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13.47. Show that if each side of equation (3), Page 356, is a constant c where c A 0, then there is no solution satisfying the boundary-value problem. 13.48. A exible string of length  is tightly stretched between points x 0 and x  on the x-axis, its ends are xed at these points. When set into small transverse vibration, the displacement Y x; t from the x-axis of @2 Y @2 Y any point x at time t is given by 2 a2 2 , where a2 T=; T tension,  mass per unit length. @t @x (a) Find a solution of this equation (sometimes called the wave equation) with a2 4 which satis es the conditions Y 0; t 0; Y ; t 0; Y x; 0 0:1 sin x 0:01 sin 4x; Yt x; 0 0 for 0 < x < ; t > 0. (b) Interpret physically the boundary conditions in (a) and the solution. Ans. (a) Y x; t 0:1 sin x cos 2t 0:01 sin 4x cos 8t 13.49. (a) Solve the boundary-value problem @2 Y @2 Y 9 2 subject to the conditions Y 0; t 0; Y 2; t 0, 2 @t @x Y x; 0 0:05x 2 x ; Yt x; 0 0, where 0 < x < 2; t > 0. (b) Interpret physically. Ans: a Y x; t
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1 1:6 X 1 2n 1 x 3 2n 1 t cos sin 3 2 2  n 1 2n 1 3
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13.50. Solve the boundary-value problem
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@U @2 U 2 ; U 0; t 1; U ; t 3; U x; 0 2. @t @x
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[Hint: Let U x; t V x; t F x and choose F x so as to simplify the di erential equation and boundary conditions for V x; t :
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Ans:
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U x; t 1
13.51. Give a physical interpretation to Problem 13.50. 13.52. Solve Problem 13.49 with the boundary conditions for Y x; 0 and Yt x; 0 interchanged, i.e., Y x; 0; Yt x; 0 0:05x 2 x , and give a physical interpretation. Ans: Y x; t
1 3:2 X 1 2n 1 x 3 2n 1 t sin sin 2 2 34 n 1 2n 1 4
13.53. Verify that the boundary-value problem of Problem 13.24 actually has the solution (14), Page 357. MISCELLANEOUS PROBLEMS 13.54. If  < x <  and 6 0; 1; 2; . . . ; prove that  sin x sin x 2 sin 2x 3 sin 3x 2 sin  12 2 22 2 32 2 13.55. If  < x < , prove that  2  b 2 a sinh x sin x 2 sin 2x 3 sin 3x sinh  2 12 2 23 2 32 cosh x 1 cos x cos 2x sinh  2 2 12 2 22 ! x2 1 2 2 4x2 3 2 ! ! x2 1 3 2 ! 4x2 5 2
x2 13.56. Prove that sinh x x 1 2  4x2 2
! 1
! 1
13.57. Prove that cos x 1
[Hint: cos x sin 2x = 2 sin x : 13.58. Show that p 2 1 3 5 7 9 22 13 15 . . . 2 2 6 6 10 10 14 14 . . . 2   p 4 4 8 8 12 12 16 16 . . . (b)  2 4 3 5 7 9 11 13 15 17 . . . (a) Show that
13.59. Let r be any three dimensional vector. (a) r i r j @ r ;
2 2 2
b r i r j 2 r k 2 r2
)2 b( 1 X f x cn n x dx is a minimum when 13.60. If fn x g; n 1; 2; 3; . . . is orthonormal in (a; b), prove that a n 1 b cn f x n x dx
and discusse these with reference to Parseval s identity.
Discuss the relevance of this result to Fourier series.
Fourier Integrals
Fourier integrals are generalizations of Fourier series. The series representation 1 a0 Xn nx nxo bn sin an cos of a function is a periodic form on 1 < x < 1 obtained by genL L 2 n 1 erating the coe cients from the function s de nition on the least period L; L . If a function de ned on the set of all real numbers has no period, then an analogy to Fourier integrals can be envisioned as letting L ! 1 and replacing the integer valued index, n, by a real valued function . The coe cients an and bn then take the form A and B . This mode of thought leads to the following de nition. (See Problem 14.8.)
THE FOURIER INTEGRAL Let us assume the following conditions on f x : 1. f x satis es the Dirichlet conditions (Page 337) in every nite interval L; L . 1 j f x j dx converges, i.e. f x is absolutely integrable in 1; 1 . 2.