# 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 $$x=(0,L)$$.