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So we were given this problem in mathematical physics (Context is that we're learning about Sturm-Liouville):

Consider the 1D Helmholtz equation with $k^2>0$ $$\frac{\partial ^2y}{\partial x^2}+k^2 y=0$$ on the interval $0\leq x \leq L$ subject to the boundary conditions $$ay(0)+by'(0)=0$$ $$ay(L)+by'(L)=0$$ What are the forms of the eigenfunctions and eigenvalues?

It was easy enough to get that the eigenfunctions are of the form

$$A \left(\sin (k x)-\frac{b k }{a}\cos (k x)\right)$$

with eigenvalues

$$k^2=\left(\frac{\pi n}{L}\right)^2$$

There's this $A$ left out in front. I get how in quantum mechanics maybe this is useful as a normalizer for probabilities. But is there a physical significance beyond the fact that it makes the math nicer? What does this result mean when applied to a classical system, like a string? Or does it just not apply?

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  • $\begingroup$ I'm somewhat confused by your question. Are you asking what the amplitude of the solution represents? $\endgroup$
    – J. Murray
    Nov 13, 2020 at 19:51

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One physical interpretation of the Helmholtz equation is, as you mention, a string which is fixed at $x=0$ and $x = L$ (if $b=0$ in your equations) with an oscillation amplitude $y(x)$. If you knew your material of the string, you would know the propagation speed $c$ and you could calculate the eigenfrequency $f$ from $k = \frac{2\pi f}{c}$

The solution $y(x)$ that you state has an amplitude $A$, which can in principle be anything, as long nobody told you how the string was excited initially. As an example, if I hold the string in a position $y(x) = 2\sin(\pi \frac{x}{L})$ and release, the string will oscillate with an amplitude $A = 2$ and with a frequency $f = \frac{ck}{2\pi} = \frac{ c}{2 L}$. That is, a half standing wave between $x=0$ and $x=L$.

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