Originally Posted by cojam
I can see what your saying, but grahamef that's an analogy, not a mathematical proof. same for the equation you state that is derived from a mathematical proof defined elsewhere. As said by the op, he wanted proof, and the only way for this IS to understand the math. There is rigor in mathematics for this very reason, otherwise everyone would be selling analogies, those are called hypothesis. While a starting place they not the end. Otherwise we'd all still think the world is flat, wouldn't we.
I didn't see the OP asking for rigorous proof, just the basic maths involved. In any case, you can't really prove anything about physical phenomena. A formal mathematical proof would need to presume some axioms, and the physical world in which water waves exist has no axioms -- instead it has theories derived from experiment and observation. Any formal proof would get stuck at the point where you need to prove that real-world water waves obey that
particular kind of math. Case in point: Newtonian versus Einsteinian physics.
In fact that formula doesn't even derive from a proof. It's just a restatement of the definition of velocity as change in distance divided by change in time, with wavelength replacing distance and frequency replacing 1/time.
The rest of my post was not meant as an analogy, but an example to illustrate the point more intuitively. What it really boils down to is: If people are stepping onto a moving sidewalk at, say, 10 per minute, then they can't be stepping off the other end at 11 per minute -- after a short time, there'd be a negative number of people on the sidewalk.