Understanding Wave Equations and Solutions in Classical Mechanics

PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF online or (occasionally) in Olin 103 Discussion for Lecture 19 Chap. 7 (F&W) Solutions of differential equations 1. The wave equation 2. Sturm-Liouville equation 3. Green s function solution methods 10/7/2019 PHY 711 Fall 2029 -- Lecture 19 1

10/7/2019 PHY 711 Fall 2029 -- Lecture 19 2

Next week, it is likely that we will have a take home exam instead of homework. Perhaps distributed Monday 10/12/2020 due Monday 10/19/2020 10/7/2019 PHY 711 Fall 2029 -- Lecture 19 3

Physics Colloquium Thursday, October 8, 2020 10/7/2019 PHY 711 Fall 2029 -- Lecture 19 4

Schedule for weekly one-on-one meetings Nick 11 AM Monday (ED/ST) Tim 9 AM Tuesday Zhi 9 PM Tuesday Jeanette 11 AM Wednesday Derek 4 PM Wednesday Bamidele 7 PM Thursday Derek 12 PM Friday? 10/7/2019 PHY 711 Fall 2029 -- Lecture 19 5

Your questions From Tim 1. What does the extra potential energy density have to do with motion on a string?When you say an applied force, is that like plucking the string or somehow putting a force on the string? From Nick 1. Can you elaborate on slide 14. I think I m missing something on how Cm minimizes eps^2. From Gao 1. In slide 14, is Cn expression from what transformation? Similar to Fourier transformation? Thank you. 10/7/2019 PHY 711 Fall 2029 -- Lecture 19 6

One-dimensional wave equation representing longitudinal or transverse displacements as a function of x and t , an example of a partial differential equation -- For the displacement function, , the wave equation has the form: (x,t) 2 2 = 2 0 c 2 2 t x Note that for any function satisfies the wave equation. or ct : f(q) g(q) = + + ( ) ( ) (x,t) f x ct g x 10/7/2019 PHY 711 Fall 2029 -- Lecture 19 7

The wave equation and related linear PDEs One dimensional wave equation for : (x,t) 2 2 = = 2 2 0 where c c 2 2 t x Generalization for spacially dependent tension and mass density plus an extra potential energy density: ( , ) ( ) ( ) t x 2 ( , ) x t x x t + ( ) ( , ) v x = 0 x x x t 2 Factoring time and spatial variables: ( , ) ( ) cos( Sturm-Liouville equation for spatial function: d d dx d = + ) x t x t + ( ) x x = ( ) ( ) x 2 ( ) ( ) x ( ) x x v x 10/7/2019 PHY 711 Fall 2029 -- Lecture 19 8

Linear second-order ordinary differential equations Sturm-Liouville equations d dx d dx + = Inhomogenous problem: ( ) x ( ) v x ( ) x ( ) x ( ) F x applied force given functions When applicable, it is assumed that the form of the applied force is known. solution to be determined Homogenous problem: F(x)=0 10/7/2019 PHY 711 Fall 2029 -- Lecture 19 9

Your question -- What does the extra potential energy density have to do with motion on a string? Comment In my opinion, v(x) has nothing to do with motion on a spring, but F & W are using the one- dimensional wave equation to motivate a more general discussion of second order differential equations. In this lecture, we will briefly review/introduce many related ideas. These will be also (and perhaps more systematically) covered in PHY 712. 10/7/2019 PHY 711 Fall 2029 -- Lecture 19 10

Examples of Sturm-Liouville eigenvalue equations -- d dx d dx + = ( ) x ( ) v x ( ) x ( ) x 0 Bessel functions: 1 x ( ) = = = = ( ) = 2 ( ) v x ( ) x ( ) x x x x x J Legendre functions = : ( ) = = = + ( ) = 2 ( ) x 1 ( ) v x 0 ( ) x 1 ( 1 ) ( ) x l l x P x l F ou rier functions: ( ) 1 ( ) x = = = = ( = 2 2 0 ( ) 1 ) sin( ) v x x n x n x 10/7/2019 PHY 711 Fall 2029 -- Lecture 19 11

Solution methods of Sturm-Liouville equations (assume all functions and constants are real): : problem Homogenous d d + = ( ) ( ) ( ) ( ) 0 x v x x x 0 dx dx d d + = Inhomogeno problem us : ( ) ( ) ( ) ( ) ( ) x v x x x F x dx dx Eigenfunct d Orthogonality of eigenfunctions: ions d : + = ( ) ( ) ( ) ( ) ( ) x v x f x x f x n n n dx dx b ( ) x f x f = ( ) ( ) x dx , N n m nm n a b 2 where Completeness of eigenfunctions: ( ) n n f x f x x N ( )( x ( )) f x . N dx n n a ( ') ( ) ( ) = ' x x n n 10/7/2019 PHY 711 Fall 2029 -- Lecture 19 12

Why all of the fuss about eigenvalues and eigenvectors? a. They are always necessary for solving differential equations b. Not all eigenfunctions have analytic forms. c. It is possible to solve a differential equation without the use of eigenfunctions. d. Eigenfunctions have some useful properties. 10/7/2019 PHY 711 Fall 2029 -- Lecture 19 13

Comment on orthogonality of eigenfunctions d dx d dx d dx d dx + = ( ) x ( ) v x ( ) ( ) x f x ( ) f x n n n + = ( ) x ( ) v x ( ) x ( ) x f ( ) x f m m m d dx d dx d dx d dx + + ( ) x ( ) x ( ) v x ( ) ( ) x ( ) x ( ) v x ( ) x f f x f f m n n m ( ) ( ) x f x f = ( ) ( ) x n m n m ( ) x x d ( ) d d df df x ( ) ) ( ) ( ) x f x ( ( ) ( ) x = ( ) ( ) n m dx f x x f x f x m n n m n m x 10/7/2019 PHY 711 Fall 2029 -- Lecture 19 14

Comment on orthogonality of eigenfunctions -- continued ( ) dx ( ) x dx d dx df x df ( ) ( ) x f x f ( ) x ( ) x ( ) x ( ) x = ( ) ( ) x n m f f m n n m n m Now consider integrating both sides of the equation in the interval : a x b b b ( ) dx ( ) x dx df x df ( ) = ( ) x ( ) x ( ) x ( ) ( ) ( ) x f ( ) n m f f x dx x f x m n n m n m a a Vanishes for various boundary conditions at x=a and x=b 10/7/2019 PHY 711 Fall 2029 -- Lecture 19 15

Comment on orthogonality of eigenfunctions -- continued b b ( ) dx ( ) x dx df x df ( ) = ( ) x f x f ( ) x ( ) x ( ) x ( ) x ( ) ( ) x n m f f dx m n n m n m a a Possibl 1. ( ) m f e boundary values for Sturm-Liouville equations: ( ) 0 ( ) ) ( ) a b dx dx df a f b = dx = = a f b m x ( df df x = = 2. ( ) x 0 m m x ( ) ( ) dx df b = 3. ( ) a ( ) an d m m f m m In any of these cases, we can conclude that : b = ( ) ( ) ( ) 0 for d x x f x f x n m n m a 10/7/2019 PHY 711 Fall 2029 -- Lecture 19 16

Comment on completeness It can be shown that for any reasonable function h(x), defined within the interval a < x <b, we can expand that function as a linear combination of the eigenfunctions fn(x) ( ) ( ), n n n These ideas lead to the notion that the set of eigenfunctions fn(x) form a ``complete'' set in the sense of ``spanning'' the space of all functions in the interval a < x <b, as summarized by the statement: ( ) ( ) ' ( ) n n N h x C f x 1 b = ( ) ( ) x h where ' ' ( ) x d ' '. C x f x n n N a n f x f x = ( ' ). x x x n n 10/7/2019 PHY 711 Fall 2029 -- Lecture 19 17

Comment on completeness -- continued ( ) ( ), n n n h x C f x 1 b = ( ) ( ) ' h x where ' ( ) x ' '. C x f dx n n N a n Consider the squared err or of the expansion: 2 b = 2 ( ) x ( ) h x ( ) dx C f x n n n a 2 can be min mized: i b 2 = = 0 2 ( ) x ( ) h x ( ) ( ) x dx C f x f n n m C n m a b 1 = ( ) ( ) x h ( ) x C dx x f m m N m a 10/7/2019 PHY 711 Fall 2029 -- Lecture 19 18

Your question -- Can you elaborate on slide 14. I think I m missing something on how Cm minimizes eps^2. Also -- In slide 14, is Cn expression from what transformation? Similar to Fourier transformation? Comment This could be similar to a Fourier transformation if the eigenfunctions fm(x) were sinusoidal (a particular choice of the Sturm-Liouville form). About the minimization of epsilon^2 solving for the 0 of the derivative of the expression is a necessary condition for finding a minimum. Consider the squared error of the expansion: 2 b = 2 ( ) x ( ) h x ( ) dx C f x n n n a 2 can be minimized: b 2 = = 0 2 ( ) x ( ) h x ( ) ( ) x dx C f x f n n m C n m a 10/7/2019 PHY 711 Fall 2029 -- Lecture 19 19

Variation approximation to lowest eigenvalue In general, there are several techniques to determine the eigenvalues nand eigenfunctions fn(x). When it is not possible to find the ``exact'' functions, there are several powerful approximation techniques. For example, the lowest eigenvalue can be approximated by minimizing the function , h h d dx d dx h S h + ( ) S x ( ) x ( ) v x 0 ( ) h x where is a variable function which satisfies the correct boundary values. The ``proof'' of this inequality is based on the notion that can in principle be expanded in terms of the (unknown) exact eigenfunctions fn(x): where the coefficients Cn can be ( ) ( ), n n n ( ) h x = h x C f x assumed to be real. 10/7/2019 PHY 711 Fall 2029 -- Lecture 19 20

Estimation of the lowest eigenvalue continued: From the eigenfunction equation, we know that ( ) ( ) ( ) n It follows that: ( ) ( ) ( ) a It also follows that : = = ( ) ( ) x f x ( ). S x h x S x C f x C n n n n n n b = = N 2 | C | . h S h h x S x h x dx n n n n b = = ( ) ( ) ( ) h x 2 | C | , h h x h x dx N n n a n 2 | C | N h S h n n n = Therefore . n 0 2 | C | N h h n n n 10/7/2019 PHY 711 Fall 2029 -- Lecture 19 21

Rayleigh-Ritz method of estimating the lowest eigenvalue h S h , 0 h h 2 d dx = = = Example: ( ) x ( ) with (0) x ( ) 0 f f f f a n n n n n 2 = trial function ( ) x ( ) f x x a trial 2 9.869604404 a = = Exact value of 0 2 2 a 2 d dx ( ) ( ) x a x x a x 10 a 2 = Raleigh-Ritz estimate: ( ) ( x x a ) 2 x a x 10/7/2019 PHY 711 Fall 2029 -- Lecture 19 22

Greens function solution methods -- note the following slides were note yet covered. Suppose that we can find a Green's function defined as follows: d d x v x x G x x dx dx ( ) + = ( ) ( ) ( ) ( , ') ' x x Completeness of eigenfunctions: ( ) ( ') n n f x f x N Recall: ( ) x ( ) = ' x x n n In terms of eigenfunctions: d d x dx dx ( ) ( ') f x f x N ( ) x + = ( ) ( ) v x ( ) x ( , ') G x x n n n n ( ) ( ')/ f x f x N = ( , ') G x x n n n n n 10/7/2019 PHY 711 Fall 2029 -- Lecture 19 23

Solution to inhomogeneous problem by using Greens functions Inhomogenous problem: d d x v x dx dx : function s Green' d x dx Formal solution: + = ( ) ( ) ( ) x ( ) x ( ) F x d ( ) ' x + = ( ) ( ) ( ) ( , ) ' x v x x G x x dx L + ( ) x = ( ) x ( , ') ( ') G x x F x dx ' 0 0 Solution to homogeneous problem 10/7/2019 PHY 711 Fall 2029 -- Lecture 19 24

Example Sturm-Liouville problem: = = = = = Example : ( ) ; 1 ( ) ; 1 ( ) ; 0 and 0 x x v x a b L x = = ; 1 ( ) sin F x F 0 L Inhomogeno equation us : 2 d x = 1 ( ) sin x F 0 2 dx L 10/7/2019 PHY 711 Fall 2029 -- Lecture 19 25

Eigenvalue dx equation : 2 d = ( ) ( ) f x f x n n n 2 Eigenfunct ions Eigenvalue : s 2 2 L n x n = = ( ) sin f x n n L L Completene ss x of f n eigenfunct ions : ( ) N ( ) ' x f ( ) x ( ) = ' n x x n n 2 L ' n x n x ( ) ' x = In this example : sin sin x L L n 10/7/2019 PHY 711 Fall 2029 -- Lecture 19 26

Green' function s d x : d ( ) ' x + = ( ) ( ) ( ) ( , ) ' x v x x G x x dx dx Green' function s for the example : ' n x n x sin sin ( ) ( / ) ' 2 f x f x N L L n n = = n n n G(x,x') 2 L n n 1 L 10/7/2019 PHY 711 Fall 2029 -- Lecture 19 27

Using Green' inhomogeno solve o function t s equation us : 2 d x = 1 ( ) sin x F 0 2 dx L L ' x 0 = + ( ) ( ) ( , ) ' sin ' x x G x x F dx 0 0 L n x sin L 2 ' ' n x x L 2 n ' 0 = + ( ) sin sin x F dx 0 0 L L L n 1 L F x = + 0 ( ) sin x 0 2 L 1 L 10/7/2019 PHY 711 Fall 2029 -- Lecture 19 28

Alternate Green' function s method : 1 ( ) x ( ) x = ( , ) ' G x x g g a B W 2 d ( ) x ( ) x dg = = = 1 ( ) 0 sin( ); sin( ); g x g x g L x i a b 2 dx ( ) x ( ) x dg ( ) x ( ) x ( ) ( ) x ( ) x ( ) = = + sin cos sin cos a b W g g L x L x b a dx dx ( ) L = sin x L sin( ) ' L x x 0 = + ( ) ( ) sin( ) ' sin ' x x x F dx 0 0 sin( ) L L sin( ) ' x x x + sin( ) ' sin ' L x F dx 0 sin( ) L L F x = + 0 ( ) ( ) sin x x 0 2 L 1 L 10/7/2019 PHY 711 Fall 2029 -- Lecture 19 29

General method of constructing Greens functions using homogeneous solution : function s Green' d x dx d ( ) ' x + = ( ) ( ) ( ) ( , ) ' x v x x G x x dx Two homogeneous solutions d d x dx dx + = = ( ) ( ) v x ( ) x ( ) 0 for , g x i a b i Let 1 = ( , ') ( ) ( ) G x x g x g x a b W 10/7/2019 PHY 711 Fall 2029 -- Lecture 19 30

For 0: + + ' ' x x d dx d dx ( ) + = ( ) x ( ) v x ( ) x ( , ' ) ' dx G x x dx x x + ' ' x x ' x 1 d dx d dx W = ( ) x ( ) ( ) 1 dx g x g x a b ' x + ' x ( ) x W ( ') x W d dx d dx d d = ( ) ( ) ( ') x ( ') ( ') x ( ' x ) g x g x g g x g g a b a b b a x ' x d d d dxg = ( ') x ( ') x ( ') ( ') x ( ') x W g g x g a b b a x dW dx = Note -- (Wronskian) is constant, since 0. W ' Useful Green's function construction in one dimension: 1 ( , ' ( ) ( ) ) a b G x x g x x g W = 10/7/2019 PHY 711 Fall 2029 -- Lecture 19 31

d dx d dx + = ( ) x ( ) v x ( ) x ( ) x ( ) F x Green's function solution: x u ( ) x = + ( ) x ( , ') ( ') G x x F x dx ' 0 x l x x ( ) x W ( ) x W g g u + + = ( ) x ( ' ) ( ') ' ( ') ( ') ' b a g x F x dx g x F x dx 0 a b x x l 10/7/2019 PHY 711 Fall 2029 -- Lecture 19 32