# Analytic Solution¶

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

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

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

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

and so

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

and the corresponding eigenfunctions are

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