User:IssaRice/One-dimensional wave equation

why are people so bad at explaining this?

• stein and shakarchi just say "We further assume that the force (or tension) coming from the right of the $n$th particle is proportional to $(y_{n+1}-y_{n})/h$" -- but... why? making the force proportional to the distance between the two adjacent particles makes sense, but why divide by h?
• this video by Christopher Lum doesn't explain why we get to say that $T_{1}\cos \alpha =T_{2}\cos \beta =T$, but not the same thing with sines. By analogy to his argument, we should be saying that $\sin \alpha =\sin \beta =0$ since the angles are small. (Seth Whittington points this out in comments, but the only response is "it's complicated so I'll explain it over a video call".) Also, tan(alpha) should be -slope instead of slope? there's also this inconsistency in modeling where we only consider two points instead of three points (one point and its two adjacent points). Also, why is the tension in the string in the direction tangent to the string at each point? Tension T is a force, so it can be positive or negative; why do we get to write c^2? also like, what even is tension?