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## The Simple String and Poisson's Equation

Consider a simple string with a force applied to it. For such a string let

net transverse force acting on the string segment between and due to tension only.

vertical force component

so that

Let applied force density .

If the string is in equilibrium then there is no acceleration. Consequently, the total force density is zero:

or

For constant tension one obtains

 (413)

This is the one-dimensional Poisson equation.

Example

Consider a cable whose linear mass density is and which is suspended between two horizontal points, and , in a uniform gravitational field whose acceleration is .

The force density on such a cable is . If the tension in the cable is , then the equilibrium profile is governed by

The solution is evidently

where the integration constants are determined by and . It follows that the cable's profile away from the straight horizontal is

Exercise 43.1 (ADJOINT OF AN OPERATOR)
Find the adjoint differential operator and the space on which it acts if
(a)
where

(b)
where

Assume that the scalar product is

Let be a differential operator defined over that domain of functions which satisfy the given homogeneous boundary condition and . Let be the corresponding adjoint operator defined on the domain of functions which satisfy the corresponding adjoint boundary conditions, and .

Let be an eigenfunction of :

Similarly let be an eigenfunction of :

(i)
Make a guess as to the relationship between the eigenvalues of and the eigenvalues of and give a reason why.
(ii)
Prove: If then . i.e. An eigenfunction of corresponding to the eigenvalue is orthogonal to every eigenfunction of which does not correspond to . Here the overline means complex conjugate, of course.

Exercise 43.3 (BESSEL OPERATORS)
Find the Green's function for the Bessel operators
(a)
(b)
with finite and ,
i.e. solve the equations with the given boundary conditions.

Exercise 43.4 (DIFFERENT ENDPOINT CONDITIONS)
1. Find the Green's function for the operator with

and a fixed constant. i.e. solve with the given boundary conditions.
2. Does this Green's function exist for all values of ? If NO, what are the exceptional values of ?
3. Having found the Green's function in part (1), suppose one wishes to find the Green's function for the same differential equation, but with different end point conditions, namely and . How would one find this new Green's function with a minimal amount of work? Go ahead, find it.

Exercise 43.5 (ADJOINT FOR GENERIC ENDPOINT CONDITIONS)
Suppose that where

and

1. Find and the space on which it acts if one uses the scalar product .
2. For what values of the constants is the operator self adjoint?

Lecture 29

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Ulrich Gerlach 2010-12-09