Analytic Solution

The stretched-string BVP is straithgforward to solve analytically. General solutions of the wave equation take the form of traveling waves,

\[y(x,t) = A \exp [\ii (k x - \sigma t) ],\]

where \(A\) an arbitrary constant, and the frequency \(\sigma\) and wavenumber \(k\) are linked by the dispersion relation

\[\sigma^{2} = c^{2} k^{2}.\]

The phase velocity of these waves is \(\sigma/k = \pm c\).

To satisfy the boundary condition at \(x=0\), we combine traveling-wave solutions with opposite-sign wavenumbers

\[y(x,t) = A \exp [\ii (k x - \sigma t) ] - A \exp [\ii (- k x - \sigma t) ] = B \sin(k x) \exp ( - \ii \sigma t),\]

where \(B = 2A\). For the boundary condition at \(x=L\) to be satisfied simultaneously,

\[\sin(k L) = 0,\]

and so

\[k L = n \pi\]

where \(n\) is a non-zero integer (we exclude \(n=0\) because it leads to the trivial solution \(y(x,t)=0\)). . Combining this with the dispersion relation, we find the normal-mode eigenfrequencies of the stretched-string BVP are

(1)\[\sigma = n \frac{\pi c}{L},\]

and the corresponding eigenfunctions are

(2)\[y(x,t) = B \sin \left( \frac{n \pi x}{L} \right) \exp ( - \ii \sigma t).\]

The index \(n\) uniquely labels the modes, and \(y(x,t)\) exhibits \(n-1\) nodes in the open interval \(x=(0,L)\).