I was told the following argument as to why successive eigenfunctions tend to have more oscillations:
Suppose (without worrying about why) that the first eigenfunction has the least oscillation.
The second eigenfunction is orthogonal to the first, thus it must have both positive and negative parts on the region where the 1st eigenfunction is positive, and similarly for the region corresponding to the negative part of the 1st eigenfunction (if any).
Thus each eigenfunction has more oscillations than the previous.
Although I certainly believe the conclusion is true, I do not quite see that this is a solid argument. Suppose the first eigenfunction is $\sin(x)$. Then the second eigenfunction can be $-\sin(x)$, which is orthogonal to the first. So, one can find an orthogonal function without using step 2 in the argument, so (to me) the argument fails. [Edit: a big error here-- I don't know how I was thinking that sin, -sin are orthogonal, they certainly are not]
Is there a better intuitive argument for ``successive eigenfunctions have more oscillation''? (Or, point out the flaw in my thinking or description)