vibrating string with variable density
The unidimensional wave’s problem may be stated as
with initial conditions
which may be specialized to a string’s motion if we physically interpret as the ratio between the string’s initial tension and its linear density. We will discuss the free string’s vibrations (i.e. ) given by the string’s problem
and boundary conditions
Without loss of generality, we assume unitary the natural undeformed string’s length. The solution of this problem approaches to a string’s motion whose linear density is proportional to . The method of separation of variables (i.e. ) gives the equations
with boundary conditions , and
with initial conditions , . In these equations, is a constant parameter.
In (2), we are dealing with a Sturm-Liouville problem. To find out the eigenvalues, one searches the solution of (2) on the form , as we realize that (2) outcomes the associated characteristic equation
In order to satisfying , let us choose
Thus, the boundary condition becomes
We next study all the possible cases for the eigenvalue in the last above equation.
. Then is real, and the equation does not have solution.
. Then the pair of solutions, above indicated, will not be independent. Indeed the functions and are linearly independent solutions of (2), in . Nevertheless, although the last one satisfies the boundary condition at , it does not vanish at . Hence, is not an eigenvalue.
. Then is imaginary. We may even get two solutions by setting ()
being the real and imaginary parts of two linearly independent solutions. For satisfying one sets
then the boundary condition gives
To these eigenvalues correspond the eigenfunctions
The completeness above mentioned and the Fourier series converges absolutely and uniformly to in , only if , and is finite. 11A result due to Green-Parseval-Schwarz (GPS) and Bessel’s inequality.
On the other hand, for satisfying (3) and its initial conditions, we choose the eigenfunction
Thus, a solution of (1) is given by
So that, the string’s problem (1) has the formal solution
This series converges uniformly, and hence satisfies the initial and boundary conditions, as the series for converges uniformly. However, in order to assure continuous derivatives and the partial differential equation (1) to be satisfied, we need suppose that the series for converges uniformly, i.e. we must suppose that to be regular 22i.e. ., that , and that to be finite. 33By GPS, again.
|Title||vibrating string with variable density|
|Date of creation||2013-03-22 17:26:42|
|Last modified on||2013-03-22 17:26:42|
|Last modified by||perucho (2192)|