I understand tension in a straight string as a reaction force to a weight, which acts along the string, ultimately resulting from the attractive forces between the constituent particles of the string. I'm not sure how to understand tension in a curved string, such as a vibrating guitar string. I understand that an ideally elastic string is meant to have tension acting at tangent to the string, but that's about it. Is the magnitude of the tension vector equal throughout the string?
I've been going through various derivations of the wave equation for transverse motion of a string fixed at both ends and they all use the assumption that the tensions at each end of an infinitesimal segment of the string are equal in magnitude. I don't understand how this assumption is warranted.
The formula speed of wave propagation in a string involves this tension $T$ in which it is a constant. Can tension in a string be constant when it's oscillating?