. Compute the Fourier Transform of a top-hat function of height h and width 2a, whichis centred at x = d = a. Sketch the real and imaginary parts of the FT. Printed: September 22, 203 9
Fourier Analysis Workshop 5: Dirac Delta Functions Professor John A. Peacock School of Physics and Astronomy jap@roe.ac.uk Session: 203/4 22nd & 25th October 203. By using the result that if, for all functions f(x), then g(x) =h(x), show that Z f(x)g(x)dx = Z (a) ( x) = (x) Hint: show that Z f(x) ( x)dx = f(0) = (b) (ax) = (x) a (c) (x 2 a 2 )= (d) x (x) =0. (x a)+ (x+a) 2 a. f(x)h(x)dx Z f(x) (x)dx 2. Evaluate (a) R f(x) (2x 3) dx (b) R 2 f(x) (x 3) dx 3. Evaluate Z +0. 0. Z 4 dx dy (sin x) (x 2 y 2 ) 4. Show that the derivative of the Dirac delta function has the property that Z d (t) f(t) dt = dt df dt t=0 5. What are the Fourier Transforms of: (a) (x) (b) (x d) Printed: September 22, 203 0
(c) (2x)? By writing (x) asanintegral(i.e.asaninversefouriertransform)showthat (d) (x) = (x) 6. Evaluate Z where a is a constant. Z t 2 e iax e itx dxdt 7. Compute Z where b is a constant. Z Z Z e ixy e ixz e ibz e iyt e t3 dxdydzdt 8. A one-dimensional harmonic oscillator with natural frequency! 0 is driven with a driving acceleration a(t), so obeys d 2 z dt 2 +!2 0z = a(t). (a) Take the Fourier Transform of this equation (from t to!) andshowthat z(!) = ã(!)! 2 0! 2 (b) Hence show that z(t) = Z 2 ã(!)! 2 0! 2 ei!t d!. (c) If a(t) =sin 2 t, findã(!). (d) Hence find a solution for z(t) (ignoresolutionstothehomogeneousequation). Printed: September 22, 203
Fourier Analysis Workshop 6: Convolutions, Correlations Professor John A. Peacock School of Physics and Astronomy jap@roe.ac.uk Session: 203/4 29th October & st November 203 Handin Deadline: 4 p.m. Friday 8th November 203. Show that convolving a signal f(t) withagaussiansmoothingfunction g(t) = p t 2 exp 2 2 2 results in the Fourier Transform being low-pass filtered with a weight exp( 2! 2 /2). 2. Show that the FT of a product h(x) =f(x)g(x) isaconvolutionink-space: h(k) = 2 f(k) g(k) = Z dk 0 2 f(k 0 ) g(k k 0 ). 3. Show that the convolution of a Gaussian of width with a Gaussian of width 2 gives another Gaussian, and calculate its width. (A Gaussian of width has the form N exp[ x 2 /(2 2 )]). 4. Show that the FT of the cross-correlation h(x) of f(x) and g(x), h(x) = Z f (x 0 )g(x 0 + x)dx 0 is h(k) = f (k) g(k). 5. Asignalf(x) =e x for x>0andzerootherwise. (a) Show that the Fourier Transform is f(k) =(+ik). (b) Using Parseval s theorem, relate the integral of the power f(k) 2 to an integral of f(x) 2. (c) Hence show that Z dk +k 2 =. (d) If the signal is passed through a low-pass filter, which sets the Fourier transform coe - cients to zero above k = k 0,calculatek 0 such that the filtered signal has 90% of the original power. Printed: September 22, 203 2
6. (a) Compute the Fourier Transform of h(t) = (b) A system obeys the di erential equation e bt t 0 0 t<0. dz dt +! 0z = f(t) By using Fourier transforms, show that a solution of the equation is a convolution of f(t) with e! 0 t t 0 g(t) = 0 t<0 i.e. z(t) = Z and write down the full expression for z(t). f(t 0 )g(t t 0 ) dt 0, 7. (a) Show that the FT of h(t) =e a t,fora>0is h(!) = (b) A system obeys the di erential equation 2a a 2 +! 2. d 2 z dt 2! 2 0z = f(t). Calculate z(!) intermsof f(!). (c) By considering the form of z(!), show using the convolution theorem that a solution of the equation is the convolution of f(t) withsomefunctiong(t). (d) Using your answer to (a), find the function g(t) andwritedownexplicitlyasolutionto the equation. 8. Compute the Fourier Transform of x apple h(x) = 0 otherwise Show that the convolution H(x) h(x) (x a) isifa <x<a+ and zero otherwise, and compute its Fourier transform directly, and via the convolution theorem. 9. AtripleslitexperimentconsistsofslitswhicheachhaveaGaussiantransmissionwithGaussian width, and theyareseparatedby adistanced. Compute the intensity distribution far from the slits, and sketch it. Printed: September 22, 203 3
Fourier Analysis Workshop 7: Sampling, and Green s Functions Professor John A. Peacock School of Physics and Astronomy jap@roe.ac.uk Session: 203/4 5th & 8th November 203. (a) Expand z as a Taylor series about z =0,orasapowerseries. (b) Hence show that where z =exp(ik x). (c) Similarly, show that (d) Finally, show that if z 6=, X j=0 0X j= e ijk x = z e ijk x = X j= /z. e ijk x =0. 2. Letting p = dy/dt, andthenusinganintegratingfactor,showthatthegeneralsolutionto d 2 y dt + dy 2 dt =0 is y(t) =A + Be t,wherea and B are constants. 3. Show that the Green s function for the range x 0, satisfying @ 2 G(x, z) @x 2 + G(x, z) = (x z) with boundary conditions G(x, z) =@G(x, z)/@x =0atx =0is cos z sin x sin z cos x x > z G(x, z) = 0 x<z Printed: September 22, 203 4
4. Consider the equation, valid for t 0 d 2 f dt 2 +5df dt +6f = e t, subject to boundary conditions f =0,df /dt =0att = 0. Find the Green s function G(t, z), showing is is zero for t<z,andfort>zit is G(t, z) =e 2z 2t e 3z 3t. (You may find the complementary function (homogeneous solution) by using a suitable trial function). Hence show that the solution to the equation is f(t) = 2 e t e 2t + 2 e 3t. 5. (a) Show that the Green s function for the equation, valid for t 0 d 2 y dt + dy 2 dt = f(t), with y =0anddy/dt =0att =0,is 0 t<t G(t, T )= e T t t>t. (b) Hence show that if f(t) =Ae 2t,thesolutionis y(t) = A 2 2e t + e 2t. 6. The equation for a driven, damped harmonic oscillator is d 2 y dt 2 +2dy dt +(+k2 )y = f(t) (a) If the initial conditions are y =0anddy/dt =0att =0,showthattheGreen sfunction, valid for t 0, is A(T )e G(t, T )= t cos kt + B(T )e t sin kt 0 <t<t C(T )e t cos kt + D(T )e t sin kt t > T (b) Show that A = B =0andsoG(t, T )=0fort<T. (c) By matching G(t, T )att = T,andrequiringdG/dt to have a discontinuity of there, show that, for t>t G(t, T )= et k (d) Hence if f(t) =e t,findthesolutionfory(t). t ( sin kt cos kt +coskt sin kt). Printed: September 22, 203 5
Fourier Analysis Workshop 8: Partial Di erential Equations Professor John A. Peacock School of Physics and Astronomy jap@roe.ac.uk Session: 203/4 2th & 5th November 203 Handin Deadline: 4 p.m. Friday 22nd November 203. Find solutions u(x, y) byseparationofvariablesto (a) x @u y @u @x @y =0 (b) @u @x xy @u @y =0 2. Consider a particle of mass m which is confined within a square well 0 <x<,0<y<. The steady-state 2D Schrödinger equation inside the well (where the potential is zero) is h 2 2m r2 = E. The walls have infinite potential, so =0ontheboundaries. (a) Find separable solutions (x, y) = X(x)Y (y) and show that they are for integers r, n. (x, y) =A sin(rx)sin(ny) (b) The wavefunction is normalised so that R (x, y) 2 dx dy =. Forgivenr, n, finda. (c) Show that the energy levels corresponding to the quantum numbers m, n are E =(r 2 + n 2 ) h2 2m. 3. Show by direct substitution into the equation that u(x, t) =f(x ct)+g(x + ct) where f and g are arbitrary functions, is a solution of the D wave equation, @ 2 u @x = @ 2 u 2 c 2 @t, 2 Printed: September 22, 203 6
where the sound speed c is a constant. You may recall that the partial derivative of f(y) with respect to x (where y may be a function of several variables y(x, t,...)) is @f @x = @y df @x dy. 4. Consider the wave equation (with sound speed unity) for t>0 @ 2 u @x = @2 u 2 @t 2 with initial conditions u(x, 0) = h(x) and@u/@t t=0 = v(x). (a) Write down d Alembert s solution for u(x, t). (b)if h(x) andv(x) areknownonlyfor0<x<, then find the regions in the x, t plane for which the solution for u can be determined, and sketch it. 5. (a) Consider separable solutions for the temperature u(x, t) = X(x)T (t) ofthedheat equation @ 2 u @x = @u 2 @t and find the di erential equations which X and T must satisfy, giving your reasoning. (b) Solving for T,showthattheseparablesolutionswhicharefiniteast!are of the form [A cos(kx)+b sin(kx)] exp( k 2 t). where k 2 > 0. (c) There is one more (rather simple) permitted solution. What is it? (d) Following on from the last question, find all solutions for which u(0,t)=u(, t) =0at all times. Hint: the answer is not a single term, but rather a sum. (e) If the initial temperature (at t =0)isu(x, 0) = sin x cos x, whatisthefullsolution u(x, t)? 6. This is a question which looks hard, because it uses polar coordinates, but you can solve it in exactly the same way as the cartesian equations. Laplace s equation in polar coordinates r, is @ 2 u @r + @u 2 r @r + @ 2 u r 2 @ =0 2 (a) Show that for solutions which are separable, u(r, ) =R(r) ( ), for some constant k 2. 00 ( ) = k 2 ; r 2 R 00 (r)+rr 0 (r) k 2 R(r) =0 (b) Argue that the solution must be periodic in, andsaywhattheperiodmustbe. Printed: September 22, 203 7
(c) As a consequence, what values of k are permitted? (d) By trying power-law solutions R(r) / r,findthegeneralsolutionwhichisfiniteatthe origin. (e) Find the solution for a situation where u is fixed on a circular ring at r =tobe u(r =, )=sin 2 +2sin cos. Printed: September 22, 203 8
Fourier Analysis Workshop 9: Fourier PDEs Professor John A. Peacock School of Physics and Astronomy jap@roe.ac.uk Session: 203/4 9th & 22nd November 203. If we have a function u(x, t), we may do a partial Fourier Transform, changing x to k but leaving t in the equations. We have used the result that apple @u(x, t) FT @t Show this (you can probably fit it on one line). = @ũ(k, t) @t 2. Consider the D wave equation @ 2 u @x = @ 2 u 2 c 2 @t, 2 with boundary conditions at t =0thatu(x, t) =e a x for some a>0, and @u(x, t)/@t =0. (a) By applying a Fourier Transform with respect to x, show that the FT of the general solution is of the form ũ(k, t) =A(k)e ikct + B(k)e ikct. (b) Show that at t =0, ũ(k, 0) = 2a a 2 + k. 2 (c) Hence, applying the boundary conditions, show that ũ(k, t) = a a 2 + k 2 e ickt + e ickt. Note that you will need to argue that the boundary condition on @u/@t also applies to each Fourier component individually. (d) Finally deduce that u(x, t) = 2 e a x ct + e a x+ct 3. Consider the D heat equation for the temperature u(x, t), @ 2 u @x = apple@u 2 @t where the initial condition is that u(x, t =0)= (x). Printed: September 22, 203 9
(a) Take the Fourier Transform with respect to x, i.e. ũ(k, t) = Z u(x, t)e ikx dx Note that the transform is still a function of t. Showthatitobeys @ũ(k, t) @t = k2 ũ(k, t). apple (b) Now fix the value of k for now, and use an integrating factor to find ũ(k, t) = f(k)e k2 t/apple for some (arbitrary) function f(k). (c) From the initial condition u(x, 0) = (x) show that f(k) = (k) ) ũ(k, t) = (k)e k2 t/apple. (d) Using the result that the FT of e applex2 /(4t) is p 4 t/applee k2t/apple,showusingtheconvolution theorem that the general solution for u(x, t) intermsof (x) is u(x, t) = p apple p 4 t Z e apple(x x0 ) 2 /(4t) (x 0 ) dx 0 (e) If (x) = (x ), what is u(x, t)? 4. Using d Alembert s method, show that the solution to the wave equation with the boundary conditions @ 2 u @t 2 = c2 @2 u @x 2 is u(x, t =0)=h(x) =0 u(x, t) = 4c (x + ct) 2 @u (x, t =0)=v(x) =x @t (x ct) 2. If v(x) =x only for 0 <x< (and is zero otherwise), what is the solution? Note - you will have to consider many di erent combinations depending on the values of x ct and x + ct - be guided by the spacetime diagram which is in the notes. 5. The charge density and the electrostatic potential are related by Poisson s equation r 2 (x) = (x) 0 Printed: September 22, 203 20
where we assume that there is no time-dependence. Treating this as a one-dimension problem (so r 2! d 2 /dx 2 ), show using a Fourier Transform that a Gaussian potential (x) = e x2 /(2 2 ) is sourced by a charge density field (x) = 0 2 e x2 /(2 2 ) x 2 2. In doing this, you will demonstrate that Z dk 2 k2 e k2 2 /2 e ikx = p e x2 /(2 2 ) 2 3 x 2 2. You can use this result without proof in handin question 6. Verify the solution by direct di erentiation of (x). You may assume that the Fourier Transform of e x2 /(2 2) is p 2 e k2 2 /2,andthat R e u2 /2 du = p 2. (Thismethodisofmorepracticaluseif is known and you want, when the direct method here cannot be employed). 6. (Hint: do all of this in cartesian coordinates - do not be tempted to use spherical polars, despite the symmetry of the problem). The charge density and the electrostatic potential are related by Poisson s equation r 2 (r) = (r) 0 where we assume that there is no time-dependence. Treating this now as a 3D problem (so r 2! @ 2 /@x 2 + @ 2 /@y 2 + @ 2 /@z 2 ), show using a Fourier Transform that a Gaussian potential (r) = e r2 /(2 2 ) (where r 2 = x 2 + y 2 + z 2 ) has a charge density FT given by where k 2 = k 2 x + k 2 y + k 2 z. (k) =(2 ) 3/2 3 0 k 2 e k2 2 /2 and so the potential is sourced by a charge density field (r) = 0 r 2 3 e r2 /(2 2). 2 2 You may assume that the Fourier Transform (w.r.t. x,! k x )ofe x2 /(2 2) is p 2 e k2 x 2 /2, and that R e u2 /2 du = p 2. You can also assume the inverse FT of k 2 e k2 2 /2 which you proved in question 5. Hint: you will be faced with an integral with 3 terms in it (involving k 2 x + k 2 y + k 2 z). Do one of them only, and engage brain to write down the answer for the other two without doing more algebra. Printed: September 22, 203 2
Fourier Analysis Workshop 0: Revision: Green s functions and convolutions Professor John A. Peacock School of Physics and Astronomy jap@roe.ac.uk Session: 203/4 26th & 29th November 203 Marks out of 25 (like exam paper). There are no hand-ins from this revision workshop.. On the mysterious and enigmatic planet Pendleton, a planetary explorer vehicle falls o a cli at t =0. Theaccelerationduetogravityisaconstant g, andthevehicleattempts to slow down its motion by applying an upward acceleration f(t). Unfortunately the fuel in the vehicle rapidly runs out, so the upward acceleration decays with time according to f(t) =ae t,foraconstanta. The equation of motion for the height z(t) isevidently d 2 z(t) dt 2 = f(t) g F (t). (a) If the top of the cli is at z =0,thenevidentlyz(t = 0) = 0. What is dz/dt at t =0? (b) Write down the equation for the Green s function G(t, T ). (c) Hence show that G(t, T )=0fort<T. (d) Show that the solution for G(t, T )fort>t is G(t, T )=t T t > T. (e) Hence show that the general solution for the vertical height as a function of time is z(t) =a(e t +t) 2 gt2. (f) Verify your answer by directly integrating the equation, with the appropriate boundary conditions. (g) Show that the early time behaviour is z(t) =(a g)t 2 /2+O(t 3 ). 2. The e ects of magnet errors in a synchrotron require the solution of the equation d 2 y( ) +! 2 y( ) =g( ) d 2 where is an angle which lies in the range 0 apple apple 2 and! is fixed. The solution is periodic, so the boundary conditions are y(0) = y(2 ); dy = dy. d =0 d =2 Printed: September 22, 203 22
(a) Write down the equation which the Green s function G(, z) satisfies. (b) Write down the solutions for <zand >zin terms of complex exponentials. (c) Applying the supplied boundary conditions and continuity of G(, z) at = z, simplify the Green s function to G(, z) = A(z) e i! e i! +2i!z 2 i! for some arbitrary function A(z). (d) Similarly show that G(, z) =A(z)(e 2 i!+i! + e 2i!z i! ) >z (e) Applying the boundary condition for the derivative of G at = z, show that A(z) = Hence show that the Green s function is 2i!(e 2 i! ) G(, z) = G(, z) = e i!z 2i!(e 2 i! ) ei! + e i! +2i!z 2 i! <z e i!z 2i!(e 2 i! ) e 2 i!+i! + e 2i!z i! >z 3. Find the Fourier transform of the aperture function of a double slit. This consists of a function which is two top hats, centred on x = a and x = a. Each has a width 2b and height. i.e. the function h(x) isequaltozero,exceptfor a b<x< a + b and a b<x<a+ b, whereitisequaltounity. Do this using convolutions. i.e. note that h(x) istheconvolutionofthesumoftwodelta functions, f(x) = (x a)+ (x + a) withatophatg(x) =if x <b,andg =0otherwise. (a) Show that the FT of f(x) is f(k) =2cos(ka). (b) Show that the FT of g(x) is g(k) = 2sin(kb). k (c) Hence write down (assuming the convolution theorem) the Fourier transform of h(x). 4. In this question, the logic is the important part, not the algebra so much, so write plenty of words so it is clear that you understand what is going on. Consider the equation dy dt + y = f(t) for some as yet unspecified function f(t), for t>0. The function f(t) =0fort<0. Printed: September 22, 203 23
(a) Solve this first by using an integrating factor to show that where A is a constant. y(t) = Z t e (t 0 t0) f(t 0 )dt 0 + Ae t (b) This is a convolution of f with what function (answer carefully)? (c) Now solve the equation using a Fourier Transform w.r.t. t: to show that where f(!) is the FT of f(t). ỹ(!) = Z ỹ(!) = y(t)e i!t dt f(!) i! + (d) We see that this is a multiplication in Fourier space. solution in real (t) space? What does this mean for the (e) By inspecting the solution in (a) obtained with an integrating factor, can you guess what function has the Fourier Transform (+i!)?supportyouranswerbyexplicitlycalculating the FT of the function. (f) Hence write down the general solution obtained by the Fourier Transform method. It is not quite the same as in part (a). Pay careful attention to the limits, which you should justify. (g) Why are we allowed to add an extra term Ae t to the FT solution? (h) If f(t) =e 2t,andy(0) =, find the full solution. Printed: September 22, 203 24