Eigenvalues and Eigenfunctions

An eigenvalue here is simply one of the special numbers $\lambda$ that let a boundary problem have a nonzero solution, and the eigenfunction is the shape that solution takes. ("Eigen" is German for "own": these values and shapes belong to the problem itself, like a string's own natural notes.) All we're really asking is: which waves fit on a string that's tied down at both ends? The formula $\lambda_n = \left(\frac{n\pi}{L}\right)^2$ looks scary, but it's just a few simple parts: $L$ (the length of the string), $n$ (how many half-waves fit, $1, 2, 3, \dots$) and $\lambda_n$ (the number in the equation that makes that many half-waves fit exactly). The matching shapes $\sin\frac{n\pi x}{L}$ are just sine waves stretched so they hit zero exactly at both ends. It's the guitar string, written in symbols; bear with it.

In the last lesson, $y'' + y = 0$ with $y(0) = y(\pi) = 0$ had infinitely many solutions, $c_2\sin x$, while most other boundary problems had exactly one. With both ends at zero, $y = 0$ always works, so the only interesting question is whether there's a nonzero solution too. Now put a parameter in the equation, $y'' + \lambda y = 0$, and ask: for which $\lambda$ does a nonzero solution exist?

Why only special values. For $\lambda > 0$, $y = c_1\cos\sqrt\lambda x + c_2\sin\sqrt\lambda x$. The condition $y(0) = 0$ kills the cosine, and $y(L) = 0$ needs $\sin\sqrt\lambda L = 0$ (or $c_2 = 0$, which gives only the zero solution). So $\sqrt\lambda L = n\pi$. In pictures: the sine starts at $0$, and it only comes back to $0$ at $x = L$ if a whole number of half-waves fits between the ends.

For $\lambda \le 0$, only the zero solution fits these boundary conditions. At $\lambda = 0$ the solutions are straight lines, and the only line that's $0$ at both ends is $y = 0$. For $\lambda < 0$ they're exponentials, which can't come back to $0$ once they've left it.

A common mistake. It's tempting to include $n = 0$, since it fits the formula $\lambda_n = \left(\frac{n\pi}{L}\right)^2$. It isn't an eigenvalue here: $\sin(0\cdot x)$ is the zero function, and an eigenfunction has to be nonzero. So the list starts at $n = 1$.

Different boundary conditions give different eigenfunctions: with $y'(0) = y'(L) = 0$ (insulated ends), they're cosines, including the constant $\lambda_0 = 0$. There $n = 0$ does count, because $\cos(0\cdot x) = 1$ isn't zero.

This is the same idea as matrix eigenvalues, with a differential operator in place of a matrix: $-y'' = \lambda y$ looks just like $A\vec v = \lambda\vec v$. It's also the foundation of solving the heat and wave equations, which we'll build toward over the rest of this unit. There's an animation further down that shows the first few eigenfunctions vibrating as the modes of a string.

A guitar string's harmonics are eigenfunctions: the fundamental is $n = 1$, and overtones are $n = 2, 3, \dots$, at frequencies proportional to $\sqrt{\lambda_n}$.