All waves can be approximated to any desired degree of accuracy over an arbitrary interval by an assembly of sine waves. Whether this means they are made of sine waves is a philosophical question.
I'm not sure that it is a philosophical question. At some level, perhaps, but at a more practical level I would suggest it is a confusion between a physical system and a mathematical model of one.
Consider a sound wave created by pulling a sticky surface (such as a violin bow) past the edge of a steel plate. You'll get a sawtooth wave as the plate is drawn back, slips and springs forward, is again drawn back and so on.
There's no doubt that the physical pressure wave has a sawtooth form. You can model this as a weighted sum of a lot of sine waves but those are imaginary, you can't physically measure any sine waves without filtering (i.e. applying a mathematical transformation).
Why I think it's important to pick up on this, is that I have seen, more than once, people confuse mathematical models with reality. To the extent that people start to argue
from the mathematical model rather than from the underlying modelled system. Maths is a way to describe things, it is not, with the exception of pure mathematics, the thing being described.
That pure mathematics can have properties that correlate with physical reality is the point where it goes back to becoming a subject for philosophy.
Please edify.
You've got to love somebody who wishes to be edified.