I am told that I can model the vibration of a guitar string of length $l >0$ by the following Sturm-Liouville equation $ -u'' = \lambda u \: \: \text{ on } [0,l],$ with boundary conditions $u(0)=0 \: \text{ and } \: u(l)=0.$
And the vibrations of air in a pipe (with one end closed and the other open) by $-u'' = \lambda u $ with BCS $ u(0)=0 \: \text{ and } u'(l)=0. $
It is easy to see that the eigenvalues of the first and second problem are $\lambda_k = \left( \frac{k \pi} {l}\right)^2$ and $\lambda_k = \left( \frac{(2k-1) \pi} {2l}\right)^2$, respectively.
My question is how does this explain the difference in sound from a plucked guitar string and a pipe even when they are tunes to the same pitch?
Sorry, maybe this is obvious but I don't have the physical intuition and I am curious...
Thx!