In deriving the governing equation for a vibrating string, there are several assumptions that are made. One of the assumptions that I had a hard time understanding was the following. Once the string is split into n particles, the force of tension on each particle from the particles in the right and the left is assumed to be proportional to the ratio of the vertical displacement to the horizontal displacement. Could you please explain to me how this assumption is correct. What are the reasons behind it. Thanks.
Generally, one assumes that the tension is constant throughout the string. You need the transverse component of the tension--which is the restoring force tending to pull the string back to its equilibrium position. At any point, the string makes some angle θ. The transverse component of the tension = T sinθ, which for small angles ≈ T tanθ = T Δy/Δx. (One should derive this, as above, not just assume it.)
I see. I can only suppose that this follows the assumption that the string is of uniform density. What if this is not the case? How would one proceed.
You could assume otherwise, but why? If uniformity gives simple solutions that match reality, isn't all you need? It's a hypothesis that works out to be correct, an example of successful science. Assuming non-uniform tension would be the next step, in the case that the solutions didn't match reality. It would also complicate the math tremendously. First in that you would have to make another guess as how the tension behaves (which function T(x) ?).