Mathematical Methods for Physicists: A concise introduction - Site Map

VIBRATING STRINGS
Since the left hand side is a function of x only and the right hand side is a function
of time only, they must be equal to a common separation constant, which we will
call 2 . Then we have
and
d 2 X=dx 2 ˆ 2 X; X…0† ˆX…L† ˆ0 …4:16a†
d 2 T=dt 2 ˆ 2 v 2 T dT=dt ˆ 0 at t ˆ 0: …4:16b†
Both of these equations are typical eigenvalue problems: we have a di€erential
equation containing a parameter , and we seek solutions satisfying certain
boundary conditions. If there are special values of for which non-trivial solutions
exist, we call these eigenvalues, and the corresponding solutions eigensolutions
or eigenfunctions.
The general solution of Eq. (4.16a) can be written as
Applying the boundary conditions
and
X…x† ˆA 1 sin…x†‡B 1 cos…x†:
X…0† ˆ0 ) B 1 ˆ 0;
X…L† ˆ0 ) A 1 sin…L† ˆ0
A 1 ˆ 0 is the trivial solution X ˆ 0 (so y ˆ 0); hence we must have sin…L† ˆ0,
that is,
and we obtain a series of eigenvalues
and the corresponding eigenfunctions
L ˆ n; n ˆ 1; 2; ...;
n ˆ n=L; n ˆ 1; 2; ...
X n …x† ˆsin…n=L†x; n ˆ 1; 2; ... :
To solve Eq. (4.16b) for T…t† we must use one of the values n found above. The
general solution is of the form
T…t† ˆA 2 cos… n vt†‡B 2 sin… n vt†:
The boundary condition leads to B 2 ˆ 0.
The general solution of Eq. (4.14) is hence a linear superposition of the solutions
of the form
y…x; t† ˆX1
nˆ1
A n sin…nx=L† cos…nvt=L†;
…4:17†
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