Generalization of Wave Equation: Mathematical Techniques
Explore the generalization of the one-dimensional wave equation through various mathematical problems and techniques such as orthogonal function expansions, Fourier series, Fourier transforms, and Fast Fourier transforms. Dive into the properties of eigenvalues and eigenfunctions of Sturm-Liouville equations, understanding their orthogonality. Discover the Fourier series representation of functions within specific intervals and how Fourier transforms play a crucial role in mathematical analyses.
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PHY 711 Classical Mechanics and Mathematical Methods 9-9:50 AM MWF Olin 107 Plan for Lecture 23: Read Chapter 7 & Appendices A-D Generalization of the one dimensional wave equation various mathematical problems and techniques including: 1. Orthogonal function expansions 2. Fourier series 3. Fourier transforms 4. Fast Fourier transforms 10/25/2017 PHY 711 Fall 2017 -- Lecture 23 1
10/25/2017 PHY 711 Fall 2017 -- Lecture 23 2
10/25/2017 PHY 711 Fall 2017 -- Lecture 23 3
Eigenvalues and eigenfunctions of Sturm-Liouville equations d d x v x f x dx dx Properties: n + = ( ) ( ) ( ) ( ) x f x ( ) n n n Eigenvalues are real b ( ) x f x f = Eigenfunctions are orthogonal: ( ) ( ) x dx , N n m nm n a b 2 where ( )( x ( )) f x . N dx n n a = = ( ) ( ) x = Special case: ( ) x 1 0 v x 2 d dx = = = ( ) ( ) for 0 f x , wi th (0) n f ( ) 0 f x x a f a n n n n 2 10/25/2017 PHY 711 Fall 2017 -- Lecture 23 4
= = ( ) ( ) x = Special case: ( ) x 1 0 v x 2 d dx = = = ( ) ( ) for 0 f x , with (0) ( ) 0 f x x a f f a n n n n n 2 2 2 a n x a n = = ( ) sin f x n n a Fourier series representation of function ( ) in the interval 0 h x : x a 2 a n x a = ( ) sin h x A n = 1 n a 2 a ' n x a = ' ( ') sin dx h x A n 0 h x = = *Note that if ( ) does not vanish at 0 and , t he more general x x a 2 a 2 a n x a n x a = + expression applies: ( ) sin cos h x A B n n = = 1 0 n n (with some restrictions). 10/25/2017 PHY 711 Fall 2017 -- Lecture 23 5
Example + + + sin( ) 2sin(2 4 ) sin( n ) x x n x = 2 sinh + ... ( 1) + n ( ) x sinh( ) (1) ... h x n 2 2 2 2 1 1 1 ( ) h x n = 1..4 n = 1..8 10/25/2017 PHY 711 Fall 2017 -- Lecture 23 6
Fourier series representation of function ( ) in the interval 0 : h x x a 2 a n x a = ( ) sin h x A n = 1 n a 2 a ' n x a = ' ( ') sin dx h x A n 0 Can show that the series conve piecewise continu rges provided that ( ) h x is ous. Generalization to infinite range Examples in time domain -- 10/25/2017 PHY 711 Fall 2017 -- Lecture 23 7
Fourier transforms A useful identity ( ) ( ) i t = 2 dt e 0 0 Note that ( ) 2sin T T ( ) ( ) 0 i t = 2 dt e 0 0 T 0 T 10/25/2017 PHY 711 Fall 2017 -- Lecture 23 8
Definition of Fourier Transform function a for ( : ) t f = i t ( ) ( ) f t d F e Backward transform : 1 = i t ) F( dt f(t) e 2 Check : 1 = ' i t i t ( ) ' ' f t d dt f(t ) e e 2 1 ( ) = = ' i t t ( ) ' ' ' ' ' ( ) f t dt f(t ) d e dt f(t ) t t 2 Note: The location of the 2 factor varies among texts. 10/25/2017 PHY 711 Fall 2017 -- Lecture 23 9
Properties of Fourier transforms -- Parseval's theorem: ( ) ( ) * * = ( ) ( ) 2 dt f(t) f(t) d F F * ( ) ( ) ( ) * = ' i t i t Check: ' ' dt f(t) f(t ) dt d F e d F e ( ) ( ) ( ) ' i t * = ' ' d F d F dt e ( ) ( ) ( ) * = ' ' 2 d F d F ( ) ( ) F * = 2 d F 10/25/2017 PHY 711 Fall 2017 -- Lecture 23 10
Use of Fourier transforms to solve wave equation 2 2 u u = 2 Wave equation : 0 c 2 2 t x ~ F ~ F i = where ) satisfies ) More generally: t Suppose ( , c ) ( , ( , equation the : u x t e x x ~ F 2 2 ( x , ) x ~ F ~ F = 2 ( , ) ( , ) x k x 1 2 2 i t = , ) ( , ) u x t ( d F x e 2 boundary fixed that assume Further = conditions = apply : 0 x L ~ ~ with , 0 ( ) 0 and ( , ) 0 F 3 , 2 , 1 = F L For n n x n ~ = = ( , ) sin n F x k k n n L L c ( ) ( ) ( ) ( ) + ik x ik x ik x ct ik x ct e e e e ( ) n n n n = = = i t i t ( , ) sin u x t e k x e n n n 2 2 i i 10/25/2017 PHY 711 Fall 2017 -- Lecture 23 11
Use of Fourier transforms to solve wave equation -- continued 2 2 u u = 2 0 c 2 2 t x ~ F = i t Using superposit ion Suppose : ( , ) ( , ) u x t C e x n n n n n ~ F 2 2 n ( x , ) x ~ F ~ F = 2 n ( , ) ( , ) n n x k x n n 2 2 c n x n ~ F = = For ( , ) sin n x k k n n L L c ( ) C ( ) = = i t i t ik x ik x ( , ) sin n u x t C e k x e e e n n n n n n 2 i n n ( ) C ( ) ( ) + = + + ik x ct ik x ct ( ) ( ) n e e f x ct g x ct n n 2 i n 10/25/2017 PHY 711 Fall 2017 -- Lecture 23 12
Fourier transform for a time periodic function: Suppose ( ) f t nT f t + = ( ) for any integer = n T 1 1 ( ) t nT + i t i ( ) = F dt f(t)e dt f(t)e 2 2 = n 0 Note that: 2 T ( ) in T = , where e = = n Details: ( sin ) ( ) + sin T N 1 2 N in T in T = = lim lim e e ( ) N N T 1 2 = = n n N 10/25/2017 PHY 711 Fall 2017 -- Lecture 23 13
( sin ) ( ) + sin T N 1 2 ( ) T 1 2 4 T 2 T = = = 0 Note that: 2 T ( ) in T = , where e = = n 10/25/2017 PHY 711 Fall 2017 -- Lecture 23 14
Some details : ( sin ( ) T ) + 2 sin M T M 1 = n = in T 2 e ( ) 1 M ( sin ( ) T ) + 2 T sin M T 2 1 ( ) ( ) = 2 = lim 2 T T ( ) 1 M 2 ( ) = = = in T where , e T n T 0 1 ( ) = ( = = i t i t ) F dt f(t)e dt f(t)e 2 Thus, for a time periodic function 1 ( ) 2 = ( ) i = t f t F e 10/25/2017 PHY 711 Fall 2017 -- Lecture 23 15
Example: t nT T ( ) ( ) + = for 1 1 ; 0,2,4,6 n T t n T n = Suppose: ( ) f t 0 otherwise Note, in this case the repeat period is 2 and the convenient sample time interval is . T t T T T 1 T 2 2 2 2 t t t ( ) ( ) = = sin ( ) 2 sin F i dt f t F 2 T T T = 1 T 10/25/2017 PHY 711 Fall 2017 -- Lecture 23 16
Example: Suppose: ( ) 1 ( ) ( ) 2 2 t nT + / i a = = t f t e F e a = = n 2 T 1 ( ) 2 2 2 = /4 a where and F e 2 f(t) F( ) 10/25/2017 PHY 711 Fall 2017 -- Lecture 23 17
1 ( ) ( ) 2 2 t nT + / i a = = t Continued: ( ) f t e F e a = = n Note: 2 T F( ) = M ( ) i t ( ) f t F e = M =M =-M M mT M ( ) m + = 2 /(2 1) i m M f F e = For : t T + 2 1 + 2 1 M = M 10/25/2017 PHY 711 Fall 2017 -- Lecture 23 18
periodic a for Thus, function ( ) = suppose = i t ( ) f t F e Now that the transform function ed bounded; is ( ) a periodic transform for F Define N function function ~ ~ ( ) ( ( ( ) ) ) on time + domain + ' 2 1 F Effect F N : 2 N T ~ ~ ( ) ( ) = = = = i t i t ( ) f t F e F e t ( ) + + 2 1 2 1 N N N 10/25/2017 PHY 711 Fall 2017 -- Lecture 23 19
Doubly periodic T functions t + 2 1 N ~ f 1 N ~ F ( ) + = 2 / 2 1 i N e + 2 1 N = N ~ f N ~ F ( ) 1 + = 2 / 2 i N e = N 10/25/2017 PHY 711 Fall 2017 -- Lecture 23 20
convenient More + notation 2 1 N M ~ 1 1 M = ~ = 2 / i M f F e M = that 0 ~ M ~ = 2 / i M F f e 0 i = 2 / M Note for W e ~ f ~ f ~ f ~ f ~ F = + + + + 0 0 0 0 W W W W 0 0 1 W 2 W 3 W ~ f ~ f ~ f ~ f ~ F = + + + + 0 1 2 3 W 1 0 1 2 f 3 ~ f ~ f ~ f ~ ~ F = + + + + 0 2 4 6 W W W W 2 0 1 2 3 10/25/2017 PHY 711 Fall 2017 -- Lecture 23 21
i = 2 / M Note that for W e ~ f ~ f ~ f ~ f ~ F = + + + + 0 0 0 0 W W W W 0 0 1 W 2 W 3 W ~ f ~ f ~ f ~ f ~ F = + + + + 0 1 2 3 W 1 0 1 2 f 3 ~ f ~ f ~ f ~ ~ F = + + + + 0 2 4 6 W W W W 2 0 1 2 3 ( ) M = = 2 / M i M However, 1 W e ( ) / 2 M = = / 2 2 / M i M and 1 W e Cooley-Tukey algorithm: J. W. Cooley and J. W. Tukey, An algorithm for machine calculation of complex Fourier series Math. Computation 19, 297-301 (1965) 10/25/2017 PHY 711 Fall 2017 -- Lecture 23 22
