Question

(a) A string of length L is fixed at one end (in other words, y(0) = 0) and free at the other end (y 0 (L) = 0). Consider the position-dependent wave equation for the string. What are f, g, w, α1, β1, α2 and β2 of this Sturm-Liouville system? Find the eigenvalues and eigenvectors of this system. How may these be useful for solving problems? (Aside: this is known as the one-dimensional Helmholtz equation.)

(b) Consider a string that is fixed at position x = 0 and is free at it’s other end, a distance x = L away. If the string is lifted up at the very end that is free and is kept taut, and is released at time t = 0, what is its subsequent motion?

(c) Plot the string’s position at a couple of different times to illustrate it’s motion. Does your answer make sense? (using Mathematica)

Answer 1

ans (a);   

Recall that equation for the oscillation of a string strecthed between two fixed points.

where   are liner density of string and tension in the string.

for a hanging string , only upper end is fixed. the bottem end is free. the tension comes from gravity.

Let us establish a coordinate system in the vertical direction:

1) the free end of the hanging string is defined as y = 0;

2) the upward direction is the positive direction:

3) the fixed end is y = L

the tension is a function of y.

i.e,

where g is the gravitational accelaration.

the governing equation for a hanging string is

Again , for convenience we denote y by x.

The initial boundery value problem for a hanging string is

  

The oscillation is independent of the line density .  

Now we solve the initial boundaryvalue problem by separation of varia bles.

we try u(x,t)= A(x)B(t).

Substituting into the differential equation , we get

The Sturm-Liouville problem is

This is a Sturm-Liouville problem with p(x) = x, q(x) = 0, r(x) = 1.

It is a singular Strum-Liouville problem because p(0)= 0.

Eigen values and Eigenfunctions:

First , we point out that all eigenvalues are positive.

the equation in the eigenvalue problem :

We compare it with the Bessel equation of order 0.

We need to use a scaling to connect these two differential equations and the scalimg needs to be non-liner

let

consider a function Z based on function A(x):

W(z)=A(x)

A(x)= W(z)=

by using boundary conditions eigenvalue s are A(L)=0

=>J_o (squrt(4 lamda L) = 0

=> Squrt(4 lamda L) = Z_n n=0,1,2,....

the eigenvalues are